Most operational amplifier circuits work in the deep negative feedback state. When analyzing such circuits, we often use the idealized model of operational amplifier (i.e. using virtual short virtual break Technology). In fact, this idealized model ignores the impact of the open-loop gain of operational amplifier and the non idealization of input and output resistance on the operational amplifier circuit. Therefore, we use a more approximate method, the equivalent negative feedback model, to analyze the operational amplifier circuit.
1. Equivalent negative feedback model of operational amplifier circuit
The in-phase amplifier shown in Fig. 1 is analyzed, which is a typical negative feedback system. It is equivalent to the basic structure of the negative feedback circuit shown in Fig. 2. among α Is the forward gain of the amplifier, which is called the open-loop gain of the operational amplifier circuit. β The gain of the feedback network is called the feedback coefficient of the operational amplifier circuit. In order to find B, remove all input sources, cut off the operational amplifier and replace it with its input resistance RD and output resistance R0 to maintain the same load condition. Then, add a test source VT via R0, as shown in Fig. 3, and calculate the difference VD across Rd, then
This formula is easy to organize into
Now let’s analyze the negative feedback system of Fig. 2. According to the knowledge of control theory, the closed-loop gain of the system can be obtained
Substitute equation (1) into equation (3) and make Rd →∞, R0 → 0 to obtain the ideal case aideal = (R1 + R2) / R1
This is consistent with the results obtained by the ideal model.
For the inverting operational amplifier circuit, the feedback coefficient can be obtained in the same way, and its negative feedback circuit model (this model is different from the in-phase circuit because the input is added to different points of the same circuit) can be established, so as to obtain the values of a and aideal.
3. Influence of loop gain on closed-loop parameters of operational amplifier circuit
Define loop gain t= αβ， As can be seen from equation (2), the closed-loop gain can be expressed in the following insightful form:
It can be seen from the above formula that when designing the operational amplifier circuit, we should make t as large as possible.
Next, the mature operational amplifier model is used to derive the expressions of in-phase closed-loop input resistance and output resistance.
In order to calculate RI, add the test voltage V in Fig. 4, calculate the current I from the positive end of the test source, and then make RI = V / I. The solution is as follows:
For a large enough a, the last item can be omitted, and in a well-designed circuit, R0《
Referring to Fig. 5, the test voltage technology can also be obtained
Typical RD is megaohm or even larger, R1 and R2 are kiloohm, and R0 is hundred ohm, so R0 / R1, R0 / RD and R2 / RD can be omitted
For the inverted structure, the input resistance and output resistance can be obtained by the same method, and the approximate expression is given directly here
RI R1 (RL is the resistance between the signal source and the inverting terminal)
3. The influence of loop gain on the stability of operational amplifier circuit
Using the frequency characteristic of the loop gain of the negative feedback amplifier circuit, we can judge whether the circuit has self-excited oscillation after closed-loop, that is, whether the circuit is stable. As can be seen from Fig. 2, when the circuit generates self-excited oscillation,
When the frequency characteristic t (f) of the loop gain of the operational amplifier circuit meets the above conditions, the operational amplifier circuit will produce self-excited oscillation. In order to make the circuit work stably, the condition of self-excited oscillation must be eliminated, which is not discussed here.
To sum up, the loop gain t plays a core role in the negative feedback circuit composed of operational amplifier. For a given open-loop gain value of operational amplifier α， The larger the loop gain t is, the closer the closed-loop parameters are to the ideal value. At the same time, the frequency characteristic of t also determines whether the negative feedback circuit of an operational amplifier is stable or oscillates on the contrary. These have important reference value for us in the design of operational amplifier circuit.
Responsible editor: GT