Abstract:
An amplifier system comprises an amplifier, a comparator connectable to the amplifier to compare an incoming signal with a number of reference potentials. A switching stage is controlled by the output of the comparator and arranged to select a voltage level to be applied to one side of a load, such as a loudspeaker system. A subtractor is included to subtract from the incoming signal a percentage of the selected voltage level to be applied to the load. The amplifier receives the output signal of the subtractor, amplifies that signal and applies it to a second side of the load. There is also disclosed a method for amplifying a signal.

Description:
FIELD OF THE INVENTION 
   The present invention relates generally to an amplifier system and method for amplifying a signal, for use, for example in an audio amplifier circuit. 
   BACKGROUND OF THE INVENTION 
   Environmental considerations are becoming an increasingly important design parameter for product developments. For amplifier designs, considerable research has been directed towards high efficiency amplifiers which have less energy loss and various topologies exist that perform at a higher efficiency than conventional Class B amplifiers. Examples of such amplifiers are Class D, Class G and Class H amplifiers. Such amplifiers require smaller heat sinks than conventional Class B amplifiers, resulting in lighter and more compact products. 
   For a Class B amplifier having a fixed primary power supply, the theoretical maximum efficiency is 78.5% when it is driven to saturation by a sine wave, but in actual practice the efficiency does not exceed 70%. 
   The basic concept of conventional Class D amplifiers is to convert the input signal into a Pulse Width Modulated (PWM) signal and to recover the analog output signal through a low pass filter before delivering the signal to the load. A theoretical efficiency of 100% may be achievable with conventional Class D amplifiers and, in practice, over 90% efficiencies have been achieved using Class D Digital Amplifiers. Disadvantages of Class D amplifiers are the EMI issues which arise from the high level switching of the PWM signal. Furthermore, the typical requirement of the output low pass filter adds costs to the implementation of Class D amplifiers. At the same time, this output low pass filter also affects the output characteristics of the amplifier as a fixed load impedance of, for example, 6 ohms has to be assumed during its design. 
   This is not, however, the case for typical loudspeaker loads. In typical loudspeaker systems, when connected to a load which may range from 2 ohms to some tens of ohms for typical loudspeakers, the output characteristics will deviate from the target design. 
   Conventional Class G amplifiers generally achieve higher efficiencies than conventional Class B amplifiers by switching the power supply level at different fixed levels to have the power supply closer to the output signal. In theory, 100% efficiency may be achieved by increasing the number of fixed power supply levels indefinitely. However, as several devices have to be used to switch among the different supply voltages, this will create losses in practice. 
   Conventional Class H amplifiers use the input signal to control the power supply and vary the power supply by Pulse Width Modulation (PWM) to form an envelope over the output signal. Although a very high efficiency may be achieved in such amplifiers, such as theoretically up to 100%, the transient response of conventional Class H amplifiers may be poor and the bandwidth may be extremely limited due to the slow response of the power supply. Moreover, EMI issues will be a major concern in such amplifiers. 
   SUMMARY OF THE INVENTION 
   In general terms, the invention provides an amplifier system and method in which the voltage signal level in an amplifier is reduced by applying a voltage at various fixed levels to a first side of a load, obtaining a difference signal derived by subtracting a proportion of the voltage applied to the load from the input signal and applying the difference signal to the other side of the load through the amplifier. This results in the reconstruction of the signal across the load. The voltage swing obtainable across the load is greater than the output voltage swing obtainable from the amplifier alone, thus the efficiency of the amplifier system and power handling capacity of the system are improved. 
   According to a first aspect of the invention there is provided an amplifier system comprising:
         an amplifier;   a comparator connectable to said amplifier to compare an incoming signal with a number of reference potentials, said comparator having an output;   a switching stage controllable by said output of said comparator to select a voltage level to be applied to one side of a load;   a subtractor for subtracting from said incoming signal a percentage of said selected voltage level to be applied to said load to provide an output signal of said subtractor;   said amplifier being arranged to receive said output signal of said subtractor and to amplify said output signal to be applied to a second side of said load.       

   According to a second aspect of the invention there is provided an audio system comprising the amplifier system defined above. 
   According to a third aspect of the invention there is provided a method for amplifying a signal in an amplifier system comprising:
         comparing in a comparator an incoming signal with a number of reference potentials, said comparator having an output;   selecting in a switching stage a voltage level to be applied to one side of a load, said switching stage being controllable by said output of said comparator;   subtracting from said incoming signal a percentage of said selected voltage level to be applied to said load to provide an output signal of a subtractor;   receiving in an amplifier said output signal of said subtractor;   amplifying said output signal of said subtractor; and   applying said amplified signal to a second side of said load.       

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     Preferred features of the invention will now be described, for the sake of illustration only, with reference to the following Figures in which: 
       FIG. 1  is a schematic circuit diagram of a conventional Class B amplifier; 
       FIG. 2  is a schematic circuit diagram of an amplifier according to a preferred embodiment of the invention; 
       FIG. 3  is a waveform of an input signal in the form of a sine wave; 
       FIG. 4  is a waveform of a signal at node  1  in the circuit of  FIG. 2  for a sine wave input signal of the type shown in  FIG. 3 ; 
       FIG. 5  is a waveform of a signal at node  3  in the circuit of  FIG. 2  for a sine wave input signal of the type shown in  FIG. 3 ; 
       FIG. 6  is a graph of the efficiency against normalised power for the amplifier of  FIG. 2  compared with a conventional Class B amplifier; 
       FIG. 7  is a graph of power loss against normalised output power for the amplifier of  FIG. 2  compared with a conventional Class B amplifier; 
       FIG. 8  is a schematic circuit diagram according to a further preferred embodiment of the invention; 
       FIG. 9  is a waveform of a signal at node  1  in the circuit of  FIG. 8  for a sine wave input signal; 
       FIG. 10  is a waveform of a signal at node  3  in the circuit of  FIG. 8  for a sine wave input signal; 
       FIG. 11  is a graph of the efficiency against normalised power for the amplifier of  FIG. 8  compared with a conventional Class B amplifier; 
       FIG. 12  is a graph of power loss against normalised output power for the amplifier of  FIG. 8  compared with a conventional Class B amplifier; 
       FIG. 13  is a graph of the efficiency against normalised power for the amplifiers of  FIGS. 2 and 8  compared with a conventional Class B amplifier; and 
       FIG. 14  is a graph of power loss against normalised output power for the amplifiers of  FIGS. 2 and 8  compared with a conventional Class B amplifier. 
   

   DESCRIPTION OF PREFERRED EMBODIMENTS 
     FIG. 1  shows a circuit diagram of a conventional Class B amplifier. An input signal  10  is applied between ground and the positive input terminal of an amplifier  12  which has a gain of Gv. Negative feedback is applied to the amplifier  12  by dividing the output signal therefrom in a potential divider comprising two resistors  14 ,  15  connected in series from the output of the amplifier  12  to ground. The junction  16  of the two resistors  14 ,  15  is connected to the negative terminal of the amplifier  12 , the two resistors  14 ,  15  defining the gain of the amplifier  12 . The amplifier  12  is powered by twin power supplies having power supply voltages which are fixed at V cc  and −V cc  respectively. Thus, the output signal from the amplifier  12  will vary within the range −V cc  to +V cc . 
   Assuming that there is no clipping, if the peak amplitude of the input signal  10  is V ref , giving a corresponding peak output of V cc , the gain of the amplifier  12  may be denoted by 
             G   v     =       V   cc         V   ref     .             
The theoretical efficiency of such an amplifier  12  based on a sinusoidal input of amplitude V 0  may be denoted as
 
             π   ⁢           ⁢     V   0         4   ⁢     V   cc             
and the power loss as
 
                 2   ⁢     V   cc     ⁢     V   0         π   ⁢           ⁢   R       -       V   0       2   ⁢   R         ,         
where R is a resistive load.
 
   An amplifier circuit  18  according to a first preferred embodiment of the present invention is shown in  FIG. 2 . In this embodiment, the amplifier circuit  18  comprises an input signal  19 , a comparator  20 , a control circuit  21 , a multi-way switch  22  and a differential amplifier  23 . The input signal  19  is applied via a resistor R 7  to the negative input of the differential amplifier  23 . A resistor R 8  is connected between the output of the differential amplifier  23  at node  3  and the negative input of the differential amplifier  23 . The input signal  19  is also applied to the input of the comparator  20 . The output terminals of the comparator  20  are connected to the multi-way switch  22  and the output of the multi-way switch is connected at a node  1  to the positive terminal of a loudspeaker  24  and also to a potential divider comprising two resistors R 9  and R 10  connected in series between node  1  and ground. The junction (node  2 ) of the resistors R 9  and R 10  is connected to the positive input of the differential amplifier  23 . 
   The comparator  20  compares the input signal  19  with the reference voltages −2/3V ref , −1/3V ref , +1/3 V ref  and +2/3 V ref , which are fixed by resistors R 1 , R 2 , R 3 , R 4 , R 5  and R 6  within the comparator  20 . Preferably,
 
R1=R2=R3=R4=R5=R6
 
 V   cc1   =V   cc /3
 
V cc2 =2× V   cc1 
 
 R 8/ R 7= Gv 
 
 R 10/( R 9+ R 10)=1 /Gv 
 
Also, |V ref | corresponds to the peak amplitude of the input signal  19 .
 
   When the amplitude of the input signal  19  is below |V ref /3|, the ground potential is applied to node  1  at the positive terminal of the load  24 . When the input signal  19  amplitude exceeds the reference voltages, a corresponding signal is sent to the control circuit  21  to indicate the level of the input signal  19 . Via the multi-way switch  22 , the control circuit  20  then switches on the appropriate DC voltage, at −V cc2 , −V cc1 , V cc1  or V cc2  respectively, to the positive terminal of the load  24 , where 
                    V   cc1          =            V   cc     3            ,            V   cc2          =            2   ⁢     V   cc       3                  
and V cc  refers to the peak output voltage, which is also the power supply voltage of a conventional Class B amplifier delivering the same output power as the embodiment of the present invention illustrated in  FIG. 2 .
 
     FIG. 3  shows an input waveform  19  comprising a sinusoidal waveform, the peak amplitude of which is equal to the maximum voltage present in the comparator  20 . 
   When the waveform of  FIG. 3 , namely a sinusoidal input of amplitude V ref , is applied to the comparator  20 , a stepped waveform resembling the sinusoidal waveform will appear at the positive terminal of the load, as shown in  FIG. 4 . 
   The gain of the differential amplifier  23  is chosen such that |Gv|=|V cc |/|V ref | and its power supply voltages are +V cc1  and −V cc1 . A proportion of the DC voltage applied to the load from node  2 , the amplitude of which may be fixed by selecting appropriate values for the resistors R 9  and R 10 , is fed to the positive input of the differential amplifier  23 . This amplitude may be selected to be 1/Gv of the supplied DC voltage to the load. As the DC voltages are fixed at −Vcc 2 , −Vcc 1 , GND, Vcc 1  and Vcc 2 , the signal to the positive input of the differential amplifier  23  will be correspondingly −2/3 Vref, −1/3 Vref, GND, +1/3 Vref and +2/3 Vref. 
   The input signal  19  is fed to the negative input of the differential amplifier  23  through resistor R 7 . The difference between the positive input and the negative input of the differential amplifier  23  is amplified and fed to the negative terminal, Node  3 , of the load, as shown in  FIG. 5 . That is, the differential amplifier acts as a subtractor to subtract the input signal  19  from the signal level presented at the positive input, and as an amplifier to amplify that difference and output it to the negative terminal of the load. The differential amplifier  23  essentially compensates for the difference between the signal fed to the positive terminal of the load and the desired signal. Thus, the overall signal as seen by the load will be equivalent to the input signal amplified by a gain of Gv. 
   On the positive side of the load  24 , which is essentially switching among several DC voltage levels, the efficiency is theoretically 100%. On the negative side of the load, a conventional Class B amplifier design may be used for the differential amplifier  23 . The efficiency of such a differential amplifier will be the same as a conventional Class B amplifier. However, since the supplied voltage is +V cc /3 and −V cc /3, the overall efficiency of the amplifier circuit of  FIG. 2  as compared to a conventional Class B amplifier delivering the same output power, may be seen to be significantly higher. The efficiency and power loss calculations are described as follows. 
   It is proposed that amplifiers embodying the present invention are to be named the Class μ Amplifier because of the resultant shape of the curve when the graph of Efficiency against Output Power is plotted (as shown in  FIG. 6  described below). 
   SYMBOL DEFINITIONS 
   
       
       ε Efficiency 
       ε B  Efficiency of normal (conventional) Class B Amplifier 
       ε 1  Efficiency of Stage  1  of Class μ Amplifier 
       ε 2  Efficiency of Stage  2  of Class μ Amplifier 
       ε 3  Efficiency of Stage  3  of Class μ Amplifier 
       f Frequency 
       I Current 
       P in  Input power 
       P out  Output power 
       P total-in-2+  Total input power for positive side of Stage  2  of Class μ Amplifier 
       P total-in-2−  Total input power for negative side of Stage  2  of Class μ Amplifier 
       P total-in-3+  Total input power for positive side of Stage  3  of Class μ Amplifier 
       P total-in-3−  Total input power for negative side of Stage  3  of Class μ Amplifier 
       P ave-in  Average input power 
       P ave-out  Average output power (same for both Class B and μ in this application) 
       P ave-in-B  Average input power of normal Class B Amplifier 
       P ave-in-1  Average input power of Stage  1  of Class μ Amplifier 
       P ave-loss-1  Average power loss of Stage  1  of Class μ Amplifier 
       P ave-in-2  Average input power of Stage  2  of Class μ Amplifier 
       P ave-loss-2  Average power loss of Stage  2  of Class μ Amplifier 
       P ave-in-3  Average input power of Stage  3  of Class μ Amplifier 
       P ave-loss-3  Average power loss of Stage  3  of Class μ Amplifier 
       R Impedance, assumed resistive for this case 
       t time 
       t 1  Moment (or the time) that V cc1  is switched on 
       t 2  Moment (or the time) that V cc2  is switched on 
       T Period 
       V out  Output voltage 
       V 0  Amplitude of output voltage (=|V out |) 
       V cc  Peak output voltage, which is also the supply voltage for a normal Class B Amplifier 
       V cc1  First stage output voltage to positive side of load for Class μ Amplifier 
       V cc2  Second stage output voltage to positive side of load for Class μ Amplifier 
       V ref  Reference voltage, which corresponds to the peak input voltage to give the peak output voltage at V cc    
       ω Angular frequency (=2πf) 
     
  
   In the calculations below, the positive side of the load  24  refers to the side with the DC voltage switching between −V cc1 , −V cc2 , GND, +V cc1  and +V cc2  and the negative side of the load  24  is connected to the differential amplifier  23  with a supply voltage of +V cc1  and −V cc1 . The load R is taken to be resistive. 
   The following calculations are based on sinusoidal inputs. 
   
     
       
         
           
             
               
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   For both an amplifier of the type illustrated in  FIG. 2  (hereinafter referred to as a Class μ amplifier) and a conventional Class B Amplifier: 
   
     
       
         
           
             
               
                 
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                   efficiency 
                 
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                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               n 
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       = 
                         
                       ⁢ 
                       
                         
                           
                             V 
                             0 
                             2 
                           
                           
                             2 
                             ⁢ 
                             R 
                           
                         
                         × 
                         
                           
                             π 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             R 
                           
                           
                             2 
                             ⁢ 
                             
                               V 
                               cc 
                             
                             ⁢ 
                             
                               V 
                               0 
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       = 
                         
                       ⁢ 
                       
                         
                           π 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             V 
                             0 
                           
                         
                         
                           4 
                           ⁢ 
                           
                             V 
                             cc 
                           
                         
                       
                     
                   
                 
               
             
           
           
             
               
                 
                   Average 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   power 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   loss 
                 
                 , 
               
             
             
               
                 
                   
                     
                       
                         P 
                         
                           ave 
                           - 
                           loss 
                           - 
                           1 
                         
                       
                       = 
                         
                       ⁢ 
                       
                         
                           P 
                           
                             ave 
                             - 
                             
                               i 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               n 
                             
                             - 
                             1 
                           
                         
                         - 
                         
                           P 
                           
                             ave 
                             - 
                             out 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       = 
                         
                       ⁢ 
                       
                         
                           
                             2 
                             ⁢ 
                             
                               V 
                               cc 
                             
                             ⁢ 
                             
                               V 
                               0 
                             
                           
                           
                             π 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             R 
                           
                         
                         - 
                         
                           
                             V 
                             0 
                             2 
                           
                           
                             2 
                             ⁢ 
                             R 
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
   
   For a Class μ Amplifier—Stage  1 : For sinusoidal output voltage |V 0 |&lt;V cc1 |: 
   For this stage of operation, the ground potential will be supplied to the positive side of the load  24 . Thus, only the differential amplifier  23  is supplying the signal to the load  24  at the negative side thereof. 
   
     
       
         
           
             
               
                 Thus 
                 , 
                 
                   average 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   input 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   power 
                 
                 , 
               
             
             
               
                 
                   
                     
                       
                         P 
                         
                           ave 
                           - 
                           
                             i 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             n 
                           
                           - 
                           1 
                         
                       
                       = 
                         
                       ⁢ 
                       
                         
                           2 
                           ⁢ 
                           
                             V 
                             cc1 
                           
                           ⁢ 
                           
                             V 
                             0 
                           
                         
                         
                           π 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           R 
                         
                       
                     
                   
                 
                 
                   
                     
                       = 
                         
                       ⁢ 
                       
                         
                           
                             
                               2 
                               ⁢ 
                               
                                 V 
                                 cc 
                               
                               ⁢ 
                               
                                 V 
                                 0 
                               
                             
                             
                               3 
                               ⁢ 
                               π 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               R 
                             
                           
                           ⁢ 
                           
                               
                           
                           ∵ 
                           
                             V 
                             cc1 
                           
                         
                         = 
                         
                           
                             V 
                             cc 
                           
                           3 
                         
                       
                     
                   
                 
               
             
           
           
             
               
                 Efficiency 
                 , 
               
             
             
               
                 
                   
                     
                       
                         ɛ 
                         1 
                       
                       = 
                         
                       ⁢ 
                       
                         
                           π 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             V 
                             0 
                           
                         
                         
                           4 
                           ⁢ 
                           
                             V 
                             cc1 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       = 
                         
                       ⁢ 
                       
                         
                           3 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           π 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             V 
                             0 
                           
                         
                         
                           4 
                           ⁢ 
                           
                             V 
                             cc 
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
   
   Comparing the above with a conventional Class B Amplifier, during this stage of operation, 
   
     
       
         
           
             
               ɛ 
               1 
             
             
               ɛ 
               B 
             
           
           = 
           
             
               
                 
                   3 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   π 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     V 
                     0 
                   
                 
                 
                   4 
                   ⁢ 
                   
                     V 
                     cc 
                   
                 
               
               × 
               
                 
                   4 
                   ⁢ 
                   
                     V 
                     cc 
                   
                 
                 
                   π 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     V 
                     0 
                   
                 
               
             
             = 
             
               
                 3 
                 ⁢ 
                 
                   
 
                 
                 ⇒ 
                 
                   ɛ 
                   1 
                 
               
               = 
               
                 3 
                 ⁢ 
                 
                   ɛ 
                   B 
                 
               
             
           
         
       
     
     
       
         
           
             
               
                 
                   Average 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   power 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   loss 
                 
                 , 
               
             
             
               
                 
                   
                     
                       
                         P 
                         
                           ave 
                           - 
                           loss 
                           - 
                           1 
                         
                       
                       = 
                         
                       ⁢ 
                       
                         
                           P 
                           
                             ave 
                             - 
                             
                               i 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               n 
                             
                             - 
                             1 
                           
                         
                         - 
                         
                           P 
                           
                             ave 
                             - 
                             out 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       = 
                         
                       ⁢ 
                       
                         
                           
                             2 
                             ⁢ 
                             
                               V 
                               cc 
                             
                             ⁢ 
                             
                               V 
                               0 
                             
                           
                           
                             3 
                             ⁢ 
                             π 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             R 
                           
                         
                         - 
                         
                           
                             V 
                             0 
                             2 
                           
                           
                             2 
                             ⁢ 
                             R 
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
   
   Class μ Amplifier Stage  2 : For sinusoidal output voltage |V cc1 |&lt;V 0 |&lt;|V cc2 |: 
   For this stage, it is necessary to determine the time t 1 , when V out =V cc1 , which is the moment that V cc1  is switched on. 
   
     
       
         
           
             
               
                 
                   
                     
                       
                         Output 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         voltage 
                       
                       , 
                     
                   
                   
                     
                       
                         
                           V 
                           out 
                         
                         = 
                         
                           
                             V 
                             0 
                           
                           ⁢ 
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ω 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           t 
                         
                       
                       ⁢ 
                       
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         At 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           t 
                           1 
                         
                       
                       , 
                       
                         
                           V 
                           out 
                         
                         = 
                         
                           V 
                           cc1 
                         
                       
                       , 
                     
                   
                   
                     
                       
                           
                       
                       ⁢ 
                       
                         ⇒ 
                         
                           
                             
                               
                                 
                                   V 
                                   cc1 
                                 
                                 = 
                                   
                                 ⁢ 
                                 
                                   
                                     V 
                                     0 
                                   
                                   ⁢ 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       2 
                                       ⁢ 
                                       π 
                                     
                                     T 
                                   
                                   ⁢ 
                                   
                                     t 
                                     1 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     
                                       2 
                                       ⁢ 
                                       π 
                                     
                                     T 
                                   
                                   ⁢ 
                                   
                                     t 
                                     1 
                                   
                                 
                                 = 
                                   
                                 ⁢ 
                                 
                                   
                                     sin 
                                     
                                       - 
                                       1 
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       V 
                                       cc1 
                                     
                                     
                                       V 
                                       0 
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   t 
                                   1 
                                 
                                 = 
                                   
                                 ⁢ 
                                 
                                   
                                     T 
                                     
                                       2 
                                       ⁢ 
                                       π 
                                     
                                   
                                   ⁢ 
                                   
                                     sin 
                                     
                                       - 
                                       1 
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       V 
                                       cc 
                                     
                                     
                                       3 
                                       ⁢ 
                                       
                                         V 
                                         0 
                                       
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 1 
                 ) 
               
             
           
         
       
     
   
   For positive side, total input power for ¼ period, 
   
     
       
         
           
             
               
                 
                   P 
                   
                     total 
                     - 
                     
                       i 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       n 
                     
                     - 
                     2 
                     + 
                   
                 
                 = 
                 
                   
                     ∫ 
                     
                       t 
                       1 
                     
                     
                       T 
                       4 
                     
                   
                   ⁢ 
                   
                     
                       
                         
                           V 
                           cc1 
                         
                         ⁢ 
                         
                           V 
                           0 
                         
                         ⁢ 
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         ω 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         t 
                       
                       R 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ⅆ 
                       t 
                     
                   
                 
               
             
           
           
             
               
                 = 
                 
                   
                     
                       
                         V 
                         cc1 
                       
                       ⁢ 
                       
                         V 
                         0 
                       
                     
                     R 
                   
                   ⁢ 
                   
                     
                       ∫ 
                       
                         t 
                         1 
                       
                       
                         T 
                         4 
                       
                     
                     ⁢ 
                     
                       sin 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           2 
                           ⁢ 
                           π 
                         
                         T 
                       
                       ⁢ 
                       t 
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                 
               
             
           
           
             
               
                 = 
                 
                   
                     
                       
                         V 
                         cc 
                       
                       ⁢ 
                       
                         V 
                         0 
                       
                     
                     
                       3 
                       ⁢ 
                       R 
                     
                   
                   ⁢ 
                   
                     
                       
                         T 
                         
                           2 
                           ⁢ 
                           π 
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             - 
                             cos 
                           
                           ⁢ 
                           
                             
                               2 
                               ⁢ 
                               π 
                             
                             T 
                           
                           ⁢ 
                           t 
                         
                         ] 
                       
                     
                     
                       t 
                       1 
                     
                     
                       T 
                       4 
                     
                   
                 
               
             
           
           
             
               
                 = 
                 
                   
                     
                       
                         V 
                         cc 
                       
                       ⁢ 
                       
                         V 
                         0 
                       
                       ⁢ 
                       T 
                     
                     
                       6 
                       ⁢ 
                        
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       R 
                     
                   
                   ⁢ 
                   cos 
                   ⁢ 
                   
                     
                       2 
                       ⁢ 
                       π 
                     
                     T 
                   
                   ⁢ 
                   
                     t 
                     1 
                   
                 
               
             
           
         
       
     
   
   For the negative side, total input power for ¼ period, 
   
     
       
         
           
             
               
                 
                   
                     
                         
                     
                   
                   
                     
                       
                         
                           
                             
                               P 
                               
                                 total 
                                 - 
                                 
                                   i 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   n 
                                 
                                 - 
                                 2 
                                 - 
                               
                             
                             = 
                               
                             ⁢ 
                             
                               
                                 ∫ 
                                 0 
                                 
                                   T 
                                   4 
                                 
                               
                               ⁢ 
                               
                                 
                                   
                                     
                                       V 
                                       cc1 
                                     
                                     ⁢ 
                                     
                                       V 
                                       0 
                                     
                                     ⁢ 
                                     sin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     ω 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     t 
                                   
                                   R 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ⅆ 
                                   t 
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             = 
                               
                             ⁢ 
                             
                               
                                 
                                   
                                     V 
                                     cc1 
                                   
                                   ⁢ 
                                   
                                     V 
                                     0 
                                   
                                 
                                 R 
                               
                               ⁢ 
                               
                                 
                                   ∫ 
                                   0 
                                   
                                     T 
                                     4 
                                   
                                 
                                 ⁢ 
                                 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       2 
                                       ⁢ 
                                       π 
                                     
                                     T 
                                   
                                   ⁢ 
                                   t 
                                   ⁢ 
                                   
                                     ⅆ 
                                     t 
                                   
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             = 
                               
                             ⁢ 
                             
                               
                                 
                                   
                                     V 
                                     cc 
                                   
                                   ⁢ 
                                   
                                     V 
                                     0 
                                   
                                 
                                 
                                   3 
                                   ⁢ 
                                   R 
                                 
                               
                               ⁢ 
                               
                                 
                                   
                                     T 
                                     
                                       2 
                                       ⁢ 
                                       π 
                                     
                                   
                                   ⁡ 
                                   
                                     [ 
                                     
                                       
                                         - 
                                         cos 
                                       
                                       ⁢ 
                                       
                                         
                                           2 
                                           ⁢ 
                                           π 
                                         
                                         T 
                                       
                                       ⁢ 
                                       t 
                                     
                                     ] 
                                   
                                 
                                 0 
                                 
                                   T 
                                   4 
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             = 
                               
                             ⁢ 
                             
                               
                                 
                                   V 
                                   cc 
                                 
                                 ⁢ 
                                 
                                   V 
                                   0 
                                 
                                 ⁢ 
                                 T 
                               
                               
                                 6 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 R 
                               
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         Average 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         input 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         power 
                       
                       , 
                     
                   
                   
                     
                       
                         
                           
                             
                               P 
                               
                                 ave 
                                 - 
                                 
                                   i 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   n 
                                 
                                 - 
                                 2 
                               
                             
                             = 
                             
                               
                                 
                                   P 
                                   
                                     total 
                                     - 
                                     
                                       i 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       n 
                                     
                                     - 
                                     2 
                                     + 
                                   
                                 
                                 + 
                                 
                                   P 
                                   
                                     total 
                                     - 
                                     
                                       i 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       n 
                                     
                                     - 
                                     2 
                                     + 
                                   
                                 
                               
                               
                                 T 
                                 4 
                               
                             
                           
                         
                       
                       
                         
                           
                             = 
                             
                               
                                 
                                   2 
                                   ⁢ 
                                   
                                     V 
                                     cc 
                                   
                                   ⁢ 
                                   
                                     V 
                                     0 
                                   
                                 
                                 
                                   3 
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   R 
                                 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   
                                     cos 
                                     ⁢ 
                                     
                                       
                                         2 
                                         ⁢ 
                                         π 
                                       
                                       T 
                                     
                                     ⁢ 
                                     
                                       t 
                                       1 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       Efficiency 
                       , 
                     
                   
                   
                     
                       
                         
                           
                             
                               ɛ 
                               2 
                             
                             = 
                               
                             ⁢ 
                             
                               
                                 
                                   V 
                                   0 
                                   2 
                                 
                                 
                                   2 
                                   ⁢ 
                                   R 
                                 
                               
                               × 
                               
                                 
                                   3 
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   R 
                                 
                                 
                                   2 
                                   ⁢ 
                                   
                                     V 
                                     cc 
                                   
                                   ⁢ 
                                   
                                     
                                       V 
                                       0 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         1 
                                         + 
                                         
                                           cos 
                                           ⁢ 
                                           
                                             
                                               2 
                                               ⁢ 
                                               π 
                                             
                                             T 
                                           
                                           ⁢ 
                                           
                                             t 
                                             1 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             = 
                               
                             ⁢ 
                             
                               
                                 3 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   V 
                                   0 
                                 
                               
                               
                                 4 
                                 ⁢ 
                                 
                                   
                                     V 
                                     cc 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       1 
                                       + 
                                       
                                         cos 
                                         ⁢ 
                                         
                                           
                                             2 
                                             ⁢ 
                                             π 
                                           
                                           T 
                                         
                                         ⁢ 
                                         
                                           t 
                                           1 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 2 
                 ) 
               
             
           
         
       
     
   
   Comparing the above with a conventional Class B Amplifier, 
   
     
       
         
           
             
               
                   
               
             
             
               
                 
                   
                     ɛ 
                     2 
                   
                   
                     ɛ 
                     B 
                   
                 
                 = 
                 
                   
                     
                       3 
                       ⁢ 
                       π 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         V 
                         0 
                       
                     
                     
                       4 
                       ⁢ 
                       
                         
                           V 
                           cc 
                         
                         ⁡ 
                         
                           ( 
                           
                             1 
                             + 
                             
                               cos 
                               ⁢ 
                               
                                 
                                   2 
                                   ⁢ 
                                   π 
                                 
                                 T 
                               
                               ⁢ 
                               
                                 t 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   × 
                   
                     
                       4 
                       ⁢ 
                       
                         V 
                         cc 
                       
                     
                     
                       π 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         V 
                         0 
                       
                     
                   
                 
               
             
           
           
             
               
                   
               
             
             
               
                 
                   
                     ɛ 
                     2 
                   
                   
                     ɛ 
                     B 
                   
                 
                 = 
                 
                   3 
                   
                     ( 
                     
                       1 
                       + 
                       
                         cos 
                         ⁢ 
                         
                           
                             2 
                             ⁢ 
                             π 
                           
                           T 
                         
                         ⁢ 
                         
                           t 
                           1 
                         
                       
                     
                     ) 
                   
                 
               
             
           
           
             
               
                 
                   Average 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   power 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   loss 
                 
                 , 
               
             
             
               
                 
                   
                     
                       
                         P 
                         
                           ave 
                           - 
                           loss 
                           - 
                           2 
                         
                       
                       = 
                         
                       ⁢ 
                       
                         
                           P 
                           
                             ave 
                             - 
                             
                               i 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               n 
                             
                             - 
                             2 
                           
                         
                         - 
                         
                           P 
                           
                             ave 
                             - 
                             out 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       = 
                         
                       ⁢ 
                       
                         
                           
                             
                               2 
                               ⁢ 
                               
                                 V 
                                 cc 
                               
                               ⁢ 
                               
                                 V 
                                 0 
                               
                             
                             
                               3 
                               ⁢ 
                               π 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               R 
                             
                           
                           ⁢ 
                           
                             ( 
                             
                               1 
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                   
                                     2 
                                     ⁢ 
                                     π 
                                   
                                   T 
                                 
                                 ⁢ 
                                 
                                   t 
                                   1 
                                 
                               
                             
                             ) 
                           
                         
                         - 
                         
                           
                             V 
                             0 
                             2 
                           
                           
                             2 
                             ⁢ 
                             R 
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
   
   Class μ Amplifier Stage  3 : For sinusoidal output voltage |V cc2 |&lt;|V 0 |&lt;|V cc |: 
   From Equation (1) above, 
   
     
       
         
           
             t 
             1 
           
           = 
           
             
               T 
               
                 2 
                 ⁢ 
                 π 
               
             
             ⁢ 
             
               sin 
               
                 - 
                 1 
               
             
             ⁢ 
             
               
                 V 
                 cc 
               
               
                 3 
                 ⁢ 
                 
                   V 
                   0 
                 
               
             
           
         
       
     
   
   At the time t 2 , when V out =V cc2 , the moment that V cc2  is switched on, is given as follows: 
   
     
       
         
           
             
               
                 
                   V 
                   out 
                 
                 = 
                 
                   
                     V 
                     0 
                   
                   ⁢ 
                   sin 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   ω 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   t 
                 
               
             
           
           
             
               
                 
                   ⇒ 
                   
                     V 
                     cc2 
                   
                 
                 = 
                 
                   
                     V 
                     0 
                   
                   ⁢ 
                   sin 
                   ⁢ 
                   
                     
                       2 
                       ⁢ 
                       π 
                     
                     T 
                   
                   ⁢ 
                   
                     t 
                     2 
                   
                 
               
             
           
           
             
               
                 
                   
                     
                       2 
                       ⁢ 
                       π 
                     
                     T 
                   
                   ⁢ 
                   
                     t 
                     2 
                   
                 
                 = 
                 
                   
                     sin 
                     
                       - 
                       1 
                     
                   
                   ⁢ 
                   
                     
                       V 
                       cc2 
                     
                     
                       V 
                       0 
                     
                   
                 
               
             
           
           
             
               
                 
                   t 
                   2 
                 
                 = 
                 
                   
                     T 
                     
                       2 
                       ⁢ 
                       π 
                     
                   
                   ⁢ 
                   
                     sin 
                     
                       - 
                       1 
                     
                   
                   ⁢ 
                   
                     
                       2 
                       ⁢ 
                       
                         V 
                         cc 
                       
                     
                     
                       3 
                       ⁢ 
                       
                         V 
                         0 
                       
                     
                   
                 
               
             
           
         
       
     
   
   For the positive side of the load  24 , the total input power for ¼ period may be denoted as follows: 
   
     
       
         
           
             
               
                 
                   P 
                   
                     total 
                     - 
                     
                       i 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       n 
                     
                     - 
                     3 
                     + 
                   
                 
                 = 
                   
                 ⁢ 
                 
                   
                     
                       ∫ 
                       
                         t 
                         1 
                       
                       
                         t 
                         2 
                       
                     
                     ⁢ 
                     
                       
                         
                           
                             V 
                             cc1 
                           
                           ⁢ 
                           
                             V 
                             0 
                           
                           ⁢ 
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ω 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           t 
                         
                         R 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   + 
                   
                     
                       ∫ 
                       
                         t 
                         2 
                       
                       
                         T 
                         4 
                       
                     
                     ⁢ 
                     
                       
                         
                           
                             V 
                             cc2 
                           
                           ⁢ 
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ω 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           t 
                         
                         R 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                 
               
             
           
           
             
               
                 = 
                   
                 ⁢ 
                 
                   
                     
                       
                         
                           V 
                           cc1 
                         
                         ⁢ 
                         
                           V 
                           0 
                         
                       
                       R 
                     
                     ⁢ 
                     
                       
                         ∫ 
                         
                           t 
                           1 
                         
                         
                           t 
                           2 
                         
                       
                       ⁢ 
                       
                         sin 
                         ⁢ 
                         
                           
                             2 
                             ⁢ 
                             π 
                           
                           T 
                         
                         ⁢ 
                         t 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ⅆ 
                           t 
                         
                       
                     
                   
                   + 
                   
                     
                       
                         
                           V 
                           cc2 
                         
                         ⁢ 
                         
                           V 
                           0 
                         
                       
                       R 
                     
                     ⁢ 
                     
                       
                         ∫ 
                         
                           t 
                           2 
                         
                         
                           T 
                           4 
                         
                       
                       ⁢ 
                       
                         sin 
                         ⁢ 
                         
                           
                             2 
                             ⁢ 
                             π 
                           
                           T 
                         
                         ⁢ 
                         t 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ⅆ 
                           t 
                         
                       
                     
                   
                 
               
             
           
           
             
               
                 = 
                   
                 ⁢ 
                 
                   
                     
                       
                         
                           V 
                           cc1 
                         
                         ⁢ 
                         
                           V 
                           0 
                         
                       
                       R 
                     
                     ⁢ 
                     
                       
                         
                           T 
                           
                             2 
                             ⁢ 
                             π 
                           
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               - 
                               cos 
                             
                             ⁢ 
                             
                               
                                 2 
                                 ⁢ 
                                 π 
                               
                               T 
                             
                             ⁢ 
                             t 
                           
                           ] 
                         
                       
                       
                         t 
                         1 
                       
                       
                         t 
                         2 
                       
                     
                   
                   + 
                   
                     
                       
                         
                           V 
                           cc2 
                         
                         ⁢ 
                         
                           V 
                           0 
                         
                       
                       R 
                     
                     ⁢ 
                     
                       
                         
                           T 
                           
                             2 
                             ⁢ 
                             π 
                           
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               - 
                               cos 
                             
                             ⁢ 
                             
                               
                                 2 
                                 ⁢ 
                                 π 
                               
                               T 
                             
                             ⁢ 
                             t 
                           
                           ] 
                         
                       
                       
                         t 
                         2 
                       
                       
                         T 
                         4 
                       
                     
                   
                 
               
             
           
           
             
               
                 = 
                   
                 ⁢ 
                 
                   
                     
                       
                         V 
                         0 
                       
                       ⁢ 
                       T 
                     
                     
                       2 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       π 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       R 
                     
                   
                   ⁢ 
                   
                     { 
                     
                       
                         
                           V 
                           cc1 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               cos 
                               ⁢ 
                               
                                 
                                   2 
                                   ⁢ 
                                   π 
                                 
                                 T 
                               
                               ⁢ 
                               
                                 t 
                                 1 
                               
                             
                             - 
                             
                               cos 
                               ⁢ 
                               
                                 
                                   2 
                                   ⁢ 
                                   π 
                                 
                                 T 
                               
                               ⁢ 
                               
                                 t 
                                 2 
                               
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           V 
                           cc2 
                         
                         ⁡ 
                         
                           ( 
                           
                             cos 
                             ⁢ 
                             
                               
                                 2 
                                 ⁢ 
                                 π 
                               
                               T 
                             
                             ⁢ 
                             
                               t 
                               2 
                             
                           
                           ) 
                         
                       
                     
                     } 
                   
                 
               
             
           
         
       
     
     
       
         
           
             
               But 
               ⁢ 
               
                   
               
               ∵ 
               
                 V 
                 cc2 
               
             
             = 
             
               2 
               ⁢ 
               
                 V 
                 cc1 
               
             
           
           , 
           
             
 
           
           ⁢ 
           
             ∴ 
             
               
                 
                   
                     
                       P 
                       
                         total 
                         - 
                         
                           i 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           n 
                         
                         - 
                         3 
                         + 
                       
                     
                     = 
                       
                     ⁢ 
                     
                       
                         
                           
                             V 
                             0 
                           
                           ⁢ 
                           T 
                         
                         
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           π 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           R 
                         
                       
                       ⁢ 
                       
                         
                           V 
                           cc1 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               cos 
                               ⁢ 
                               
                                 
                                   2 
                                   ⁢ 
                                   π 
                                 
                                 T 
                               
                               ⁢ 
                               
                                 t 
                                 1 
                               
                             
                             + 
                             
                               cos 
                               ⁢ 
                               
                                 
                                   2 
                                   ⁢ 
                                   π 
                                 
                                 T 
                               
                               ⁢ 
                               
                                 t 
                                 2 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   
                     = 
                       
                     ⁢ 
                     
                       
                         
                           
                             V 
                             0 
                           
                           ⁢ 
                           
                             V 
                             cc 
                           
                           ⁢ 
                           T 
                         
                         
                           6 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           π 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           R 
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             cos 
                             ⁢ 
                             
                               
                                 2 
                                 ⁢ 
                                 π 
                               
                               T 
                             
                             ⁢ 
                             
                               t 
                               1 
                             
                           
                           + 
                           
                             cos 
                             ⁢ 
                             
                               
                                 2 
                                 ⁢ 
                                 π 
                               
                               T 
                             
                             ⁢ 
                             
                               t 
                               2 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
             
           
         
       
     
   
   For the negative side of the load  24 , from equation (2), 
   
     
       
         
           
             P 
             
               total 
               - 
               
                 i 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 n 
               
               - 
               3 
               - 
             
           
           = 
           
             
               
                 V 
                 cc 
               
               ⁢ 
               
                 V 
                 0 
               
               ⁢ 
               T 
             
             
               6 
               ⁢ 
               
                   
               
               ⁢ 
               π 
               ⁢ 
               
                   
               
               ⁢ 
               R 
             
           
         
       
     
     
       
         
           
             
               
                 
                   
                     
                       
                         Average 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         total 
                       
                     
                   
                   
                     
                       
                         input 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         power 
                       
                     
                   
                 
                 , 
               
             
             
               
                 
                   
                     
                       
                         P 
                         
                           total 
                           - 
                           
                             i 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             n 
                           
                           - 
                           3 
                         
                       
                       = 
                         
                       ⁢ 
                       
                         
                           
                             P 
                             
                               total 
                               - 
                               
                                 i 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 n 
                               
                               - 
                               3 
                               + 
                             
                           
                           + 
                           
                             P 
                             
                               total 
                               - 
                               
                                 i 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 n 
                               
                               - 
                               3 
                               - 
                             
                           
                         
                         
                           T 
                           4 
                         
                       
                     
                   
                 
                 
                   
                     
                       = 
                         
                       ⁢ 
                       
                         
                           
                             
                               
                                 4 
                                 T 
                               
                               ⁢ 
                               
                                 { 
                                 
                                   
                                     
                                       
                                         
                                           V 
                                           0 
                                         
                                         ⁢ 
                                         
                                           V 
                                           cc 
                                         
                                         ⁢ 
                                         T 
                                       
                                       
                                         6 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         π 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         R 
                                       
                                     
                                     ⁡ 
                                     
                                       [ 
                                       
                                         ( 
                                         
                                           
                                             cos 
                                             ⁢ 
                                             
                                               
                                                 2 
                                                 ⁢ 
                                                 π 
                                               
                                               T 
                                             
                                             ⁢ 
                                             
                                               t 
                                               1 
                                             
                                           
                                           + 
                                           
                                             cos 
                                             ⁢ 
                                             
                                               
                                                 2 
                                                 ⁢ 
                                                 π 
                                               
                                               T 
                                             
                                             ⁢ 
                                             
                                               t 
                                               2 
                                             
                                           
                                         
                                         ) 
                                       
                                       ] 
                                     
                                   
                                   + 
                                 
                               
                             
                           
                         
                         
                           
                             
                                 
                               ⁢ 
                               
                                 
                                   
                                     V 
                                     cc 
                                   
                                   ⁢ 
                                   
                                     V 
                                     0 
                                   
                                   ⁢ 
                                   T 
                                 
                                 
                                   6 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   R 
                                 
                               
                               } 
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       = 
                         
                       ⁢ 
                       
                         
                           
                             2 
                             ⁢ 
                             
                               V 
                               0 
                             
                             ⁢ 
                             
                               V 
                               cc 
                             
                           
                           
                             3 
                             ⁢ 
                             π 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             R 
                           
                         
                         ⁢ 
                         
                           ( 
                           
                             1 
                             + 
                             
                               cos 
                               ⁢ 
                               
                                 
                                   2 
                                   ⁢ 
                                   π 
                                 
                                 T 
                               
                               ⁢ 
                               
                                 t 
                                 1 
                               
                             
                             + 
                             
                               cos 
                               ⁢ 
                               
                                 
                                   2 
                                   ⁢ 
                                   π 
                                 
                                 T 
                               
                               ⁢ 
                               
                                 t 
                                 2 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
             
           
           
             
               
                 Efficiency 
                 , 
               
             
             
               
                 
                   
                     
                       
                         ɛ 
                         3 
                       
                       = 
                         
                       ⁢ 
                       
                         
                           
                             V 
                             0 
                             2 
                           
                           
                             2 
                             ⁢ 
                             R 
                           
                         
                         × 
                         
                           
                             3 
                             ⁢ 
                             π 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             R 
                           
                           
                             2 
                             ⁢ 
                             
                               V 
                               cc 
                             
                             ⁢ 
                             
                               
                                 V 
                                 0 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   
                                     cos 
                                     ⁢ 
                                     
                                       
                                         2 
                                         ⁢ 
                                         π 
                                       
                                       T 
                                     
                                     ⁢ 
                                     
                                       t 
                                       1 
                                     
                                   
                                   + 
                                   
                                     cos 
                                     ⁢ 
                                     
                                       
                                         2 
                                         ⁢ 
                                         π 
                                       
                                       T 
                                     
                                     ⁢ 
                                     
                                       t 
                                       2 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       = 
                         
                       ⁢ 
                       
                         
                           3 
                           ⁢ 
                           π 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             V 
                             0 
                           
                         
                         
                           4 
                           ⁢ 
                           
                             
                               V 
                               cc 
                             
                             ⁡ 
                             
                               ( 
                               
                                 1 
                                 + 
                                 
                                   cos 
                                   ⁢ 
                                   
                                     
                                       2 
                                       ⁢ 
                                       π 
                                     
                                     T 
                                   
                                   ⁢ 
                                   
                                     t 
                                     1 
                                   
                                 
                                 + 
                                 
                                   cos 
                                   ⁢ 
                                   
                                     
                                       2 
                                       ⁢ 
                                       π 
                                     
                                     T 
                                   
                                   ⁢ 
                                   
                                     t 
                                     2 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
   
   Comparing the above with a conventional Class B Amplifier, 
   
     
       
         
           
             
               
                   
               
             
             
               
                 
                   
                     ɛ 
                     3 
                   
                   
                     ɛ 
                     B 
                   
                 
                 = 
                 
                   
                     
                       3 
                       ⁢ 
                       π 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         V 
                         0 
                       
                     
                     
                       4 
                       ⁢ 
                       
                         
                           V 
                           cc 
                         
                         ⁡ 
                         
                           ( 
                           
                             1 
                             + 
                             
                               cos 
                               ⁢ 
                               
                                 
                                   2 
                                   ⁢ 
                                   π 
                                 
                                 T 
                               
                               ⁢ 
                               
                                 t 
                                 1 
                               
                             
                             + 
                             
                               cos 
                               ⁢ 
                               
                                 
                                   2 
                                   ⁢ 
                                   π 
                                 
                                 T 
                               
                               ⁢ 
                               
                                 t 
                                 2 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   × 
                   
                     
                       4 
                       ⁢ 
                       
                         V 
                         cc 
                       
                     
                     
                       π 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         V 
                         0 
                       
                     
                   
                 
               
             
           
           
             
               
                   
               
             
             
               
                 
                   
                     ɛ 
                     2 
                   
                   
                     ɛ 
                     B 
                   
                 
                 = 
                 
                   3 
                   
                     ( 
                     
                       1 
                       + 
                       
                         cos 
                         ⁢ 
                         
                           
                             2 
                             ⁢ 
                             π 
                           
                           T 
                         
                         ⁢ 
                         
                           t 
                           1 
                         
                       
                       + 
                       
                         cos 
                         ⁢ 
                         
                           
                             2 
                             ⁢ 
                             π 
                           
                           T 
                         
                         ⁢ 
                         
                           t 
                           2 
                         
                       
                     
                     ) 
                   
                 
               
             
           
           
             
               
                 
                   
                     
                       Average 
                     
                   
                   
                     
                       
                         power 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         loss 
                       
                     
                   
                 
                 , 
               
             
             
               
                 
                   
                     
                       
                         P 
                         
                           ave 
                           - 
                           loss 
                           - 
                           3 
                         
                       
                       = 
                         
                       ⁢ 
                       
                         
                           P 
                           
                             ave 
                             - 
                             
                               i 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               n 
                             
                             - 
                             3 
                           
                         
                         - 
                         
                           P 
                           
                             ave 
                             - 
                             out 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       = 
                         
                       ⁢ 
                       
                         
                           
                             
                               2 
                               ⁢ 
                               
                                 V 
                                 cc 
                               
                               ⁢ 
                               
                                 V 
                                 0 
                               
                             
                             
                               3 
                               ⁢ 
                               π 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               R 
                             
                           
                           ⁢ 
                           
                             ( 
                             
                               1 
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                   
                                     2 
                                     ⁢ 
                                     π 
                                   
                                   T 
                                 
                                 ⁢ 
                                 
                                   t 
                                   1 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                   
                                     2 
                                     ⁢ 
                                     π 
                                   
                                   T 
                                 
                                 ⁢ 
                                 
                                   t 
                                   2 
                                 
                               
                             
                             ) 
                           
                         
                         - 
                         
                           
                             V 
                             0 
                             2 
                           
                           
                             2 
                             ⁢ 
                             R 
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
   
     FIG. 6  shows a graph of Efficiency against Normalized Output Power for the amplifier circuit of  FIG. 2 , and, for comparison, a graph of Efficiency against Normalized Output Power for a conventional Class B amplifier. The graphs shown are for amplifiers operating at maximum output power with a sinusoidal input, without clipping, normalized to 1. 
     FIG. 7  shows a graph of Power Loss in the amplifier of  FIG. 2  against Normalized Output Power and, for comparison, a graph of Power Loss in a conventional Class B amplifier against Normalized Output Power. As with  FIG. 6 , the graphs shown are for amplifiers operating at maximum output power with a sinusoidal input, without clipping, normalized to 1. 
     FIG. 8  shows an alternative preferred embodiment of a Class μ Amplifier according to the present invention. The same reference numerals as those used in respect of the embodiment of  FIG. 2  have been used to denote identical components. 
   The circuit shown in  FIG. 8  is identical to that shown in  FIG. 2 , the difference being that V cc1 , which is the voltage step between inputs on the switching unit  22 , is V cc /5 in the circuit of  FIG. 8  and the DC voltages output to the positive terminal at node  1  of the load  24  will therefore be −4V cc1 , −2V cc1 , GND, 2V cc1  and 4V cc1  accordingly. 
   The resistor values in the circuit of  FIG. 8  are chosen as follows:
 
 R   3 /( R   1   +R   2   +R   3 )= R   4 /( R   4   +R   5   +R   6 )=1/5
 
( R   2   +R   3 )/( R   1   +R   2   +R   3 )=( R   4   +R   5 )/( R   4   +R   5   +R   6 )=3/5
 
 V   cc1   =V   cc /5
 
R8/R7=G V 
 
 R 10/( R 9+ R 10)=1/ G   V 
 
     FIG. 9  shows the waveform at the positive terminal (node  1 ) of the load  24  of the circuit of  FIG. 8 . 
     FIG. 10  shows the signal at node  3  of the negative terminal of the circuit of  FIG. 8 . 
   In the embodiment of  FIG. 8 , on the negative side of the load  24 , the differential amplifier, instead of giving a half swing (as shown in  FIG. 5 ), gives a full swing as shown in  FIG. 10 . 
     FIG. 11  shows a graph of Efficiency against Normalized Output Power for the amplifier of  FIG. 8 , and, for comparison, a graph of Efficiency against 
   Normalized Output Power for a conventional Class B amplifier. The graphs shown are for the amplifiers operating at maximum output power with a sinusoidal input, without clipping normalized to 1. 
     FIG. 12  shows a graph Power Loss in the amplifier of  FIG. 8  against Normalized Output Power and, for comparison, a graph of Power Loss in a conventional Class B amplifier against Normalized Output Power. As with  FIG. 11 , the graphs shown are for amplifiers operating at maximum output power with a sinusoidal input, without clipping normalized to 1. 
     FIG. 13  shows a graph of Efficiency against Normalized Output Power for each of the amplifiers of  FIGS. 2 and 8  and, for comparison, a graph of Efficiency against Normalized Output Power for a conventional Class B amplifier. The graphs shown are for amplifiers operating at maximum output power with a sinusoidal input, without clipping normalized to 1. 
     FIG. 14  shows a graph Power Loss in the amplifiers of  FIGS. 2 and 8  against Normalized Output Power and, for comparison, a graph of Power Loss in a conventional Class B amplifier against Normalized Output Power. As with  FIG. 13 , the graphs shown are for the amplifiers operating at maximum output power with a sinusoidal input, without clipping normalized to 1. 
   Although in terms of efficiency and power loss, the performance of the embodiment of  FIG. 8  is not as good as that of  FIG. 2 , an advantage of the embodiment of  FIG. 8  is that a differential amplifier having lower output voltage and also a lower output power may be used. If the maximum output voltage of the differential amplifier  23  is 12V, then the overall output voltage to the load  24  will be 60V. 
   The efficiency and power loss calculations for the embodiment of  FIG. 8  are described as follows. 
   As before, a conventional Class B amplifier with a supply voltage of V cc  and capable of delivering a maximum output voltage of amplitude V cc  is used as comparison. The symbols used have been kept the same, except for V cc1 , which is changed to V cc /5. 
   Output voltage, V out =V 0  sin ωt 
   Average output power, 
   
     
       
         
           
             P 
             
               ave 
               - 
               out 
             
           
           = 
           
             
               
                 V 
                 out 
               
               2 
             
             
               2 
               ⁢ 
               R 
             
           
         
       
     
   
   Class μ Amplifier Stage  1 : For sinusoidal output voltage |V 0 |&lt;|V cc1|:    
   For this stage of operation, the ground potential will be supplied to the positive side of the load  24 . Thus, only the differential amplifier  23  is supplying the signal to the load  24  at the negative side thereof. 
   
     
       
         
           
             
               
                 
                   Average 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   input 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   power 
                 
                 , 
               
             
             
               
                 
                   
                     
                       
                         P 
                         
                           ave 
                           - 
                           
                             i 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             n 
                           
                           - 
                           1 
                         
                       
                       = 
                         
                       ⁢ 
                       
                         
                           2 
                           ⁢ 
                           
                             V 
                             cc1 
                           
                           ⁢ 
                           
                             V 
                             0 
                           
                         
                         
                           π 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           R 
                         
                       
                     
                   
                 
                 
                   
                     
                       = 
                         
                       ⁢ 
                       
                         
                           
                             
                               2 
                               ⁢ 
                               
                                 V 
                                 cc 
                               
                               ⁢ 
                               
                                 V 
                                 0 
                               
                             
                             
                               5 
                               ⁢ 
                               π 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               R 
                             
                           
                           ⁢ 
                           
                               
                           
                           ∵ 
                           
                             V 
                             cc1 
                           
                         
                         = 
                         
                           
                             V 
                             cc 
                           
                           5 
                         
                       
                     
                   
                 
               
             
           
           
             
               
                 Efficiency 
                 , 
               
             
             
               
                 
                   
                     
                       
                         ɛ 
                         1 
                       
                       = 
                         
                       ⁢ 
                       
                         
                           π 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             V 
                             0 
                           
                         
                         
                           4 
                           ⁢ 
                           
                             V 
                             cc1 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       = 
                         
                       ⁢ 
                       
                         
                           5 
                           ⁢ 
                           π 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             V 
                             0 
                           
                         
                         
                           4 
                           ⁢ 
                           
                             V 
                             cc 
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
   
   Comparing the above with a conventional Class B Amplifier, during this stage of operation, 
   
     
       
         
           
             
               
                   
               
             
             
               
                 
                   
                     
                       
                         
                           ɛ 
                           1 
                         
                         
                           ɛ 
                           B 
                         
                       
                       = 
                       
                         
                           
                             
                               5 
                               ⁢ 
                               π 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 V 
                                 0 
                               
                             
                             
                               4 
                               ⁢ 
                               
                                 V 
                                 cc 
                               
                             
                           
                           × 
                           
                             
                               4 
                               ⁢ 
                               
                                 V 
                                 cc 
                               
                             
                             
                               π 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 V 
                                 0 
                               
                             
                           
                         
                         = 
                         5 
                       
                     
                   
                 
                 
                   
                     
                       
                         ⇒ 
                         
                           ɛ 
                           1 
                         
                       
                       = 
                       
                         5 
                         ⁢ 
                         
                           ɛ 
                           B 
                         
                       
                     
                   
                 
               
             
           
           
             
               
                 
                   Average 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   power 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   loss 
                 
                 , 
               
             
             
               
                 
                   
                     
                       
                         P 
                         
                           ave 
                           - 
                           loss 
                           - 
                           1 
                         
                       
                       = 
                         
                       ⁢ 
                       
                         
                           P 
                           
                             ave 
                             - 
                             
                               i 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               n 
                             
                             - 
                             1 
                           
                         
                         - 
                         
                           P 
                           
                             ave 
                             - 
                             out 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       = 
                         
                       ⁢ 
                       
                         
                           
                             2 
                             ⁢ 
                             
                               V 
                               cc 
                             
                             ⁢ 
                             
                               V 
                               0 
                             
                           
                           
                             5 
                             ⁢ 
                             π 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             R 
                           
                         
                         - 
                         
                           
                             V 
                             0 
                             2 
                           
                           
                             2 
                             ⁢ 
                             R 
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
   
   Class μ Amplifier Stage  2 : For sinusoidal output voltage |V cc1 |&lt;|V 0 |&lt;|3V cc1 | 
   For this stage, it is necessary to determine the time t 1  when V out =V cc1 , which is the moment that V cc1  is switched on. 
   
     
       
         
           
             
               
                 
                   Output 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   voltage 
                 
                 , 
               
             
             
               
                 
                   
                     V 
                     out 
                   
                   = 
                   
                     
                       V 
                       0 
                     
                     ⁢ 
                     sin 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     ω 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     t 
                   
                 
                 ⁢ 
                 
                     
                 
               
             
           
           
             
               
                 
                   At 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     t 
                     1 
                   
                 
                 , 
                 
                   
                     V 
                     out 
                   
                   = 
                   
                     V 
                     cc1 
                   
                 
                 , 
               
             
             
               
                 
                   
                     
                       
                         ⇒ 
                         
                           V 
                           cc1 
                         
                       
                       = 
                       
                         
                           V 
                           0 
                         
                         ⁢ 
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             2 
                             ⁢ 
                             π 
                           
                           T 
                         
                         ⁢ 
                         
                           t 
                           1 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             2 
                             ⁢ 
                             π 
                           
                           T 
                         
                         ⁢ 
                         
                           t 
                           1 
                         
                       
                       = 
                       
                         
                           sin 
                           
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           
                             V 
                             cc1 
                           
                           
                             V 
                             0 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         t 
                         1 
                       
                       = 
                       
                         
                           T 
                           
                             2 
                             ⁢ 
                             π 
                           
                         
                         ⁢ 
                         
                           sin 
                           
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           
                             V 
                             cc 
                           
                           
                             5 
                             ⁢ 
                             
                               V 
                               0 
                             
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           3 
                           ) 
                         
                       
                     
                   
                 
               
             
           
         
       
     
   
   For the positive side of the load  24 , the total input power for ¼ period is: 
   
     
       
         
           
             
               
                 
                   P 
                   
                     total 
                     - 
                     
                       i 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       n 
                     
                     - 
                     2 
                     + 
                   
                 
                 = 
                   
                 ⁢ 
                 
                   
                     ∫ 
                     
                       t 
                       1 
                     
                     
                       T 
                       4 
                     
                   
                   ⁢ 
                   
                     
                       
                         2 
                         ⁢ 
                         
                           V 
                           cc1 
                         
                         ⁢ 
                         
                           V 
                           0 
                         
                         ⁢ 
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         ω 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         t 
                       
                       R 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ⅆ 
                       t 
                     
                   
                 
               
             
           
           
             
               
                 = 
                   
                 ⁢ 
                 
                   
                     
                       2 
                       ⁢ 
                       
                         V 
                         cc1 
                       
                       ⁢ 
                       
                         V 
                         0 
                       
                     
                     R 
                   
                   ⁢ 
                   
                     
                       ∫ 
                       
                         t 
                         1 
                       
                       
                         T 
                         4 
                       
                     
                     ⁢ 
                     
                       sin 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           2 
                           ⁢ 
                           π 
                         
                         T 
                       
                       ⁢ 
                       t 
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                 
               
             
           
           
             
               
                 = 
                   
                 ⁢ 
                 
                   
                     
                       2 
                       ⁢ 
                       
                         V 
                         cc 
                       
                       ⁢ 
                       
                         V 
                         0 
                       
                     
                     
                       5 
                       ⁢ 
                       R 
                     
                   
                   ⁢ 
                   
                     
                       
                         T 
                         
                           2 
                           ⁢ 
                           π 
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             - 
                             cos 
                           
                           ⁢ 
                           
                             
                               2 
                               ⁢ 
                               π 
                             
                             T 
                           
                           ⁢ 
                           t 
                         
                         ] 
                       
                     
                     
                       t 
                       1 
                     
                     
                       T 
                       4 
                     
                   
                 
               
             
           
           
             
               
                 = 
                   
                 ⁢ 
                 
                   
                     
                       
                         V 
                         cc 
                       
                       ⁢ 
                       
                         V 
                         0 
                       
                       ⁢ 
                       T 
                     
                     
                       5 
                       ⁢ 
                       π 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       R 
                     
                   
                   ⁢ 
                   cos 
                   ⁢ 
                   
                     
                       2 
                       ⁢ 
                       π 
                     
                     T 
                   
                   ⁢ 
                   
                     t 
                     1 
                   
                 
               
             
           
         
       
     
   
   For the negative side of the load  24 , the total input power for ¼ period is: 
   
     
       
         
           
             
               
                 
                   
                     
                         
                     
                   
                   
                     
                       
                         
                           
                             
                               P 
                               
                                 total 
                                 - 
                                 
                                   i 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   n 
                                 
                                 - 
                                 2 
                                 - 
                               
                             
                             = 
                               
                             ⁢ 
                             
                               
                                 ∫ 
                                 0 
                                 
                                   T 
                                   4 
                                 
                               
                               ⁢ 
                               
                                 
                                   
                                     
                                       V 
                                       cc1 
                                     
                                     ⁢ 
                                     
                                       V 
                                       0 
                                     
                                     ⁢ 
                                     sin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     ω 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     t 
                                   
                                   R 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ⅆ 
                                   t 
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             = 
                               
                             ⁢ 
                             
                               
                                 
                                   
                                     V 
                                     cc1 
                                   
                                   ⁢ 
                                   
                                     V 
                                     0 
                                   
                                 
                                 R 
                               
                               ⁢ 
                               
                                 
                                   ∫ 
                                   0 
                                   
                                     T 
                                     4 
                                   
                                 
                                 ⁢ 
                                 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       2 
                                       ⁢ 
                                       π 
                                     
                                     T 
                                   
                                   ⁢ 
                                   t 
                                   ⁢ 
                                   
                                     ⅆ 
                                     t 
                                   
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             = 
                               
                             ⁢ 
                             
                               
                                 
                                   
                                     V 
                                     cc 
                                   
                                   ⁢ 
                                   
                                     V 
                                     0 
                                   
                                 
                                 
                                   5 
                                   ⁢ 
                                   R 
                                 
                               
                               ⁢ 
                               
                                 
                                   
                                     T 
                                     
                                       2 
                                       ⁢ 
                                       π 
                                     
                                   
                                   ⁡ 
                                   
                                     [ 
                                     
                                       
                                         - 
                                         cos 
                                       
                                       ⁢ 
                                       
                                         
                                           2 
                                           ⁢ 
                                           π 
                                         
                                         T 
                                       
                                       ⁢ 
                                       t 
                                     
                                     ] 
                                   
                                 
                                 0 
                                 
                                   T 
                                   4 
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             = 
                               
                             ⁢ 
                             
                               
                                 
                                   V 
                                   cc 
                                 
                                 ⁢ 
                                 
                                   V 
                                   0 
                                 
                                 ⁢ 
                                 T 
                               
                               
                                 10 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 R 
                               
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         Average 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         input 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         power 
                       
                       , 
                     
                   
                   
                     
                       
                         
                           
                             
                               P 
                               
                                 ave 
                                 - 
                                 
                                   i 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   n 
                                 
                                 - 
                                 2 
                               
                             
                             = 
                             
                               
                                 
                                   P 
                                   
                                     total 
                                     - 
                                     
                                       i 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       n 
                                     
                                     - 
                                     2 
                                     + 
                                   
                                 
                                 + 
                                 
                                   P 
                                   
                                     total 
                                     - 
                                     
                                       i 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       n 
                                     
                                     - 
                                     2 
                                     + 
                                   
                                 
                               
                               
                                 T 
                                 4 
                               
                             
                           
                         
                       
                       
                         
                           
                             = 
                             
                               
                                 
                                   2 
                                   ⁢ 
                                   
                                     V 
                                     cc 
                                   
                                   ⁢ 
                                   
                                     V 
                                     0 
                                   
                                 
                                 
                                   5 
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   R 
                                 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   
                                     2 
                                     ⁢ 
                                     cos 
                                     ⁢ 
                                     
                                       
                                         2 
                                         ⁢ 
                                         π 
                                       
                                       T 
                                     
                                     ⁢ 
                                     
                                       t 
                                       1 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       Efficiency 
                       , 
                     
                   
                   
                     
                       
                         
                           
                             
                               ɛ 
                               2 
                             
                             = 
                               
                             ⁢ 
                             
                               
                                 
                                   V 
                                   0 
                                   2 
                                 
                                 
                                   2 
                                   ⁢ 
                                   R 
                                 
                               
                               × 
                               
                                 
                                   5 
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   R 
                                 
                                 
                                   2 
                                   ⁢ 
                                   
                                     V 
                                     cc 
                                   
                                   ⁢ 
                                   
                                     
                                       V 
                                       0 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         1 
                                         + 
                                         
                                           2 
                                           ⁢ 
                                           cos 
                                           ⁢ 
                                           
                                             
                                               2 
                                               ⁢ 
                                               π 
                                             
                                             T 
                                           
                                           ⁢ 
                                           
                                             t 
                                             1 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             = 
                               
                             ⁢ 
                             
                               
                                 5 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   V 
                                   0 
                                 
                               
                               
                                 4 
                                 ⁢ 
                                 
                                   
                                     V 
                                     cc 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       1 
                                       + 
                                       
                                         2 
                                         ⁢ 
                                         cos 
                                         ⁢ 
                                         
                                           
                                             2 
                                             ⁢ 
                                             π 
                                           
                                           T 
                                         
                                         ⁢ 
                                         
                                           t 
                                           1 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 4 
                 ) 
               
             
           
         
       
     
   
   Comparing the above with a conventional Class B Amplifier, 
   
     
       
         
           
             
               
                   
               
             
             
               
                 
                   
                     ɛ 
                     2 
                   
                   
                     ɛ 
                     B 
                   
                 
                 = 
                 
                   
                     
                       5 
                       ⁢ 
                       π 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         V 
                         0 
                       
                     
                     
                       4 
                       ⁢ 
                       
                         
                           V 
                           cc 
                         
                         ⁡ 
                         
                           ( 
                           
                             1 
                             + 
                             
                               2 
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                 
                                   2 
                                   ⁢ 
                                   π 
                                 
                                 T 
                               
                               ⁢ 
                               
                                 t 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   × 
                   
                     
                       4 
                       ⁢ 
                       
                         V 
                         cc 
                       
                     
                     
                       π 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         V 
                         0 
                       
                     
                   
                 
               
             
           
           
             
               
                   
               
             
             
               
                 
                   
                     ɛ 
                     2 
                   
                   
                     ɛ 
                     B 
                   
                 
                 = 
                 
                   5 
                   
                     ( 
                     
                       1 
                       + 
                       
                         2 
                         ⁢ 
                         cos 
                         ⁢ 
                         
                           
                             2 
                             ⁢ 
                             π 
                           
                           T 
                         
                         ⁢ 
                         
                           t 
                           1 
                         
                       
                     
                     ) 
                   
                 
               
             
           
           
             
               
                 
                   Average 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   power 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   loss 
                 
                 , 
               
             
             
               
                 
                   
                     
                       
                         P 
                         
                           ave 
                           - 
                           loss 
                           - 
                           2 
                         
                       
                       = 
                         
                       ⁢ 
                       
                         
                           P 
                           
                             ave 
                             - 
                             
                               i 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               n 
                             
                             - 
                             2 
                           
                         
                         - 
                         
                           P 
                           
                             ave 
                             - 
                             out 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       = 
                         
                       ⁢ 
                       
                         
                           
                             
                               2 
                               ⁢ 
                               
                                 V 
                                 cc 
                               
                               ⁢ 
                               
                                 V 
                                 0 
                               
                             
                             
                               5 
                               ⁢ 
                               π 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               R 
                             
                           
                           ⁢ 
                           
                             ( 
                             
                               1 
                               + 
                               
                                 2 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                   
                                     2 
                                     ⁢ 
                                     π 
                                   
                                   T 
                                 
                                 ⁢ 
                                 
                                   t 
                                   1 
                                 
                               
                             
                             ) 
                           
                         
                         - 
                         
                           
                             V 
                             0 
                             2 
                           
                           
                             2 
                             ⁢ 
                             R 
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
   
   Class μ Amplifier Stage  3 : For sinusoidal output voltage |3V cc1 |&lt;|V 0 |V cc1 | 
   
     
       
         
           
             From 
             ⁢ 
             
                 
             
             ⁢ 
             
               ( 
               3 
               ) 
             
           
           , 
           
             
               t 
               1 
             
             = 
             
               
                 T 
                 
                   2 
                   ⁢ 
                   π 
                 
               
               ⁢ 
               
                 sin 
                 
                   - 
                   1 
                 
               
               ⁢ 
               
                 
                   V 
                   cc 
                 
                 
                   5 
                   ⁢ 
                   
                     V 
                     0 
                   
                 
               
             
           
         
       
     
   
   The time t 2 , when V out =V cc2 , the moment that 3V cc1  is switched on, is given as follows. 
   
     
       
         
           
             
               
                 
                   V 
                   out 
                 
                 = 
                 
                   
                     V 
                     0 
                   
                   ⁢ 
                   sin 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   ω 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   t 
                 
               
             
           
           
             
               
                 
                   ⇒ 
                   
                     3 
                     ⁢ 
                     
                       V 
                       cc1 
                     
                   
                 
                 = 
                 
                   
                     V 
                     0 
                   
                   ⁢ 
                   sin 
                   ⁢ 
                   
                     
                       2 
                       ⁢ 
                       π 
                     
                     T 
                   
                   ⁢ 
                   
                     t 
                     2 
                   
                 
               
             
           
           
             
               
                 
                   
                     
                       2 
                       ⁢ 
                       π 
                     
                     T 
                   
                   ⁢ 
                   
                     t 
                     2 
                   
                 
                 = 
                 
                   
                     sin 
                     
                       - 
                       1 
                     
                   
                   ⁢ 
                   
                     
                       3 
                       ⁢ 
                       
                         V 
                         cc1 
                       
                     
                     
                       V 
                       0 
                     
                   
                 
               
             
           
           
             
               
                 
                   t 
                   2 
                 
                 = 
                 
                   
                     T 
                     
                       2 
                       ⁢ 
                       π 
                     
                   
                   ⁢ 
                   
                     sin 
                     
                       - 
                       1 
                     
                   
                   ⁢ 
                   
                     
                       3 
                       ⁢ 
                       
                         V 
                         cc 
                       
                     
                     
                       5 
                       ⁢ 
                       
                         V 
                         0 
                       
                     
                   
                 
               
             
           
         
       
     
   
   For the positive side of the load  24 , the total input power for ¼ period is: 
   
     
       
         
           
             
               
                 
                   P 
                   
                     total 
                     - 
                     
                       i 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       n 
                     
                     - 
                     3 
                     + 
                   
                 
                 = 
                   
                 ⁢ 
                 
                   
                     
                       ∫ 
                       
                         t 
                         1 
                       
                       
                         t 
                         2 
                       
                     
                     ⁢ 
                     
                       
                         
                           2 
                           ⁢ 
                           
                             V 
                             cc1 
                           
                           ⁢ 
                           
                             V 
                             0 
                           
                           ⁢ 
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ω 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           t 
                         
                         R 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   + 
                   
                     
                       ∫ 
                       
                         t 
                         2 
                       
                       
                         T 
                         4 
                       
                     
                     ⁢ 
                     
                       
                         
                           4 
                           ⁢ 
                           
                             V 
                             cc1 
                           
                           ⁢ 
                           
                             V 
                             0 
                           
                           ⁢ 
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ω 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           t 
                         
                         R 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                 
               
             
           
           
             
               
                 = 
                   
                 ⁢ 
                 
                   
                     
                       
                         2 
                         ⁢ 
                         
                           V 
                           cc1 
                         
                         ⁢ 
                         
                           V 
                           0 
                         
                       
                       R 
                     
                     ⁢ 
                     
                       
                         ∫ 
                         
                           t 
                           1 
                         
                         
                           t 
                           2 
                         
                       
                       ⁢ 
                       
                         sin 
                         ⁢ 
                         
                           
                             2 
                             ⁢ 
                             π 
                           
                           T 
                         
                         ⁢ 
                         t 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ⅆ 
                           t 
                         
                       
                     
                   
                   + 
                   
                     
                       
                         4 
                         ⁢ 
                         
                           V 
                           cc1 
                         
                         ⁢ 
                         
                           V 
                           0 
                         
                       
                       R 
                     
                     ⁢ 
                     
                       
                         ∫ 
                         
                           t 
                           2 
                         
                         
                           T 
                           4 
                         
                       
                       ⁢ 
                       
                         sin 
                         ⁢ 
                         
                           
                             2 
                             ⁢ 
                             π 
                           
                           T 
                         
                         ⁢ 
                         t 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ⅆ 
                           t 
                         
                       
                     
                   
                 
               
             
           
           
             
               
                 = 
                   
                 ⁢ 
                 
                   
                     
                       
                         2 
                         ⁢ 
                         
                           V 
                           cc1 
                         
                         ⁢ 
                         
                           V 
                           0 
                         
                       
                       R 
                     
                     ⁢ 
                     
                       
                         
                           T 
                           
                             2 
                             ⁢ 
                             π 
                           
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               - 
                               cos 
                             
                             ⁢ 
                             
                               
                                 2 
                                 ⁢ 
                                 π 
                               
                               T 
                             
                             ⁢ 
                             t 
                           
                           ] 
                         
                       
                       
                         t 
                         1 
                       
                       
                         t 
                         2 
                       
                     
                   
                   + 
                   
                     
                       
                         4 
                         ⁢ 
                         
                           V 
                           cc1 
                         
                         ⁢ 
                         
                           V 
                           0 
                         
                       
                       R 
                     
                     ⁢ 
                     
                       
                         
                           T 
                           
                             2 
                             ⁢ 
                             π 
                           
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               - 
                               cos 
                             
                             ⁢ 
                             
                               
                                 2 
                                 ⁢ 
                                 π 
                               
                               T 
                             
                             ⁢ 
                             t 
                           
                           ] 
                         
                       
                       
                         t 
                         2 
                       
                       
                         T 
                         4 
                       
                     
                   
                 
               
             
           
           
             
               
                 = 
                   
                 ⁢ 
                 
                   
                     
                       
                         
                           V 
                           cc1 
                         
                         ⁢ 
                         
                           V 
                           0 
                         
                         ⁢ 
                         T 
                       
                       
                         π 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         R 
                       
                     
                     ⁢ 
                     
                       { 
                       
                         
                           cos 
                           ⁢ 
                           
                             
                               2 
                               ⁢ 
                               π 
                             
                             T 
                           
                           ⁢ 
                           
                             t 
                             1 
                           
                         
                         - 
                         
                           cos 
                           ⁢ 
                           
                             
                               2 
                               ⁢ 
                               π 
                             
                             T 
                           
                           ⁢ 
                           
                             t 
                             2 
                           
                         
                       
                       ) 
                     
                   
                   + 
                 
               
             
           
           
             
               
                   
                 ⁢ 
                 
                   
                     
                       2 
                       ⁢ 
                       
                         V 
                         cc1 
                       
                       ⁢ 
                       
                         V 
                         0 
                       
                       ⁢ 
                       T 
                     
                     
                       π 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       R 
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       cos 
                       ⁢ 
                       
                         
                           2 
                           ⁢ 
                           π 
                         
                         T 
                       
                       ⁢ 
                       
                         t 
                         2 
                       
                     
                     } 
                   
                 
               
             
           
         
       
     
     
       
         
           
             
               But 
               ⁢ 
               
                   
               
               ∵ 
               
                 V 
                 cc1 
               
             
             = 
             
               
                 V 
                 cc 
               
               5 
             
           
           , 
           
             
 
           
           ⁢ 
           
             
               ∴ 
               
                 P 
                 
                   total 
                   - 
                   
                     i 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     n 
                   
                   - 
                   3 
                   + 
                 
               
             
             = 
             
               
                 
                   
                     V 
                     cc 
                   
                   ⁢ 
                   
                     V 
                     0 
                   
                   ⁢ 
                   T 
                 
                 
                   25 
                   ⁢ 
                   π 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   R 
                 
               
               ⁢ 
               
                 ( 
                 
                   
                     cos 
                     ⁢ 
                     
                       
                         2 
                         ⁢ 
                         π 
                       
                       T 
                     
                     ⁢ 
                     
                       t 
                       1 
                     
                   
                   + 
                   
                     cos 
                     ⁢ 
                     
                       
                         2 
                         ⁢ 
                         π 
                       
                       T 
                     
                     ⁢ 
                     
                       t 
                       2 
                     
                   
                 
                 ) 
               
             
           
         
       
     
   
   For the negative side of the load  24 , from equation (4), 
   
     
       
         
           
             P 
             
               total 
               - 
               
                 i 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 n 
               
               - 
               3 
               - 
             
           
           = 
           
             
               
                 V 
                 cc 
               
               ⁢ 
               
                 V 
                 0 
               
               ⁢ 
               T 
             
             
               10 
               ⁢ 
               
                   
               
               ⁢ 
               π 
               ⁢ 
               
                   
               
               ⁢ 
               R 
             
           
         
       
     
     
       
         
           
             
               
                 
                   
                     
                       
                         Average 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         total 
                       
                     
                   
                   
                     
                       
                         input 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         power 
                       
                     
                   
                 
                 , 
               
             
             
               
                 
                   
                     
                       
                         P 
                         
                           total 
                           - 
                           
                             i 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             n 
                           
                           - 
                           3 
                         
                       
                       = 
                         
                       ⁢ 
                       
                         
                           
                             P 
                             
                               total 
                               - 
                               
                                 i 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 n 
                               
                               - 
                               3 
                               + 
                             
                           
                           + 
                           
                             P 
                             
                               total 
                               - 
                               
                                 i 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 n 
                               
                               - 
                               3 
                               - 
                             
                           
                         
                         
                           T 
                           4 
                         
                       
                     
                   
                 
                 
                   
                     
                       = 
                         
                       ⁢ 
                       
                         
                           
                             
                               
                                 4 
                                 T 
                               
                               ⁢ 
                               
                                 { 
                                 
                                   
                                     
                                       
                                         
                                           V 
                                           0 
                                         
                                         ⁢ 
                                         
                                           V 
                                           cc 
                                         
                                         ⁢ 
                                         T 
                                       
                                       
                                         5 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         π 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         R 
                                       
                                     
                                     ⁢ 
                                     
                                       ( 
                                       
                                         
                                           cos 
                                           ⁢ 
                                           
                                             
                                               2 
                                               ⁢ 
                                               π 
                                             
                                             T 
                                           
                                           ⁢ 
                                           
                                             t 
                                             1 
                                           
                                         
                                         + 
                                         
                                           cos 
                                           ⁢ 
                                           
                                             
                                               2 
                                               ⁢ 
                                               π 
                                             
                                             T 
                                           
                                           ⁢ 
                                           
                                             t 
                                             2 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   + 
                                 
                               
                             
                           
                         
                         
                           
                             
                                 
                               ⁢ 
                               
                                 
                                   
                                     V 
                                     cc 
                                   
                                   ⁢ 
                                   
                                     V 
                                     0 
                                   
                                   ⁢ 
                                   T 
                                 
                                 
                                   10 
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   R 
                                 
                               
                               } 
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       = 
                         
                       ⁢ 
                       
                         
                           
                             2 
                             ⁢ 
                             
                               V 
                               0 
                             
                             ⁢ 
                             
                               V 
                               cc 
                             
                           
                           
                             5 
                             ⁢ 
                             π 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             R 
                           
                         
                         ⁢ 
                         
                           ( 
                           
                             1 
                             + 
                             
                               2 
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                 
                                   2 
                                   ⁢ 
                                   π 
                                 
                                 T 
                               
                               ⁢ 
                               
                                 t 
                                 1 
                               
                             
                             + 
                             
                               2 
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                 
                                   2 
                                   ⁢ 
                                   π 
                                 
                                 T 
                               
                               ⁢ 
                               
                                 t 
                                 2 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
             
           
           
             
               
                 Efficiency 
                 , 
               
             
             
               
                 
                   
                     
                       
                         ɛ 
                         3 
                       
                       = 
                         
                       ⁢ 
                       
                         
                           
                             V 
                             0 
                             2 
                           
                           
                             2 
                             ⁢ 
                             R 
                           
                         
                         × 
                         
                           
                             5 
                             ⁢ 
                             π 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             R 
                           
                           
                             2 
                             ⁢ 
                             
                               V 
                               cc 
                             
                             ⁢ 
                             
                               
                                 V 
                                 0 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   
                                     2 
                                     ⁢ 
                                     cos 
                                     ⁢ 
                                     
                                       
                                         2 
                                         ⁢ 
                                         π 
                                       
                                       T 
                                     
                                     ⁢ 
                                     
                                       t 
                                       1 
                                     
                                   
                                   + 
                                   
                                     2 
                                     ⁢ 
                                     cos 
                                     ⁢ 
                                     
                                       
                                         2 
                                         ⁢ 
                                         π 
                                       
                                       T 
                                     
                                     ⁢ 
                                     
                                       t 
                                       2 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       = 
                         
                       ⁢ 
                       
                         
                           5 
                           ⁢ 
                           π 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             V 
                             0 
                           
                         
                         
                           4 
                           ⁢ 
                           
                             
                               V 
                               cc 
                             
                             ⁡ 
                             
                               ( 
                               
                                 1 
                                 + 
                                 
                                   2 
                                   ⁢ 
                                   cos 
                                   ⁢ 
                                   
                                     
                                       2 
                                       ⁢ 
                                       π 
                                     
                                     T 
                                   
                                   ⁢ 
                                   
                                     t 
                                     1 
                                   
                                 
                                 + 
                                 
                                   2 
                                   ⁢ 
                                   cos 
                                   ⁢ 
                                   
                                     
                                       2 
                                       ⁢ 
                                       π 
                                     
                                     T 
                                   
                                   ⁢ 
                                   
                                     t 
                                     2 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
   
   Comparing the above with a conventional Class B Amplifier, 
   
     
       
         
           
             
               
                   
               
             
             
               
                 
                   
                     ɛ 
                     3 
                   
                   
                     ɛ 
                     B 
                   
                 
                 = 
                 
                   
                     
                       5 
                       ⁢ 
                       π 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         V 
                         0 
                       
                     
                     
                       4 
                       ⁢ 
                       
                         
                           V 
                           cc 
                         
                         ⁡ 
                         
                           ( 
                           
                             1 
                             + 
                             
                               2 
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                 
                                   2 
                                   ⁢ 
                                   π 
                                 
                                 T 
                               
                               ⁢ 
                               
                                 t 
                                 1 
                               
                             
                             + 
                             
                               2 
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                 
                                   2 
                                   ⁢ 
                                   π 
                                 
                                 T 
                               
                               ⁢ 
                               
                                 t 
                                 2 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   × 
                   
                     
                       4 
                       ⁢ 
                       
                         V 
                         cc 
                       
                     
                     
                       π 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         V 
                         0 
                       
                     
                   
                 
               
             
           
           
             
               
                   
               
             
             
               
                 
                   
                     ɛ 
                     2 
                   
                   
                     ɛ 
                     B 
                   
                 
                 = 
                 
                   5 
                   
                     ( 
                     
                       1 
                       + 
                       
                         2 
                         ⁢ 
                         cos 
                         ⁢ 
                         
                           
                             2 
                             ⁢ 
                             π 
                           
                           T 
                         
                         ⁢ 
                         
                           t 
                           1 
                         
                       
                       + 
                       
                         2 
                         ⁢ 
                         cos 
                         ⁢ 
                         
                           
                             2 
                             ⁢ 
                             π 
                           
                           T 
                         
                         ⁢ 
                         
                           t 
                           2 
                         
                       
                     
                     ) 
                   
                 
               
             
           
           
             
               
                 
                   
                     
                       Average 
                     
                   
                   
                     
                       
                         power 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         loss 
                       
                     
                   
                 
                 , 
               
             
             
               
                 
                   
                     
                       
                         P 
                         
                           ave 
                           - 
                           loss 
                           - 
                           3 
                         
                       
                       = 
                         
                       ⁢ 
                       
                         
                           P 
                           
                             ave 
                             - 
                             
                               i 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               n 
                             
                             - 
                             3 
                           
                         
                         - 
                         
                           P 
                           
                             ave 
                             - 
                             out 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       = 
                         
                       ⁢ 
                       
                         
                           
                             
                               2 
                               ⁢ 
                               
                                 V 
                                 cc 
                               
                               ⁢ 
                               
                                 V 
                                 0 
                               
                             
                             
                               5 
                               ⁢ 
                               π 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               R 
                             
                           
                           ⁢ 
                           
                             ( 
                             
                               1 
                               + 
                               
                                 2 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                   
                                     2 
                                     ⁢ 
                                     π 
                                   
                                   T 
                                 
                                 ⁢ 
                                 
                                   t 
                                   1 
                                 
                               
                               + 
                               
                                 2 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                   
                                     2 
                                     ⁢ 
                                     π 
                                   
                                   T 
                                 
                                 ⁢ 
                                 
                                   t 
                                   2 
                                 
                               
                             
                             ) 
                           
                         
                         - 
                       
                     
                   
                 
                 
                   
                     
                         
                       ⁢ 
                       
                         
                           V 
                           0 
                           2 
                         
                         
                           2 
                           ⁢ 
                           R 
                         
                       
                     
                   
                 
               
             
           
         
       
     
   
   As mentioned above, it is proposed that amplifiers embodying the present invention are to be named the Class μ Amplifier because of the resultant shape of the curve when the graph of Efficiency against Output Power is plotted (as shown in  FIG. 6 ). From the graph of  FIG. 6 , it will be seen that for a Class μ amplifier implementing 3 stages of operation, during a proposed typical operation at ⅛ to ⅓ of the maximum output power, an efficiency of about 62% to 75% may be achieved. This is typically 1.5 times more efficient than conventional Class B amplifiers. 
   The difference in the actual power loss between the Class μ amplifier and the conventional Class B amplifier is evident from the graph of Power Loss against Output Power shown in  FIG. 7 . During an operation at ⅛ to ⅓ of maximum output power, the power loss of the Class μ amplifier may be seen to be less than ⅓ of that of a conventional Class B Amplifier. For the whole range of output power, the maximum power loss of the Class μ Amplifier may be seen to be less than half of that of a conventional Class B Amplifier. 
   If more stages of operation are used for the Class μ amplifier, higher efficiencies may be achieved. 
   In summary, one or more preferred embodiments of the invention may provide an amplifier circuit that achieves considerably greater efficiency than a conventional Class B amplifier. Another advantage of one or more preferred embodiments of the invention is that errors generated in the switching circuit may be substantially eliminated as the differential amplifier will apply the errors to the opposite side of the load  24 , thereby canceling them out.