Multistage amplifiers and coupling

A single transistor stage can give gain, or a high input resistance, or a low output resistance, but rarely all three at once, and never with the precision, bandwidth and output power that real products need. Practical amplifiers are therefore built from several stages in a chain, each chosen for one job: an input stage that accepts the signal without loading the source and rejects interference, gain stages that provide most of the amplification, and an output stage that delivers current to the load. Electronics I analyzed each stage on its own. Electronics II is about how stages work together: how they load one another, how they are coupled, how a differential input rejects noise, how feedback makes the whole chain precise, how it stays stable at high frequencies, how oscillators deliberately make it unstable, and how an output stage can push watts into a loudspeaker without overheating.

The course follows one product at Cedarline Controls in Madison, Wisconsin. Amara Osei, now in her second year as an engineer intern, has been given the analog design of the APS-2, the company's next accessible pedestrian signal push-button station. Daniel Fortin, PE, the senior hardware designer, reviews her work, and Leo Tremblay, a co-op student, builds and measures prototypes on the bench. The APS-2 listens to street noise through a remote microphone on the pole, and drives a weatherproof 8 Ω loudspeaker with a locator tone and spoken messages whose loudness tracks the ambient noise, so a pedestrian who is blind or has low vision can find the button and know when the walk interval begins.

The APS-2 signal chain

The signal chain has two halves. On the listening side, the remote microphone sits about 15 m away at the top of the pole, so its signal travels down a long cable that also picks up hum from the 60 Hz street-lighting circuits. A differential receiver (lesson 3) rejects that hum, and a preamplifier based on the noise-sensing amplifier from Electronics I brings the signal up to a level the microcontroller's analog-to-digital converter can measure. On the talking side, the microcontroller generates the 880 Hz locator tick and the voice messages, a digitally controlled attenuator sets the volume, and a power amplifier (lessons 10 to 12) drives the loudspeaker with up to 4 W. Every block in that chain is a multistage amplifier or part of one.

Two-port models and loading

The cleanest way to analyze a cascade is to replace each stage by a two-port model: an input resistance R_in, an open-circuit voltage gain A_vo and an output resistance R_out. The model hides the transistor details and keeps only what matters to the neighbors. When stage k drives stage k + 1, its Thévenin output, A_vo,k v_in,k behind R_out,k, sees the input resistance of the next stage, so only the fraction R_in,k+1/(R_in,k+1 + R_out,k) of its open-circuit output reaches the next input. The overall gain is the product of every stage's open-circuit gain and every interface divider, including the one at the source and the one at the load:

Two habits follow from the formula. First, each divider should be close to 1: a stage should have an output resistance much smaller than the next stage's input resistance, a condition designers call voltage bridging. Second, it is usually wrong to compute stage gains in isolation and multiply them, because the load on a stage changes its gain. Loading works the other way too: a stage's input resistance can depend on its load. For a CE stage with r_o neglected it does not, but for an emitter follower R_in ≈ (β + 1)(r_e + R_L'), so a heavy load lowers the input resistance and loads the stage before it.

import math
# Two-port cascade: each stage is (Rin, Avo, Rout). Units: ohms.
Rsig, RL = 2e3, 1e3
stages = [("CE 1", 12e3, -12.0, 5.1e3),
          ("CE 2", 15e3, -12.0, 5.1e3),
          ("follower", 150e3, 0.99, 46.0)]
def cascade(stages, Rsig, RL, verbose=False):
    gain, rprev = 1.0, Rsig
    for name, rin, avo, rout in stages:
        div = rin / (rin + rprev)
        gain *= div * avo
        if verbose:
            print(f"{name:9s} divider {div:5.3f}  stage gain {avo:+7.2f}  running gain {gain:+8.2f}")
        rprev = rout
    div = RL / (RL + rprev)
    gain *= div
    if verbose:
        print(f"{'load':9s} divider {div:5.3f}  overall gain {gain:+8.2f} ({20*math.log10(abs(gain)):.1f} dB)")
    return gain
g = cascade(stages, Rsig, RL, True)
naive = math.prod(s[2] for s in stages)
print(f"product of open-circuit gains: {naive:.0f}; loading loses {100*(1-g/naive):.0f} %")
# What if stage 1's collector resistor drove a follower before stage 2?
better = [stages[0], ("buffer", 150e3, 0.99, 46.0), stages[1], stages[2]]
print(f"with a buffer between the CE stages: {cascade(better, Rsig, RL):.0f}")

CE 1      divider 0.857  stage gain  -12.00  running gain   -10.29
CE 2      divider 0.746  stage gain  -12.00  running gain   +92.11
follower  divider 0.967  stage gain   +0.99  running gain   +88.19
load      divider 0.956  overall gain   +84.31 (38.5 dB)
product of open-circuit gains: 143; loading loses 41 %
with a buffer between the CE stages: 108

The last line of the output shows the price of a weak interface. A follower between the two CE stages lifts the gain from about 84 to about 108, because the 5.1 kΩ output of stage 1 then sees 150 kΩ instead of 15 kΩ. Whether the extra transistor is worth it is a design decision: here Amara would rather adjust the resistor values, because the noise-sensing channel needs only a known gain, not the largest possible one.

Ways to couple stages

Stages must pass the signal while keeping each transistor at its own bias point. There are four classic ways to connect them, each with its own trade-offs:

Discrete audio and sensor circuits often use capacitor coupling, as the Electronics I microphone amplifier did. Integrated circuits use direct coupling almost exclusively, because a capacitor large enough to pass audio frequencies would occupy more silicon than hundreds of transistors. Once stages are direct-coupled, the designer must solve two problems that capacitor coupling hid: the DC level at each node must suit the next stage, and any DC error at the input is amplified along with the signal.

Direct coupling and level shifting

In a chain of npn common-emitter stages, each collector sits several volts above its base. Coupling collector to base directly would push every stage's bias point higher until the last stage saturated against the supply. Designers shift the DC level down between stages in several ways: by alternating npn and pnp stages, since a pnp stage's collector sits below its emitter; by an emitter follower, which drops one V_BE; by a Zener diode or a string of diodes in series with the signal path; or by a current source driving a resistor, which drops a fixed voltage without affecting the signal much. Op amps use these tricks so that the output can sit at 0 V when the input is at 0 V.

Direct coupling has a cost: drift. If the first stage's collector voltage moves by 10 mV because its V_BE changes by about 2 mV/°C with temperature, the second stage amplifies that change exactly as it amplifies the signal. With 40 dB of gain after it, a 10 mV drift becomes 1 V at the output, enough to clip a small-signal stage. The cure is to make the input stage insensitive to temperature by design, which is the main reason every op amp begins with a matched differential pair: in the pair, the two transistors' V_BE drifts cancel (lesson 3). Feedback around the whole chain (lesson 7) then holds the output's DC level precisely.

Composite stages

Some combinations of two transistors are so useful that they are treated as single stages. The Darlington pair connects one emitter follower into the base of another, giving an overall current gain of about β₁β₂, so the input resistance is huge and a small base current can control amperes; its drawbacks are a doubled V_BE and slower switching. The CC-CE combination puts an emitter follower in front of a common-emitter stage, so a high-resistance source sees a high input resistance while the CE stage still provides the gain. The CE-CC combination does the opposite, buffering the CE output so it can drive a low-resistance load. The cascode, a CE stage feeding a CB stage, extends bandwidth and output resistance and is the subject of lesson 6. And the differential pair, two matched transistors sharing a current source, is the most important composite of all, because it amplifies the difference between two inputs and ignores what they have in common.

Gain distribution, noise and swing

Deciding how much gain each stage should provide is not arbitrary. Two considerations pull in different directions. Noise argues for a high-gain, low-noise first stage: every stage adds noise, but noise added after the first stage is divided by the first stage's gain when referred back to the input. If stage 1 has a gain of 10 and an input-referred noise of 4 nV/√Hz, and stage 2 adds 20 nV/√Hz at its own input, the total input-referred noise is

so the second stage contributes little once the first has some gain. Signal swing argues the other way for the last stages: the output stage handles the largest signal, so it must have enough headroom between the supply rails, and gain placed late in the chain multiplies any overload. A good plan puts modest, low-noise gain at the front, most of the gain in the middle where signals are still small, and a buffer with large swing capability at the end. It also checks, at every node, that the largest expected signal stays inside the linear range of that stage.

For the APS-2 listening channel, Amara writes the plan into a table before choosing any resistor: the remote microphone produces about 10 mV peak at the loudest street noise; the differential receiver must reject hum while adding a gain of about 4; the preamplifier adds a gain of about 25 to reach 1 V peak at the converter; and the converter's 3.3 V input range leaves more than 3 dB of headroom. Writing the plan as numbers makes the review with Daniel quick, because each stage's job is then a short list of targets.