Analog Simulation:Analysis of Loop Gain as Stability Criterion of Analog Circuits.

Analysis of Loop Gain as Stability Criterion of Analog Circuits

In [10.17] the determination of the loop gain for stability analysis of analog circuits is discussed extensively. There it is necessary to open the circuit and to take care that by this opening the electrical conditions for the operating point and loading are maintained, which can be very difficult in some cases. Here an elegant method of determining loop gain will be discussed which does not need to open the circuit. It was first published in [10.20] and is not easily understood, repeated erroneously in a Microsim Application Note [10.9]. A didactically prepared presentation with the notation used here can be found in [10.14].

Figure 10.41 shows the circuit diagram of a logarithmic amplifier taken as an example for stability analysis. The additional AC voltage sources at the output are not disturbed because of their zero inner resistance as well as the AC current source to ground because of its infinite inner resistance.

Assuming virtual ground at the inverting input of the operational amplifier with feedback one can consider at first sight the transistor Q1 as a diode: the collector is virtually the base directly connected to ground. Therefore the output voltage can be taken as

This result is verified by a DC sweep analysis Va = f (I1) over many decades. If one defines a step function for I1 (0 ... 1 μA) and a TR analysis is performed, an immediately starting oscillation of f = 200 kHz will result. If, on the other hand, the transistor Q1 is, in fact, connected as a diode by disconnecting the base from ground and connect- ing it to the collector the step response analysis will lead to the expected result: after a rise time the output voltage is approximately equal to the expected base emitter voltage for I1 = 1 μA, where part of the current is the base current.

The explanation of the instability discussed above is given by the loop gain analysis, which is per- formed by means of the parametric AC analysis in PSPICE using the additional AC sources in fig. 10.41.

If the parameter ‘VOLTS’ = 0 the current source I2 in parallel will send a current of 1 A into the system. It should be kept in mind that before an AC analysis is performed the circuit will be linearized so that saturation cannot occur. The resulting currents through the voltage sources V1 and V2 define a current transfer function

where the notation @1 signifies the first parameter value, i.e., VOLTS = 0. With VOLTS = 1 the cur- rent source is switched off and the voltage source V1 in series will be effective with a voltage of 1 V. The thus resulting voltages at point A and B define a voltage transfer function

where the notation @2 signifies the second parameter value, i.e., VOLTS = 1.

In [10.14] it is shown that the loop gain T can be determined from the two transfer function dis- cussed above by the formula

which can be easily programmed as a so called Macro in the output program PROBE of PSPICE.

Figure 10.42 shows the thus determined loop gain T with respect to magnitude and phase. One realises that in a rather large frequency range the  phase is approximately 180◦ and the magnitude (gain) much greater than 0 dB: the system must therefore be unstable. The physical explanation is given by the fact that the transistor as a common base circuit forms an additional amplifier and therefore alters the overall frequency response with respect to magnitude and phase. Table 10.12 lists values of the operating point (I1 = 1 μA) and the small signal parameters of transistor Q1.

From there the voltage gain v of the common base amplifier can be calculated:

With these values the frequency response shown in fig. 10.42 becomes plausible: the gain of 69 dB adds to the operational amplifier gain of 100 dB to approximately 170 dB. For frequencies > 1 kHz the additional pole causes an additional phase shift of 90◦ which adds to the phase shift of the operational amplifier of 90◦ in that frequency range.

This example shows the possibilities of finding a deeper understanding of electronic circuits by means of simulation, for the concept of ‘virtual ground’ favoured by many over-practical orientated engineers has its shortcomings

If the transistor Q1 in fig. 10.41 is connected as a diode, i.e., its base is disconnected from ground and connected to the collector, then the loop gain analysis will result in the frequency response shown in fig. 10.42. With this frequency response the circuit will be stable because the phase is well above 180◦ when the gain reaches 0 dB.

Figure 10.43 represents nothing else because the open loop gain of the operational amplifier itself, as one finds at the input a current source with infinite inner resistance, and the small signal equivalent series resistance of the diode is negligible compared to the high input resistance of the operational amplifier. Therefore the current transfer function Ti becomes very large and the loop gain.