Abstract:
A bipolar analog multiplier with a greatly reduced output sensitivity to temperature. The multiplier uses the difference between the multiplier input voltages and the reference voltages to generate currents. Voltages which are logarithmically dependent on the generated currents are developed and applied to inputs of bipolar variable transconductance stages. Circuits are used to reduce ringing at the output of the multiplier.

Description:
FIELD OF THE INVENTION 
     The present invention relates to analog multipliers and, more specifically, to pseudo-four-quadrant analog multipliers requiring a reduced thermal sensitivity such as required in the multiplication stage of a preamplifier of a cathode-ray tube. 
     DESCRIPTION OF THE RELATED ART 
     In analog-signal processing the need often arises for a circuit that takes two analog input signals and produces an output signal proportional in magnitude to their product. Such a circuit is called an analog multiplier. The term &#34;four-quadrant&#34; multiplier is well known in the art, and refers to a circuit capable of multiplying two signed analog signals. Four-quadrant analog multipliers are fundamental building blocks for many circuit applications, e.g. phase detectors in phase-locked loops and frequency translators. four-quadrant analog multipliers are specially useful in applications such as audio and video signal processing and adaptive filters. 
     A number of diverse circuit techniques have been developed to generate an output signal that is proportional in magnitude to the product of two input signals. One technique which is also readily suited to monolithic circuits depends upon the variations in transconductance in differential stages to perform the four-quadrant multiplication. When constructed from bipolar transistors, the technique makes use of the dependence of the transistor transconductance on the emitter current bias. 
     One analog multiplier is the so-called &#34;Gilbert Cell&#34;, described in B. Gilbert, &#34;A precise Four-Quadrant Multiplier with Subnanosecond Response&#34;, IEEE J. Solid-State Circuits, Vol. SC-3, 373-380 (December 1968). The Gilbert Cell is constructed using bipolar transistors and relies on variations in transconductance of three differential stages to perform the multiplication. The Gilbert Cell however, has a very limited input dynamic range. 
     FIG. 1 illustrates a transistor schematic representation of an analog multiplier 100 known in the prior art. The circuit employs variable-transconductance technique to generate an output voltage V M  which is the product of the three input voltages, namely V x , V y  and Video. Output voltage V M  is applied to the input terminal of output buffer 101, which has a gain of &#34;-10&#34; and which generates output voltage V out  at its output terminal. 
     The first disadvantage of analog multiplier 100 of FIG. 1 is that it is highly sensitive to temperature variation. From FIG. 1 it can be seen by inspection that 
     
         I.sub.1 -I.sub.2 =2*(V.sub.x -2)/R.sub.x2                  (1) 
    
     
         I.sub.3 -I.sub.4 =2*(V.sub.y -2)/R.sub.y2                  (2) 
    
     
         V.sub.1 =(I.sub.1 -I.sub.2)*R.sub.x1 =2*(R.sub.x1 /R.sub.x2)*(V.sub.x -2)(3) 
    
     
         V.sub.2 =(I.sub.3 -I.sub.4)*R.sub.y1 =2*(R.sub.y1 /R.sub.y2)*(V.sub.y -2)(4) 
    
     The collector currents I qc1 , I qc2 , I qd1  and I qd2  are related to voltages V 1  and V 2  according to the following equations: 
     
         V.sub.1 =V.sub.T *ln(I.sub.qd1 /I.sub.qd2)                 (5) 
    
     
         V.sub.2 =V.sub.T *ln(I.sub.qc1 /I.sub.qc2)                 (6) 
    
     V T  is the thermal voltage and is equal to kT/q which is approximately equal to 26 mv at 300° K, where 
     k =Boltzmann&#39;s constant 
     T =Temperature (in °K) 
     q =electric charge of an electron 
     The multiplier output voltage V M  is directly proportional to the terms ln(I qd1  /I qd2 ) and ln(I qc1  /I qc2 ). Consequently, variations in these two ratios directly affect the value of the multiplier output voltage. To keep these ratios constant over temperature, voltages V 1  and V 2  must follow the temperature variations of V T . Since the resistance of resistors R x1  and R x2  have a similar temperature dependence, the ratio R x1  /R x2  and consequently, output voltage V 1  have a minimal temperature sensitivity as can be seen from equation (3). Similarly, voltage V 2  has a negligible temperature dependence. Therefore, changes in temperature directly affect multiplier output voltage V M  through the thermal voltage term V T . 
     FIG. 2 illustrates a simulation result of the variation in output voltage V M  of multiplier 100 of FIG. 1 as the input voltages V x  and V y  are varied. For this simulation, input voltages V x  and V y  are set equal to one another and are swept from 0 volt to 4 volts as shown along the x-axis, and input voltage Video is kept constant at 0.7 volts. The y-axis shows the difference in the output voltage V M  as the input voltages V x  and V y  are varied. For proper operation, it is required that output voltage V M  of multiplier 100 rise with increasing temperature when input voltages V x  and V y  are above 2 volts. Similarly, it is required that output voltage V M  of multiplier 100 fall with decreasing temperature when input voltages V x  and V y  are below 2 volts. 
     FIG. 3 shows the change in output voltage V out  of FIG. 1 when temperature changes from 0° C. to 85° C., for the condition when input voltages V x  and V y  are both equal to 3 volts and input voltage Video is at 0.7 volts. From FIG. 3 it can be seen that output voltage V out  increases by 620 mv as temperature changes from 0° C. to 85° C. rendering this multiplier ineffective for many applications. 
     The second disadvantage of multiplier 100 of FIG. 2 is that it has a relatively small input dynamic range above which the multiplier would not behave in a linear fashion. 
     FIG. 4 illustrates another analog multiplier circuit 200 known in the prior art. Output voltage V M  of multiplier 200 is applied to the input terminal of output buffer 201 which has a gain of &#34;-10&#34; and which generates output voltage V out  at its output terminal. In analog multiplier circuit 200, diode-connected transistor Q a  is placed between transistor Q aa  and the supply voltage V cm1 , and diode-connected transistor Q b  is placed between transistor Q bb  and the supply voltage V CM1 . By inspection, it can be seen that 
     
         V.sub.1 =V.sub.T *ln(I.sub.a /I.sub.b)                     (7) 
    
     
         I.sub.a -I.sub.b =2*(V.sub.x -2)/R.sub.x2                  (8) 
    
     Resistor R x2  has a positive temperature coefficient. Therefore, as temperature increases the resistance of the resistor R x2  increases, thus causing a reduction in the current term (I a  -I b ) and in the ln(I a  /I b ) term of equation (8) above. The reduction in the term ln(I a  /I b ) decreases voltage V 1  &#39;s dependence on voltage V T , which is undesirable. 
     FIG. 5 shows an increase of 365 mv in the output voltage V out  of FIG. 4 when input voltages V x  and V y  are each set to 3 volts, input voltage Video is at 0.7 volts, and temperature is changed from 0° C. to 85° C. Although circuit 200 of FIG. 4 provides an improvement over circuit 100 of FIG. 1, the multiplier output voltage shift for the given temperature range is too great, thereby rendering use of this multiplier inadequate for many applications. 
     The second disadvantage of the multiplier of FIG. 4 is that it suffers from ringing problems at its output terminal. The emitter terminals of transistors Q a  and Q b  each have a high impedance when the input voltage V x  or V y  is either at 0 or 4 volts, making the output signal of the multiplier susceptible to ringing effect. 
     SUMMARY 
     An analog multiplier for multiplying three voltage signals utilizes circuitry for keeping the multiplier output voltage reasonably constant over temperature. Two semi-logarithmic voltage generating stages are used to provide input voltages to two variable transconductance circuits forming the last stage of the multiplier. Two differential stages receive level-shifted multiplier input voltages and convert them to currents. The multiplier includes devices for eliminating ringing at the output of the multiplier. 
     In accordance with the present invention the analog multiplier has a reduced temperature dependence. The multiplier has a wide dynamic range and is immune to ringing effect. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 illustrates an analog multiplier as known in the prior art. 
     FIG. 2 illustrates the required temperature characteristic of the output voltage V M  of the multiplier of FIG. 1 when input voltages V x  and V y  are set equal to one another and are varied from 0 volts to 4 volts and input voltage Video is kept constant at 0.7 volts. 
     FIG. 3 illustrates the voltage V out  at the output terminal of the output buffer of FIG. 1 when a voltage pulse of 0.7 volts is applied to the Video input terminal of the multiplier at two different temperatures, namely 0° C. and 85° C. Input voltages V x  and V y  are set to 3 volts in both cases. 
     FIG. 4 illustrates an analog multiplier as known in the prior art. 
     FIG. 5 illustrates the voltage V out  at the output terminal of the output buffer of FIG. 4 when a voltage pulse of 0.7 volts is applied to the Video input terminal of the multiplier at two different temperatures, namely 0° C. and 85° C. Input voltages V x  and V y  are set to 3 volts in both cases. 
     FIG. 6 illustrates an analog multiplier in accordance with the present invention. 
     FIG. 7 illustrates the temperature drift of the voltage V out  at the output terminal of the output buffer which receives at its input terminal the output voltage of the multiplier in accordance with the present invention. 
     FIG. 8 illustrates stage 100 of the multiplier in accordance with the present invention. 
     FIG. 9 illustrates stage 300 of the multiplier in accordance with the present invention. 
     FIG. 10 illustrates the effect of ringing at the output terminal of the output buffer of FIG. 6 when no impedance lowering devices are used. 
     FIG. 11 illustrates the diminished ringing at the output terminal of the output buffer of FIG. 6 when impedance lowering devices are used in the multiplier in accordance with the present invention. 
    
    
     DETAILED DESCRIPTION 
     An analog multiplier 600 which provides a thermally compensated output voltage in accordance with the present invention is illustrated in FIG. 6. 
     As shown in FIG. 6, multiplication of the three voltage inputs V x , V y  and Video is performed in three stages 100, 200 and 300. Stage 100 receives input voltage V x  and generates output voltages V 1   +  and V 1   -  which are applied to stage 300. Similarly, stage 200 receives input voltage V y  and generates output voltages V 2   +  and V 2   -  which are applied to stage 300. Stage 300 receives output voltages V 1   +   and V 1   -   of stage 100, and output voltages V 2   + , V 2   -   of stage 200 as well as input voltage Video and generates multiplier output voltage V M  . The supply voltages V ref1  and V ref2  of stages 100 and 200 are 7.0 volts and 8.0 volts respectively. Output buffer 400, which has a gain of &#34;-10&#34;, receives multiplier output voltage V M  at its input terminal and generates output voltage V out  at its output terminal. 
     Except for the differences noted above, stages 100 and 200 are identical to one another in construction and in function, therefore the description of the operation of stage 100 equally applies to that of stage 200 and as such only the operation of stage 100 is discussed. 
     An implementation of stage 100 is shown in FIG. 8. A Contrast-Control circuitry, not shown in the drawings (known in the Art), generates the first multiplier input voltage V x  which is applied to the base terminal of transistor 41a of stage 100. A constant 2 volts supply applied to the base terminal of transistor 41b provides the second input voltage to stage 100. Stage 100 includes four fully balanced sections 130, 160, 180 and 190. To enable a pseudo-four-quadrant multiplication, stage 100 includes a reference circuit 801 receiving a constant 2 volts supply at the base terminal of transistor 41b . This reference circuit is matched by a variable input circuit 802 for receiving input voltage V x  at the base terminal of transistor 41a. Variable input circuit 802 includes partitions L43, L44, L49, L47 and L50 and reference circuit 801 includes partitions L45, L46, L48, L51 and L52. Due to the substantially identical structure of reference circuit 801 and variable circuit 802, for values of input voltage V x  greater than 2 volts, output voltage V 1  across nodes V 1   +   and V 1   -   is positive and for values of input voltage V x  less than 2 volts, output voltage V 1  is negative. Thus, when input voltage V x  is exactly equal to 2 volts, the output voltage V 1  is zero. 
     Section 130 of stage 100 includes four DC voltage level-shifter partitions, namely L43, L44, L45 and L46. Each one of these partitions includes a current source and a bipolar transistor. For example, partition L43 includes current source 43a and transistor 41a. The current source in each partition is used to properly bias the bipolar transistor connected to that partition. Thus, the DC voltage level-shifters in partitions L43 and L44 raise the voltage at the base terminal of transistor 53a above that of signal V x  by two base-emitter (V be ) voltages (e.g. between 1.0 to 1.2 volts). Similarly the voltage at the base terminal of transistor 53b is two V be  voltages higher than 2 volts. Voltage level-shifting is needed to prevent transistor 53a from turning off when multiplication by zero is desired. 
     Section 140 of stage 100 generates a voltage between nodes N44 and N45 at the emitter terminals of transistors 54a and 54b that is semi-logarithmically dependent on the ratio of the currents I a  and I b  which flow through transistors 54a and 54b. Section 140 includes partitions L47 and L48. Partition L47 contains diode-connected transistor 54a and resistor 55a. Partition L48 includes diode-connected transistor 54b and resistor 55b. One terminal of resistor 55a is connected to the supply voltage V ref1 , the other terminal of resistor 55a is connected to the collector terminal of transistor 54a. The base and the collector terminals of transistor 54a are connected together. The emitter terminal of transistor 54a is connected to node N44. Similarly, in partition L48, the terminals of resistor 55bare connected to the supply voltage V ref1  and the collector terminal of transistor 54b. The base and the collector terminals of transistor 54b are connected together. The emitter terminal of transistor 54b is connected to node N45. Currents I a  and I b  flow through partitions L47 and L48 respectively. 
     Section 160 of stage 100 is a differential voltage to current converter. Section 160 converts the level-shifted voltages at the base terminals of transistors 53a and 53b to currents I a  and I b , respectively flowing in partitions L47 and L48 of section 140 of stage 100 and through transistors 53a and 53b of section 160 of stage 100. The base, the emitter and the collector terminals of transistor 53a are connected to nodes N43, N44 and N48 respectively. The base, the emitter and the collector terminals of transistor 53b are connected to nodes N46, N49 and N45 respectively. The terminals of resistor 60 are connected to nodes N48 and N49. The terminals of current source 52a are connected to nodes N48 and ground. The terminals of current source 52b are connected to nodes N49 and ground. 
     Section 180 of stage 100 which includes transistors 51a and 51b, reduces the impedance of nodes N44 and N45 in order to inhibit ringing at the multiplier output, which may occur at frequencies near 100 MHz and above, when either input voltage V x  or input voltage V y  is either at zero or four volts. The base and the emitter terminals of both transistors 51a and 51b are connected to ground. The collector terminals of transistors 51a and 51b are connected to nodes N44 and N45 respectively. The reduction in impedance of nodes N44 and N45 is achieved by the collector-base capacitance and the collector-substrate capacitance of transistors 51a and 51b respectively. FIGS. 10 and 11 illustrate the output voltage V out  of output buffer 400 without and with the impedance lowering devices 51a and 51b respectively. As shown in FIG. 11, the output voltage V out  has a lower ringing when section 180 is included in analog multiplier 600. 
     Section 190 of stage 100 includes two emitter-follower amplifiers whose output terminals are connected to the input terminals of the variable-transconductance section 320 of stage 300. Section 190 includes transistors 56a and 56b and current sources 57a and 57b. The collector terminals of transistors 56a and 56b are both connected to V cc  voltage supply. The emitter and the base terminals of transistor 56a are connected to nodes N90 +   and N44 respectively. The emitter and the base terminals of transistor 56b are connected to nodes N90 -   and N45 respectively. Current sources 57a and 57b are connected between nodes N90 +   and ground and nodes N90 -   and ground respectively. The near-unity gain of the emitter-follower amplifiers allows the semi-logarithmic voltage across nodes N44 and N45 to also appear across nodes N90 +   and N90 - . The emitter-follower amplifier stages serve as drive-boosters giving the emitter terminals of transistors 56a and 56b the needed capability to drive the differential input terminals V 1   +   and V 1   -   of the variable-transconductance section 320 of stage 300. 
     FIG. 9 shows stage 300 which provides the final phase of the multiplication and which includes sections 320, 330 and 340. Section 320 is a variable-transconductance stage formed by resistors 1 and 2 and an emitter-coupled differential pair consisting of transistors 11a and 11b. The semi-logarithmic voltage across emitter terminals of transistor 56a and 56b of section 180 of stage 100 is applied to the base terminals of transistors 11a and 11b. The emitter terminals of transistors 11a and 11b are connected to node N20. The collector terminal of transistor 11b provides the multiplier output voltage V M . The collector of transistor 11a is connected to node N22. The terminals of resistor 1 are connected across nodes V M  and N22 and the terminals of resistor 2 are connected across nodes V cc  and N22. 
     Section 330 is also a variable-transconductance stage formed by a differential pair consisting of transistors 12a and 12b. The semi-logarithmic voltage across terminal N190 +   and N190 -   of stage 200 (shown in FIG. 6) is applied to the base terminals of transistors 12a and 12b. The emitter terminals of transistors 12a and 12b are connected to node N30. The collector terminal of transistor 12a is connected to V cc  and the collector terminal of transistor 12b is connected to node N20 of section 320. 
     Section 340 is the variable current-sum stage and includes transistor 15, resistors 3, 4 and capacitor 14. The collector, the base and the emitter terminals of transistor 15 are connected to nodes N30, N41 and N40 respectively. The terminals of resistor 3 are connected across nodes N40 and ground and the terminals of resistor 4 are connected across nodes N41 and the input voltage terminal V ref  which is held constant at 2.2 volts. The terminals of capacitor 14 are connected across nodes N41 and input terminal Video which provides the third input voltage terminal to the multiplier 600. Section 340 sets the total current that flows through transconductance stages 320 and 330. A voltage pulse at input terminal Video, is capacitively coupled through capacitor 14 to the base terminal of transistor 15 causing an increase in the base-emitter voltage of transistor 15 and a proportional increase in the total current flow in stage 300, which in turn increases the multiplier output voltage V M . Resistor 4 is used to increase the impedance seen by node V ref . 
     As mentioned before, the output voltage V M  of multiplier 600 of the present invention is dependent on the ratio of the currents I q1  /I q2  and I q3  /I q4  flowing through the differential pairs of sections 320 and 330 of stage 300. These ratios are related to voltages V 1  and V 2  according to the following equations: 
     
         V.sub.1 =V.sub.T *ln(I.sub.q1 /I.sub.q2) 
    
     
         V.sub.2 =V.sub.T *ln(I.sub.q3 /I.sub.q4) 
    
     where 
     
         V.sub.1 =V.sub.1.sup.+ -V.sub.1.sup.- 
    
     and 
     
         V.sub.2 =V.sub.2.sup.+ -V.sub.2.sup.- 
    
     To keep the ln(I q1  /I q2 ) term and the ln(I q3  /I q4 ) term constant over a wide range of temperature, voltages V 1  and V 2  each have a temperature dependence which is similar to that of thermal voltage V T . Let the resistance of each of resistors 55a and 55b of stage 100 be R x1  ohms, and the resistance of each of resistors 155a and 155bof stage 200 be R yl  ohms, voltages V 1  and V 2  are related to the applied input voltages V x  and V x  according to the following equations 
     
         V.sub.1 =V.sub.T *ln(I.sub.a /I.sub.b)+2*(R.sub.x1 /R.sub.x2)*(V.sub.x -2)(9) 
    
     
         I.sub.a -I.sub.b =2*(V.sub.x -2)/R.sub.x2 (10) 
    
     
         V.sub.2 =V.sub.T *ln(I.sub.c /I.sub.d)+2*(R.sub.y1 /R.sub.y2)*(V.sub.y -2)(11) 
    
     
         I.sub.c -I.sub.d =2*(V.sub.y -2)/R.sub.y2                  (12) 
    
     Based on equations (9), (10), (11) and (12) it can be shown that 
     
         V.sub.1 =V.sub.T *ln(I.sub.a /I.sub.b)+2*(I.sub.a -I.sub.b)*R.sub.x1(13) 
    
     
         V.sub.2 =V.sub.T *ln(I.sub.c /I.sub.d)+2*(I.sub.c -I.sub.d)*R.sub.y1(14) 
    
     Equations (13) and (14) indicate the manner in which multiplier 600 of the present invention achieves an output voltage that remains relatively stable with varying temperature. According to equation (13), voltage V 1  is dependent on two terms (I a  -I b ) and ln(I a  /I b ), as temperature increases, the terms (I a  -I b ) and ln(I a  /I b ) decreases. The reduction in the ln(I a  /I b ) term compensates for the increase in voltage V T . However, as temperature increases, R x1  resistance also increases, more than offsetting the reduction in temperature dependence of voltage V 1  on voltage V T  (due to a reduction in the ln(I a  /I b ) term) , thus giving rise to a voltage V 1  which tracks temperature changes in voltage V T  more closely. Similarly, voltage V 2  has a temperature dependence that also closely tracks the temperature dependence of voltage V T . 
     FIG. 7 shows an increase of 130 mv in the output voltage V out  of FIG. 6 when input voltages V x  and V y  are each set to 3 volts, input voltage Video is at 0.7 volts and temperature is changed from 0° C. to 85° C. This increase in voltage is substantially smaller than the corresponding increase in the output voltage of the multipliers of the prior arts over the same temperature change. 
     One embodiment of the present invention uses square-emitters to match transistors. All resistors in that embodiment namely, resistors 55a, 55b, 155a, 155b, 1, 2, 3, 4 are made from p-base implant and have values of 50 ohms, 4 Kohms, 50 ohms, 4 Kohms, 1.5 Kohms, 500 ohms, 2 Kohms and 20 Kohms respectively.