Gain with Feedback (Closed Loop Gain)

In this section we show how feedback can be used in op-amp-type structures to control and improve circuit performance. The concepts introduced here form some of the basic ideas for op-amp design. The circuit in Fig. 5.5 feeds back some of the output to the inverting input terminal of the diff-amp at the base of Q2. In this circuit we only feedback the AC part because the DC levels at the emitter of Q4 and the base of Q2 are very different due to biasing For now this lab let's consider only midband operating frequencies where the feedback capacitor is a short for AC and open for DC. The effect of the feedback on small signal gain can be illustrated by the following analysis. We start with the basic open loop gain definition for our circuit, and revise it slightly to account for two inputs (we do not ground the base of Q2).  

$$\frac{v_o}{v_{in}-v_2} = A_{dm}$$ (130)

Where A_dm is the differential mode gain given by equation (5.25), and v₂ is the small signal at the base of Q2. Our goal for the analysis will be to eliminate v₂ from the expression, and then determine the closed loop gain $A_{cl}= \frac{v_o}{v_{in}}$.To keep things simple, and illustrate the main points, we will neglect loading effects in our analysis. (This is a good approximation as long as the input resistance into the diff-amp is much greater than the effective resistance of the feedback network, and the resistance of the feedback resistors seen at the emitter of Q4 is much greater than the open loop output resistance of the circuit.) After making these approximations, we can immediately express v₂ in terms of v_o using the expression for a voltage divider and treating the capacitor as an AC short to obtain:  

$$v_2=\frac{v_oR_{F2}}{R_{F1}+R_{F2}}$$ (131)

Substituting equation (5.32) for v₂ in equation (5.31), and re-arranging gives the following expression for the closed loop gain.  

$$A_{cl}= \frac{v_o}{v_{in}}= \frac{A_{dm}}{1+\frac{A_{dm}R_{F2}}{R_{F2}+R_{F1}}}$$ (132)

One very interesting aspect of equation (5.33) is that if

$$\frac{A_{dm}R_{F2}}{R_{F2}+R_{F1}} \gg 1$$ (133)

then equation (5.33) can be approximated as  

$$A_{cl}= \frac{v_o}{v_{in}} \approx 1+\frac{R_{F1}}{R_{F2}}$$ (134)

This expression should be recognized as that describing the gain of a noninverting amplifier op-amp circuit. In the final experiment of this lab, you will construct such a feedback circuit, and help determine when equation (5.35) is applicable.