3. Feedback Amplifiers

Negative feedback is the single most important concept in analog electronics. It trades raw gain for stability, linearity, predictability, and bandwidth. If you understand feedback, you understand op-amps, control systems, oscillators, and most of analog IC design. If you do not, you will spend years confused about why op-amp circuits behave so well.

3.1 The basic feedback equation

A feedback system takes some fraction of the output and subtracts it from the input:

If $A$ is the open-loop (no-feedback) gain and $\beta$ is the feedback factor (the fraction of output fed back), then the error signal is $e = x - \beta y$, the output is $y = A e$, and after a line of algebra:

$A_f = \frac{y}{x} = \frac{A}{1 + A\beta}$

This is the closed-loop gain. The product $A\beta$ is called the loop gain, the gain you would measure if you broke the loop and walked a signal all the way around.

When $A\beta \gg 1$, the equation simplifies dramatically:

$A_f \approx \frac{1}{\beta}$

The closed-loop gain depends only on the feedback network, not on the open-loop gain. This is profound. If $A$ is 100,000 and $\beta = 0.01$, the closed-loop gain is 100, set by $\beta$, not by $A$. Even if $A$ varies wildly with temperature or device variations or supply voltage (and it does, by factors of 2 to 10 in real circuits), the closed-loop gain barely moves, because $1/\beta$ does not depend on $A$.

Cruise-control analogy. You set 65 mph (the reference $x$). The car measures actual speed (the feedback $\beta y$). The error feeds the throttle (the open-loop gain $A$). If the car is going up a hill or the engine is weaker than usual, the cruise control automatically gives more throttle. The closed-loop speed (your actual speed) barely changes. The system "rejects" the disturbance, the hill or the weaker engine, both of which are equivalent to drift in $A$. You feel solid 65 mph regardless of the underlying chaos.

3.2 Why the closed-loop gain is so insensitive

Differentiate $A_f = A/(1+A\beta)$ with respect to $A$:

$\frac{dA_f}{dA} = \frac{(1+A\beta) - A\beta}{(1+A\beta)^2} = \frac{1}{(1+A\beta)^2}$

Compare to the relative sensitivity:

$\frac{dA_f / A_f}{dA / A} = \frac{1}{1 + A\beta}$

So a 50% change in open-loop $A$ produces only a $50\% / (1 + A\beta)$ change in closed-loop $A_f$. With $A\beta = 1000$, that 50% drift in the underlying transistor gains becomes 0.05% drift in the closed-loop gain. This is why feedback amplifiers are so stunningly accurate compared to open-loop ones.

3.3 Effects of negative feedback (the full list)

Wrapping feedback around an amplifier with high open-loop gain $A$ and feedback factor $\beta$ does all of the following simultaneously:

• Gain reduces by factor $1 + A\beta$. The price you pay.
• Bandwidth increases by factor $1 + A\beta$. The pole that was at $\omega_p$ in the open-loop response shifts out to $\omega_p (1 + A_0 \beta)$ in the closed-loop response. The gain-bandwidth product is conserved, so trading gain for bandwidth is mathematically a wash but practically a gift, because the closed-loop bandwidth is what you use.
• Sensitivity to component variations is reduced by factor $1 + A\beta$. We just derived this.
• Distortion is reduced by factor $1 + A\beta$. Nonlinearity in $A$ (which would normally show as harmonic distortion) gets divided by the loop gain. This is why audio amps with feedback have THD below 0.01% even though the underlying transistors are quite nonlinear.
• Noise generated inside the loop is reduced by the loop gain (relative to noise at the input), because noise added downstream of the high-gain part is referred back to the input divided by the gain ahead of it.
• Input/output impedances are modified. Depending on the topology (we will see four kinds), feedback can raise or lower the input or output impedance dramatically, decoupling the source-load story from the underlying transistor's intrinsic impedances.

The price of all these benefits: stability. With enough loop gain and the right phase shift around the loop, the amplifier can oscillate. We will see why and how to prevent it in section 3.6.

3.4 Why feedback widens bandwidth (the picture)

Anticipate confusion here. Why exactly does feedback widen the bandwidth, and why is the gain-bandwidth product conserved?

Suppose the open-loop amplifier has a single pole and its frequency response is

$A(j\omega) = \frac{A_0}{1 + j\omega/\omega_p}$

so it rolls off at 20 dB/decade above $\omega_p$. Substitute this into the closed-loop formula $A_f = A / (1 + A\beta)$:

$A_f(j\omega) = \frac{A_0 / (1 + j\omega/\omega_p)}{1 + A_0\beta / (1 + j\omega/\omega_p)} = \frac{A_0}{1 + A_0\beta + j\omega/\omega_p}$

Pull out the constant in the denominator:

$A_f(j\omega) = \frac{A_0/(1 + A_0\beta)}{1 + j\omega/[\omega_p(1 + A_0\beta)]}$

The closed-loop gain is $A_0/(1 + A_0\beta)$, lower by factor $(1 + A_0\beta)$. The closed-loop pole is at $\omega_p(1 + A_0\beta)$, higher by the same factor. The product of low-frequency gain and pole frequency is

$A_0 \cdot \omega_p$

before feedback, and

$\frac{A_0}{1+A_0\beta} \cdot \omega_p(1+A_0\beta) = A_0 \omega_p$

after. Conserved exactly. The pole has been pushed out, the low-frequency gain has been reduced, and the gain-bandwidth product is the same number. That conserved quantity is the unity-gain crossover frequency.

This is why op-amps are spec'd with a "gain-bandwidth product" rather than separate gain and bandwidth: you are buying a fixed amount of gain-bandwidth product that you can spend any way you want by choosing your $\beta$.

3.5 The four feedback topologies

There are four ways to sample the output (as voltage or as current) and two ways to mix it with the input (as voltage or as current). This gives four topologies:

The mnemonic: series sampling raises that impedance, shunt sampling lowers it, on each side of the loop independently. Voltage-sense-voltage-mix (V-V) gives the ideal voltage amplifier (high $Z_{in}$, low $Z_{out}$). Current-sense-current-mix (I-I) gives the ideal current amplifier. The other two are mixed: V-I is the transimpedance amplifier (current in, voltage out, lowest both impedances), and I-V is the transconductance amplifier (voltage in, current out, highest both impedances).

The op-amp non-inverting amplifier is V-V: very high $Z_{in}$ (the input goes straight into the op-amp's high-impedance non-inverting input) and very low $Z_{out}$ (the feedback senses output voltage and forces it to track the input). The inverting amplifier is V-I: input current flows into a virtual-ground summing junction, and the output voltage is set so the feedback resistor passes that same current. CE with un-bypassed emitter resistor is I-V: emitter resistor senses the collector current, and the resulting emitter voltage subtracts from the input voltage.

3.6 Why we still need open-loop gain

A natural question: if closed-loop gain is just $1/\beta$, why bother with high $A$ at all?

Two reasons.

1. The $A_f = 1/\beta$ approximation requires $A\beta \gg 1$. If $A\beta \approx 1$, the actual gain is more like $A/2$, and it depends on $A$. If $A\beta \approx 0.1$, the actual gain is essentially $A$ itself, and the feedback does almost nothing. So $A$ must be at least an order of magnitude bigger than the desired closed-loop gain.

2. All the benefits of feedback (stability, linearity, bandwidth, impedance modification, noise shaping) scale with the loop gain $A\beta$, not with the closed-loop gain. Higher loop gain equals better. So we want $A$ as big as possible, even though it does not directly affect the closed-loop gain.

Modern op-amps push $A$ above $10^5$ or $10^6$ for exactly this reason. The Texas Instruments OPA827 has 134 dB of open-loop gain, which is $A_0 \approx 5 \times 10^6$. Wrap a $\beta = 0.01$ feedback network around it (closed-loop gain 100) and the loop gain is 50,000 at DC, giving you 94 dB of distortion suppression and about 50,000× the open-loop bandwidth. Spectacular.

3.7 Stability and the Nyquist criterion in amplifiers

If you push the loop gain too hard or design the feedback poorly, the amplifier can oscillate. The classical analysis tool is the Nyquist criterion.

The closed-loop pole locations come from setting the denominator $1 + A(j\omega)\beta = 0$, which means $A(j\omega)\beta = -1$. In other words, if at some frequency the loop gain magnitude is 1 and the loop phase is 180°, the amplifier sits on the boundary between stable and oscillating.

Two convenient stability margins:

• Gain margin. At the frequency where the loop phase is $-180°$, how many dB below 0 dB is the loop gain magnitude? Positive gain margin equals stable; negative equals unstable. A typical good design has 10 to 15 dB.
• Phase margin. At the frequency where the loop gain magnitude is 0 dB (the unity-gain crossover), how far away from $-180°$ is the loop phase? A typical good design has 45° to 60° of phase margin. Less than 45° gives ringing on step responses; less than 0° gives oscillation.

The Bode-plot view is the practical tool. Plot $|A\beta|$ in dB and $\angle A\beta$ in degrees on log frequency axes. Find the unity-gain crossover. Read off the phase margin. If the crossover happens before any extra poles kick in too far, you are stable.

Cascading high-gain stages introduces cascaded poles and the phase wraps around faster, which is why three-stage amplifiers with heavy feedback often need careful compensation, typically a Miller cap from the output of the second stage back to its input. The compensation cap creates a "dominant pole" that sets the open-loop bandwidth low enough that all higher-frequency poles happen well above the unity-gain crossover, leaving the phase margin intact.

Audio tube amp anecdote. Heathkit and Dynaco vacuum tube amplifiers from the 1950s and 60s often had marginal stability. With certain speaker cables, the amplifier's load reactance combined with output transformer leakage inductance to push a pole into the loop and reduce phase margin below zero. The amp would burst into a few hundred kHz oscillation, melt fuses, and cook output tubes. Audio modders today still talk about "putting a Zobel network on the speaker output," which is exactly a small RC across the output to damp this resonance.