Abstract:
A method for controlling a switching voltage regulator that includes generating a feedback voltage that is proportional to the output voltage of the voltage regulator; generating a voltage proportional to the duty-cycle of the inductor charging and discharging phases as a function of the difference between the feedback voltage and a reference voltage; and adding a dominate pole and two zeros to the function used to generate the voltage proportional to the duty-cycle of the inductor charging and discharging phases.

Description:
BACKGROUND OF THE INVENTION 
     A voltage regulator is a circuit that provides a precise output voltage under varying load conditions from an unknown and possibly varying input voltage. Many different types of voltage regulators have been developed, each with its own set of advantages. This particular application is directed at a particular class of voltage regulator known as inductor-based switching voltage regulators. The two most common types of inductor-based switching regulators are Boost (output voltage greater than input voltage) and Buck (output voltage less than input voltage) switching regulators. Both Boost and Buck switching regulators are very important for battery powered applications such as cellphones. 
     As shown in  FIG. 1A , a traditional implementation for a Buck switching regulator includes a switch  102  connected between an input voltage (VP in this case) and a node  116 . A switch  104  is connected between the node  116  and the ground voltage (VN). An inductor  106  is connected between the node  116  and the output node (V OUT ) of the regulator. A filtering capacitor connects V OUT  to the ground voltage VN. The node V OUT  is also connected to a load represented by the resistor  110 . 
     A control circuit (described below) turns switches  102  and  104  ON and OFF in a repeating pattern. Switch  102  is driven out of phase with switch  104 . Thus, when switch  102  is ON switch  104  is OFF. This causes the Buck switching regulator to have two distinct operational phases. In the first phase, shown in  FIG. 1B , the switch  102  is ON. During this phase, called the charging phase the inductor  106  is connected between the battery and the output node V OUT . This causes current to flow from the battery to the load. In the process energy is stored in the inductor  106  in the form of a magnetic field. In the second, or discharge phase the switch  102  is opened (see  FIG. 1C ). In this phase, the inductor  106  is connected in series between ground and the load. Current supplied by the inductor&#39;s collapsing magnetic field flows to the output node V OUT  and the load. 
     As shown in  FIG. 1D , a typical Boost converter includes all of the components just described. A slightly different topology is used in which the switch  102  is placed between the inductor  106  and the output node. The Boost converter uses a similar two phase pattern of switching for its two switches. 
     SEPIC converters are another type of inductor-based switching regulators. SEPIC converters are more fully described in a copending U.S. patent application Ser. No. 11/933,402 entitled “High Voltage SEPIC Converter,” now U.S. Pat. No. 8,350,546. That disclosure is incorporated in this document by reference. 
     To maintain its output at a constant voltage, switching regulators include control circuits that modulate the duty factor of their high and low-side switches  102  and  104 , respectively. As shown in  FIG. 1A , the control circuit typically includes a resistive divider formed by resistors  112  and  114  as well as an error amplifier  118 , comparator  120  and break-before-make (BBM) circuit  112 . The resistive divider generates a feedback voltage FB proportional to the output of the regulator. The feedback voltage FB is one of the inputs to the error amplifier  118 . The second error amplifier  118  input is a reference voltage BG that is generated using any convenient technique as is well known in the relevant art. The error amplifier  118  compares the feedback voltage FB to the reference voltage BG and multiplies the difference by a gain factor to generate an output voltage EAOUT. 
     The error amplifier  118  output EAOUT is one of the inputs to the comparator  120 . The second input to the comparator  120  is a periodic ramp voltage RAMP. The output of the comparator  120  (i.e., the comparison between the ramp voltage RAMP and the output of the error amplifier EAOUT) is a periodic square wave signal CLKV. The square wave signal CLKV is passed to the BBM circuit  122 . The BBM circuit  122  generates a signal based on CLKV to drive the high-side switch  102  and a complementary signal to drive the low-side switch  104 . In general, it takes a finite amount of time to turn the high and low-side switches  102  and  104 , respectively, ON and OFF. For this reason, the act of turning a switch OFF is always done slightly in advance of the act of turning the other switch ON. This technique, known as break-before-make avoids the situation where both switches are ON at the same time and power is connected through the high and low-side switches to ground (a condition known as shoot through). 
       FIG. 1E  shows the ramp voltage RAMP along with the error amplifier output  118  EAOUT. The corresponding comparator  120  output CLKV is also shown. As may be appreciated, the duty cycle of CLKV is defined by the intersection of RAMP and EAOUT.  FIG. 1E  also shows a higher error amplifier  118  output (labeled EAOUT′) and the effect that it has on the duty cycle of the periodic square wave signal CLKV. This is the basic feedback mechanism for the Buck regulator of  FIG. 1A : decreases in the output voltage cause the feedback voltage FB to fall. This causes the error amplifier  118  output EAOUT to increase. The increase in EAOUT causes CLKV to have an increased duty cycle. This increases the duty cycle of the high-side switch  102  and decreases the duty cycle of the low-side switch  104 . Thus, if the output voltage increases or decreases, the duty cycle of the high and low-side switches are adjusted in a way that compensates for the increased or decreased output. 
     The control loop just described is an example of what is generally referred to as voltage mode control (i.e., regulator output is regulated as a function of output voltage). In this control loop, the gain of the error amplifier determines the accuracy of regulation. A high gain amplifier keeps the deviations of the output voltage relatively small and close to ideal. A lower gain amplifier allows larger deviations to occur. 
     The control loop must maintain stability, that is to say, must not oscillate which would cause the output voltage to oscillate. Feedback theory provides criteria for this stability. If the gain of the control loop is plotted as a function of frequency, an element of the control loop must reduce the gain below one at some frequency. This frequency is called the gain-bandwidth (GBW) product or unity gain frequency. 
     A large GBW product control loop indicates that the control loop is fast and can respond to fast transients. For example, in modern microprocessors, the processor can turn on rapidly so that the supply current takes a large fast step in times approaching the switching speed of the microprocessor. A large GBW product allows the voltage regulator to respond quickly to such changes. (If the circuit does not have a large GBW product, then large output capacitors are needed to sustain the output voltage until the loop responds). 
     Control theory says that the phase shift around the control loop must not be greater than 180 degrees at the unity gain frequency. In fact, the circuit is not really useable if the phase shift of the control loop is near 180 degrees. It is preferable to be near 90 degrees, but in many cases 140 to 130 degrees of phase shift is acceptable. 
     In a voltage mode converter, the inductor—capacitor pair introduce a 180 degree phase shift by themselves at their resonant frequency: ½π*(L*C) 1/2 . As a result, any control loop must take this into account by removing about 90 degrees of phase shift starting at the resonant frequency. 
     In the parlance of control loop theory, the removal of 90 degrees of phase shift is accomplished by adding a “zero” to the control loop. If 90 degrees of phase shift is added, a “pole” is added to the control loop. The LC filter of the buck converter adds a “double pole” at the resonant frequency, to get the 180 degree phase shift. 
     If nothing were done except adding a wide band amplifier for control, the voltage mode converter would be unstable because of the double pole adding 180 degrees of phase shift at the unity gain frequency. For good compensation a zero must be added at the resonant frequency of the output filter to add back 90 degrees of phase shift. 
     In the prior art, voltage mode compensation has been generally accomplished three ways as shown in  FIG. 2 . The first way is placing capacitor  202  in the feedback loop. This adds a zero and a pole which are generally too close together in frequency for most cases we want to consider. This makes this technique helpful but not very useful. 
     Another prior art is using the parasitic resistance  204  of the filter capacitor  108  as the zero forming element. For a 20 uf filter capacitor  108 , and a 30 kHz zero, this yields a parasitic resistance  204  value of 0.26 ohms which is large (for most cases) and may produce large ripple. To get to a reasonable ESR, large values of capacitance must be used, but still the ripple is a problem. Generally tantalum or other electrolytic capacitors are needed for this type of compensation. Ceramic capacitors, in general, have too low an ESR to be effective. Tantalum capacitors are generally more expensive than ceramic. 
     In  FIG. 2  a box is shown connecting the error amplifier  118  output to the feedback node. This feedback network might be used to create a three pole, two zero circuit which can be effective to stabilize the voltage mode circuit. The resultant gain transfer curve is shown in  FIG. 3 . A dominant pole is introduced at about 30 Hz. At about 20 k Hz a double zero is introduced, just below the resonance of the LC circuit. At about the switching frequency of the regulator, about 1 MHz, another double pole is introduced which rolls the gain off to the unity gain point above 10 MHz. 
     The error amplifier  118  output and FB nodes are brought to an external compensation network  206  where a dominant pole and two zeros are introduced. In order to make the system stable, a double pole must be introduced at a high frequency to roll the gain off to make the system stable. In this example, the gain bandwidth product is near 50 MHz for the amplifier. It can also be seen that the second zero&#39;s effectiveness is less than a decade. If the GBW product of the amplifier is reduced, the whole curve must be shifted to a lower frequency, which makes the regulator slower and uses larger external components. 
     A third compensation scheme places a low pole in the compensation network  206  such that the unity gain is reached well before the double pole of the output filter. This makes a very slow control loop. 
     SUMMARY OF THE INVENTION 
     This disclosure describes an internal compensation network and associated compensation method for inductor-based switching regulators (see  FIG. 4 ). The compensation network adds a pole and two zeros to compensate high-frequency voltage mode operation. An example of an inductor-based switching regulator that uses the compensation network  402  includes a high-side switch  410  connected between an input supply (VP) and a node  424 . The node  424  is connected to a ground voltage (VN) by a low-side switch  412 . An inductor  414  connects the node  424  to an output node. The output node is further connected to the ground voltage VN by an output or filter capacitor  416 . A load  418  is connected between the output node and the ground voltage VN in parallel with the output capacitor  416 . 
     A control circuit is used to drive the high and low-side switches  410  and  412 , respectively, in a repeating sequence that includes an inductor charging phase and an inductor discharging phase. During the inductor charging phase, the control switch activates the high-side switch  410  to connect the node  424  to the input voltage VP. This causes current to flow from the input supply, through the inductor  414  to the load  418 . During the inductor discharging phase, the control switch activates the low-side switch  412  (and deactivates the high-side switch  410 ). This connects the node  424  to the ground voltage VN. Current continues to flow to the load  418  as the magnetic field of the inductor  414  collapses. The control circuit modulates the duty cycle of the high and low-side switches  410  and  412 , respectively, (i.e., the relative duration of activation of the high and low-side switches) to regulate the voltage at the output node. 
     To perform the required modulation, the control circuit uses a resistive divider to generate a feedback voltage FB that is proportional to the voltage difference between the output node and the ground voltage VN. 
     Referring to  FIG. 8B , the feedback voltage FB is passed, via a resistor  820  to the positive input of an error amplifier  408 . A node  814  located between the resistor  820  and the error amplifier  408  is connected via a filter capacitor  826  to the ground voltage VN. A reference voltage BG is passed, via a resistor  426  to the negative input of the error amplifier  408 . A node  816  located between the resistor  426  and the error amplifier  408  is connected via a filter capacitor  828  to the ground voltage VN. A compensation network is connected between the node  816  and the output of the error amplifier  408 . 
     Within the compensation network, a series connection of a capacitor  802  and a resistor  804  connect the node  816  to an internal node  810 . The node  810  is connected, by a series connection of the resistor  812  and the capacitor  806  to the ground voltage VN. 
     A resistor  808  connects the node  810  to an internal node  818 . The node  818  is connected by a second filter capacitor  830  to the ground voltage VN. The node  818  is connected via a resistor  822  to the output of the error amplifier. The node  818  is also connected by a resistor  824  to the output node EAOUT of the compensation network. 
     Referring back to  FIG. 4 , the output node EAOUT is connected an input of a comparator  404 . The second input to the comparator  404  is a periodic ramp voltage RAMP. The output of the comparator  404  (i.e., the comparison between the ramp voltage and the output of the error amplifier EAOUT) is a periodic square wave signal CLKV. The square wave signal CLKV is passed to the BBM circuit  406 . The BBM circuit generates a signal based on CLKV to drive the high-side switch  410  and a complementary signal to drive the low-side switch  412 . 
     As the switching regulator operates, the error amplifier  408  generates a voltage proportional to the duty cycle of the high and low-side switches  410  and  412 , respectively. The compensation network  402  adds a dominate pole and two zeros to the gain product of the error amplifier  408  to compensate high-frequency voltage mode operation. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1A  is a block diagram of a prior art Buck switching regulator. 
         FIG. 1B  is a block diagram showing the prior art Buck switching regulator of  FIG. 1  during the charge phase of operation. 
         FIG. 1C  is a block diagram showing the prior art Buck switching regulator of  FIG. 1  during the discharge phase of operation. 
         FIG. 1D  is a block diagram of a prior art Boost switching regulator. 
         FIG. 1E  is graph showing the feedback and ramp voltages used to control typical prior art switching regulators. 
         FIG. 2  is a block diagram of a prior art Buck switching regulator with a compensation network. 
         FIG. 3  is a plot showing the gain transfer associated with prior art switching regulators. 
         FIG. 4  is a block diagram of an inductor-based switching regulator that includes an embodiment of the compensation network of the present invention. 
         FIG. 5  is a plot showing the LC filter response associated with the switching regulator of  FIG. 4 . 
         FIG. 6  is a gain plot of the compensation network of  FIG. 4 . 
         FIG. 7  shows the amplifier gain for the switching regulator of  FIG. 4  as well as the associated LC filter gain and the product of the amplifier gain and LC filter gain. 
         FIG. 8A  is a block diagram of a simplified embodiment of the compensation network of the present invention. 
         FIG. 8B  is a block diagram of an embodiment of the compensation network of the present invention. 
         FIG. 8C  is a block diagram of an embodiment of the compensation network of the present invention. 
         FIG. 9  is a Bode plot that has been generated for the local feedback circuit of the switching regulator of  FIG. 4 . 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     This disclosure describes an internal compensation network for use in inductor-based switching regulators as well as a related compensation method and inductor-based switching regulators that use the compensation network and method. The compensation network adds a pole and two zeros to compensate high-frequency voltage mode operation.  FIG. 4  shows an example of an inductor-based switching regulator  400  that uses an implementation of the compensation network  402 . Switching regulator  400  includes a high-side switch  410  connected between an input supply (VP) and a node  424 . The node  424  is connected to a ground voltage (VN) by a low-side switch  412 . An inductor  414  connects the node  424  to an output node which is further connected to the ground voltage VN by an output capacitor  416 . A load  418  is connected between the output node and the ground voltage VN in parallel with the output capacitor  416 . A resistive divider formed by resistors  420  and  422  is used to generate a feedback voltage FB that is proportional to the voltage difference between the output node and the ground voltage VN. 
     The feedback voltage FB is connected to the first input of an error amplifier  408 . A second input of the error amplifier  408  is connected, via a resistor  426  to the reference voltage BG. The output of the error amplifier  408  is labeled EAOUT. A compensation network  402  connects the output EAOUT of the error amplifier  408  to a node between the resistor  426  and the error amplifier  408 . 
     The EAOUT voltage is connected an input of a comparator  404 . The second input to the comparator  404  is a periodic ramp voltage RAMP. The output of the comparator  404  (i.e., the comparison between the ramp voltage and the output of the error amplifier  408 ) is a periodic square wave signal CLKV. The square wave signal CLKV is passed to the BBM circuit  406 . The BBM circuit  406  generates a signal based on CLKV to drive the high-side switch  410  and a complementary signal to drive the low-side switch  412 . 
     Stability is a crucial aspect of the Buck converter of  FIG. 4 . At the heart of the problem is the LC filter response, as shown in  FIG. 5  with its double pole at the resonant frequency. It can be seen that it is a double pole roll-off of 40 db per decade starting at the resonant frequency of the filter. This means that there is a 180 degree phase shift in the response curve. If the amplifier had infinite bandwidth, the LC filter phase shift would mean that the system would be very ringy if not unstable. The peak is the result of the poles being near the imaginary axis. A small amount of series resistance, in the switches or the inductor, or losses in the inductor, will keep the peak within reasonable bounds. 
     To compensate this circuit, a dominant pole is introduced to roll off the gain starting at low frequency. Then a first zero is introduced to cancel the effects of the dominant pole. A second zero must be overlaid on the double pole of the LC filter. This will compensate one of the double poles and allow the system to be stable. The gain plot of an amplifier compensation circuit with such a pole and two zeros is shown in  FIG. 6 . 
     
       
         
               
             
               
               
               
             
           
               
                 TABLE 1 
               
               
                   
               
               
                 LC Filter Parameters 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 L 
                 1 uH resonance 35.6 kHz 
               
               
                   
                 C 
                  20 uF 
               
               
                   
                 R 
                 0.2 ohm 
               
               
                   
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
             
           
               
                 TABLE 2 
               
               
                   
               
               
                 Poles and Zeros of the gain block 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 z1 
                 10 kHz 
               
               
                   
                 z2 
                 30 kHz 
               
               
                   
                 p1 
                 0.05 kHz dc gain 1000 
               
               
                   
                   
               
             
          
         
       
     
       FIG. 7  shows the amplifier gain, the LC filter gain and their product. It can be seen that the system gain has been compensated by the second zero the amplifier. The problem to be solved is how to do this second zero while making the system stable. The line labeled “product” is the resultant. It crosses the zero DB line with 20 db per decade slope showing that the system can be made stable. 
     To add the dominant pole and two zeros, a simplified version of the compensation network  402  is configured as shown in  FIG. 8A . Compensation network  402  includes a capacitor  802 , and resistors  804  and  808  connected in series between the second input of the error amplifier  408  and the error amplifier output. A capacitor  806  is connected between the ground voltage VN and a node  810  where the node  810  is between the resistors  804  and  808 . 
     As shown in more detail in  FIG. 8B , the compensation network is configured to include an error amplifier  408 , three filter capacitors ( 826 ,  828  and  830 ), two capacitors ( 802  and  806 ), seven resistors  426 ,  804 ,  808 ,  812 ,  820 ,  822 ,  824  and four internal nodes  810 ,  814 ,  816 ,  818 . The resistor  820  connects the feedback voltage FB to the internal node  814 . The node  814  is connected, in turn to the error amplifier  408  and via the filter capacitor  826  to the feedback voltage FB to the ground voltage VN. The capacitors  826 ,  828  and  830  are filter capacitors to filter out the switching frequency of  410  and  412 . Likewise,  820 ,  822  and  824  are the resistors which aid the filter capacitors to work. 
     The resistor  426  connects a reference voltage BG to the internal node  816 . The reference voltage BG is generated using any convenient technique as is well known in the relevant art. The node  816  is connected, in turn to the second input of the error amplifier and via the filter capacitor  828  to the ground voltage VN. 
     A series connection of the capacitor  802  and the resistor  804  connect the node  816  to the internal node  810 . The node  810  is connected, by a series connection of the resistor  812  and the capacitor  806  to the ground voltage VN. 
     The dominant pole of the amplifier is set by the miller multiplied capacitance of  802  against the resistor  426  in the reference circuit. Typically this might be set at 50 Hz. As the frequency is increased, the impedance of  802  becomes small compared to the resistors. This sets the minimum gain of the amplifier, the AC gain. The AC gain is set by the resistors  426 ,  804  and  808 , which is the sum of the resistance values of resistors  804  and  808  divided by the resistance of resistor  426 . 
     The frequency of the first zero, Z 1 , is set by resistors  804  and  808  and capacitor  802 . It occurs when the sum of the resistance of resistors  804  and  808  is greater than the impedance of capacitor  802 . The gain flattens out to the AC gain. As the frequency increases, capacitor  806  starts to be effective, shorting out the feedback signal to ground, so the gain of the amplifier starts to increase through the positive input. At some point, the whole signal is shorted out and the gain of the amplifier approaches the DC gain. 
     This second zero, Z 2 , is set by the parallel impedance of resistors  808  and  804  in addition to capacitor  806  which fully determines the compensation network. The feedback signal is uncoupled from the compensation network. Resistor  812  adds a high frequency pole which was added heuristically to improve performance. 
     In  FIG. 8C , a second implementation of the compensation network is shown. Compensation network includes an error amplifier  408 , three filter capacitors ( 826 ,  828  and  830 ), two capacitors ( 802  and  806 ), seven resistors  426 ,  804 ,  808 ,  812 ,  820 ,  822 ,  824  and four internal nodes  810 ,  814 ,  816 ,  818 . The resistor  820  connects the feedback voltage FB to the internal node  814 . The node  814  is connected, in turn to the error amplifier  408  and via the filter capacitor  826  to the ground voltage VN. The capacitors  826 ,  828  and  830  are filter capacitors to filter out the switching frequency of switches  410  and  412 . Likewise,  820 ,  822  and  824  are the resistors which aid the filter capacitors to work. 
     The resistor  426  connects a reference voltage BG to the internal node  816 . The reference voltage BG is generated using any convenient technique as is well known in the relevant art. The node  816  is connected, in turn to the second input of the error amplifier  408  and via the filter capacitor  828  to the ground voltage VN. 
     A series connection of the capacitor  802  and the resistor  804  connect the node  816  to the internal node  810 . The node  810  is connected, by a series connection of the resistor  812  and the capacitor  806  to the ground voltage VN. 
     A resistor  808  connects the node  810  to the fourth internal node  818 . The node  818  is connected by the second filter capacitor  830  to the ground voltage VN. The output of the error amplifier  408  is connected by a resistor  822  to the node  810 . The node  818  is connected by a resistor  824  to the output node EAOUT of the compensation network. 
     It must be noted that the feedback signal comes into the positive input to the error amplifier and that the reference is attached to the negative input. For the loop to operate a 180 degree phase shift is introduced at the comparator in the next stage which inverts the sign of the signals. 
     In this configuration the signal path is not utilized in the compensation network except as a filter for the switching frequency. All of the signal shaping is done in the feedback path which is not in the direct signal path. 
     The DC gain is just the DC gain of the amplifier, itself, which can be seen by opening all of the capacitors in the feedback path. The dominant pole of the amplifier is set by the miller multiplied capacitance of  802  against the resistor in the reference circuit,  426 . Typically this might be set at 50 Hz. As the frequency is increased, the impedance of  802  becomes small compared to the resistors. This sets the minimum gain of the amplifier, the AC gain. The AC gain is set by the resistors  426 ,  804  and  808 , which is the sum of the resistance values of resistors  804  and  808  divided by the resistance of resistor  426 . 
     The frequency of the first zero, Z 1 , is set by resistors  804  and  808  and capacitor  802 . It occurs when the sum of the resistance of resistors  804  and  808  is greater than the impedance of capacitor  802 . The gain flattens out to the AC gain. As the frequency increases, capacitor  806  starts to be effective, shorting out the feedback signal to ground, so the gain of the amplifier starts to increase through the positive input. At some point, the whole signal is shorted out and the gain of the amplifier approaches the DC gain. 
     This second zero, Z 2 , is set by the parallel impedance of resistors  804  and  808  in addition to capacitor  806  which fully determines the compensation network. The feedback signal is uncoupled from the compensation network. Resistor  812  adds a high frequency pole which was added heuristically to improve performance. 
     This circuit has been found to work well because at low frequency the feedback loop is open, no feedback, because of the capacitors being high impedance. At high frequency, the feedback circuit is again open, being shorted out by capacitor  806 . If the feedback loop is open, then it can not oscillate. It will be noted that there is no sign of instability of the error amplifier in any simulations whether switching or linearized. 
     To demonstrate unconditional stability a Bode plot was generated for the local feedback circuit. In order to do this, the loop must be broken. An analysis technique has been developed by Middlebrook to obtain accurate gain and phase response without breaking the loop. It requires two sources be introduced into the feedback loop, a voltage source and a current source. Two transfer ratios are measured from these two cases, Tv and Ti, which are then used to get the total transfer curve, the Bode plot, T as follows: 
     
       
         
           
             T 
             = 
             
               
                 ( 
                 
                   
                     Ti 
                     * 
                     Tv 
                   
                   - 
                   1 
                 
                 ) 
               
               
                 ( 
                 
                   Ti 
                   + 
                   Tv 
                   + 
                   2 
                 
                 ) 
               
             
           
         
       
     
     The results of this analysis, shown in  FIG. 9 , are that the local loop compensation circuit and error amplifier combination are unconditionally stable for these circuit elements. The dashed plot is gain in db and the phase is solid line. Aside from the fact that SPICE reflects phase at 180 degrees, it can be seen that the phase hovers around 180 degrees out to 100 Mhz. It can be seen that the gain reaches 1 over a range of frequencies, but the phase is always 180 degrees, or close, while the gain is near unity. There is only a small observation, that at 20 Mhz, there is a small disturbance in the phase, but the gain has dropped below unity at this point. 
     This analysis confirms the observation that the transient simulation makes, that this circuit is very stable and exhibits no tendency to oscillate.