Negative feedback and the four topologies

Transistor gain is large but unreliable: β varies by a factor of three between parts of one type, g_m follows current and temperature, and every stage distorts. Negative feedback turns that large, unreliable gain into a smaller gain set by resistors, with lower distortion, wider bandwidth and input and output resistances chosen by the designer. Harold Black's insight in the 1920s, made to solve distortion in long-distance telephone repeaters, is that if part of the output is subtracted from the input, the amplifier spends its excess gain correcting its own errors. This lesson develops the general feedback equation, quantifies each benefit and its cost, and classifies feedback amplifiers into four topologies by what they sense at the output and how they mix at the input.

Every amplifier in the APS-2 uses feedback. The receiver's gain of 4 is set by resistor ratios around an op amp; the preamplifier's gain of 25 likewise. The power amplifier, which Amara will design in lesson 12, wraps feedback around an output stage whose open-loop distortion would be clearly audible, so that the locator tone and voice messages come out clean. Before she can design it, she needs to know how much loop gain buys how much improvement.

The feedback equation

A feedback amplifier has a forward amplifier with gain A and a feedback network with transmission β. The feedback network samples the output x_o and returns x_f = βx_o, which is subtracted from the input x_s, leaving the error x_i = x_s − x_f to drive the amplifier. Then x_o = A(x_s − βx_o), so

The quantity T = Aβ, the loop gain, is the gain around the loop: a signal leaving the summing junction is amplified by A and returned by β. The quantity 1 + T, the amount of feedback, appears in every result. When T is large, the closed-loop gain approaches 1/β and depends only on the feedback network, usually a pair of resistors, whose ratio can be held to 0.1 percent or better. In a non-inverting op amp circuit with feedback resistor R_f and ground resistor R_g, β = R_g/(R_f + R_g), so the familiar gain 1 + R_f/R_g is simply 1/β.

Desensitivity and the cost of feedback

Differentiating the feedback equation shows how changes in A are suppressed:

A 1 + T of 1,000 makes the closed-loop gain a thousand times less sensitive to the forward gain. The price is gain itself: A_f is smaller than A by the same factor 1 + T. Feedback therefore makes sense only when gain is cheap, which it is: an op amp gives 100 dB for a few cents, and spending 80 dB of it on precision and linearity is a bargain. The second price is the possibility of instability, because the loop gain changes phase with frequency; lesson 8 deals with it.

Bandwidth extension

If the forward amplifier has a single pole, A(s) = A_0/(1 + s/ω_H), substituting into the feedback equation gives

Feedback divides the gain by 1 + T and multiplies the bandwidth by the same factor, leaving the gain-bandwidth product unchanged. That is why op amp circuits are designed with the GBW in mind: a 10 MHz op amp gives a 909 kHz bandwidth at a gain of 11 and a 100 kHz bandwidth at a gain of 100. Lower-frequency poles in the loop are pushed up the same way, which is how feedback flattens the response of an output stage.

Reduced distortion

Distortion generated inside the forward amplifier can be modeled as an unwanted signal added at its output. Feedback treats it like any other error and divides it by 1 + T. Equivalently, the amplifier needs only a small error signal to produce its output, so it operates over a small part of its nonlinear transfer curve. The improvement holds only while T is large, which means at frequencies where the loop gain is still high, and only while nothing clips: once the output stage saturates, A collapses, T collapses with it, and feedback cannot help. The code below models an amplifier whose output saturates smoothly, like a real output stage, and compares its distortion open-loop and with feedback at the same 8 V peak output.

import math
Vsat, A = 12.0, 2000.0          # smooth saturation at 12 V; small-signal forward gain 2000
beta = 1 / 11
f = lambda e: Vsat * math.tanh(A * e / Vsat)
N = 256
def thd(out):
    """Harmonic distortion of one period of samples via a direct DFT."""
    mags = []
    for k in range(1, 8):
        re = sum(out[n] * math.cos(2 * math.pi * k * n / N) for n in range(N))
        im = sum(out[n] * math.sin(2 * math.pi * k * n / N) for n in range(N))
        mags.append(2 * math.hypot(re, im) / N)
    return mags[0], math.sqrt(sum(m * m for m in mags[1:])) / mags[0]
def closed(x):
    lo, hi = -Vsat, Vsat         # solve y = f(x - beta*y) by bisection
    for _ in range(60):
        y = (lo + hi) / 2
        if y - f(x - beta * y) > 0: hi = y
        else: lo = y
    return y
def peak_for(fn, target):
    lo, hi = 0.0, 5.0
    for _ in range(50):          # find input amplitude giving the target output fundamental
        a = (lo + hi) / 2
        out = [fn(a * math.sin(2 * math.pi * n / N)) for n in range(N)]
        if thd(out)[0] > target: hi = a
        else: lo = a
    return a, out
a_ol, out_ol = peak_for(f, 8.0)
a_cl, out_cl = peak_for(closed, 8.0)
T = A * beta
print(f"loop gain T = {T:.0f}, 1 + T = {1+T:.0f}")
print(f"open loop:   input {a_ol*1e3:7.3f} mV peak, THD = {100*thd(out_ol)[1]:.2f} %")
print(f"closed loop: input {a_cl*1e3:7.1f} mV peak, THD = {100*thd(out_cl)[1]:.4f} %")
print(f"improvement = {thd(out_ol)[1]/thd(out_cl)[1]:.0f} times")

loop gain T = 182, 1 + T = 183
open loop:   input   4.550 mV peak, THD = 4.20 %
closed loop: input   731.9 mV peak, THD = 0.0305 %
improvement = 138 times

Open loop, an 8 V output from an amplifier that saturates at 12 V carries about 4 percent distortion, mostly third harmonic. With a loop gain of about 182, the same output carries about 0.03 percent, roughly 140 times less. The reduction falls short of 1 + T = 183 because the small-signal loop gain is not the gain at the peaks of the waveform, where the amplifier is already compressing; near clipping, the incremental A falls and so does the improvement, which is why feedback amplifiers clip abruptly rather than gradually.

What feedback does not do

Feedback cannot reduce noise that enters at the input along with the signal, because the loop cannot tell them apart; it also cannot reduce the input-referred noise of the first stage, since that noise is indistinguishable from an input signal. What it can reduce is noise and interference injected later in the loop. Hum on the output stage's supply rail, for example, is an error at the output and is divided by 1 + T, which is one reason a power amplifier with heavy feedback can run from a simple unregulated supply. Feedback also cannot fix an amplifier that is overdriven, and it cannot create gain where the loop gain has already fallen to 1.

The four topologies

A feedback amplifier senses either the output voltage (shunt sampling, the feedback network connected across the output) or the output current (series sampling, the network in series with the output loop), and mixes the feedback with the input either as a voltage in series with the input loop (series mixing) or as a current into the input node (shunt mixing). The four combinations stabilize four different gains, and each changes the input and output resistances by the factor 1 + T in a predictable direction:

The pattern is easy to remember: series connections, whether at the input or the output, multiply the resistance there by 1 + T, and shunt connections divide it. A voltage amplifier should have a high input and low output resistance, so it uses series mixing and shunt sampling. A current source should have a high output resistance, so it samples current in series. The emitter-degenerated CE stage from Electronics I is a series–series feedback amplifier with β = R_E: its transconductance is g_m/(1 + g_m R_E), approaching 1/R_E, and its input resistance rises from r_π to r_π(1 + g_m R_E), which is the resistance-reflection rule seen as feedback.

Analyzing a feedback amplifier

Real feedback networks are not ideal: they load the amplifier's input and output, and the amplifier may pass a little signal backward. The systematic method handles this by splitting the circuit into an A circuit and a β circuit. First identify the topology from how the feedback network connects at each end. Then form the A circuit by including the loading of the feedback network: at a shunt connection, the feedback network's port is left loading the node with the other port shorted; at a series connection, it is left with the other port open. Compute A, β and T, then A_f = A/(1 + T), and adjust the input and output resistances by 1 + T in the directions the table gives. For op amp circuits with large T, the shortcut A_f ≈ 1/β is usually all that is needed; the full method matters for transistor-level feedback pairs and for checking how much loop gain a design really has.