What sits still, and what wiggles.

A microphone gives you a few millivolts, swinging above and below zero. Your circuit runs from a single +12 V supply and a ground. There is no negative voltage anywhere in it.

So connect the microphone straight to the base of a common-emitter amplifier and see what happens.

Two separate things are broken

1. Half the signal is simply gone. Below 0.6 V the transistor is cut off. That is chapter 3 again, in a different circuit — and no amount of amplification recovers information that never entered.

2. The half that survives is distorted. The transistor obeys an exponential, and a sine fed into an exponential comes out as something else entirely. You are hearing the curve, not the music.

The fix is to lift the whole signal up first, so that it sits in the middle of the transistor’s useful range and never leaves it. Press bias it properly and watch the same circuit produce a recognisable, inverted, amplified copy instead of a notch.

And notice it is still not perfect. Look closely at the biased trace: the positive and negative halves are not quite the same size. With the emitter grounded, the transistor is still obeying an exponential, so even a small signal comes out slightly lopsided. Bias solved the big problem; it did not make the device linear. The cure for the remaining crookedness is a resistor in the emitter, and it arrives in lesson 5.

And notice the price. You are now burning current continuously, with no signal present, purely to hold the transistor in a sensible place. Every analogue amplifier on Earth does this. It is why a hi-fi amplifier is warm before you play anything.

With no signal applied, the circuit still sits somewhere. A particular collector current is flowing and the collector is at a particular voltage. That resting state is the quiescent point — the Q point — and choosing it is the first decision in any amplifier design.

Two walls, and the room between them

The distance from the Q point to the nearer wall is your headroom. Park at 10 V and you have 2 V going up and 10 V coming down — so the usable swing is 2 V, and the other 10 V is wasted. Symmetry is the whole argument for the middle.

Watch the load line as you drag. That straight line is not the transistor — it is the collector resistor and the supply, drawn as the set of all the places the circuit is allowed to sit. The transistor picks one point on it. The signal slides the point up and down the line, and the line’s ends are the two walls.

The two clippings look different, and that is diagnostic

Push the bias too far up and the bottom of the output flattens, because the transistor saturates. Too far down and the top flattens, because it cuts off. On a real bench, which half is flat tells you which way your bias is wrong — a genuinely useful piece of fault-finding you now have for free.

This is the lesson the whole chapter is named for, and it comes with an unusually strong piece of evidence behind it.

Researchers gave 410 students — all of whom had completed a course on transistors — three amplifier circuits and asked which produced the biggest output signal. Between 10% and 40% answered by computing DC voltages instead. Correct answers ranged from 7% to 42%.

Then they asked a second group the same question, split into two explicit parts: first work out the DC operating point; now work out the signal behaviour. Correct answers jumped to 50% on all parts, with 86% getting the DC part right and 64% the signal part. Statistically that is not a nudge — it is one of the largest effects in the whole literature, from an intervention that costs nothing.

Why the supply rail is a signal ground

Put a scope on the +12 V rail while the amplifier is running. The trace is flat. The supply is held there by a low-impedance regulator and its decoupling capacitors, and it does not move.

Now ask what “ground” means to a signal. It means: a node that does not move. By that definition the +12 V rail is a ground — it just happens to sit twelve volts up. As far as your signal is concerned, a resistor to +12 V and a resistor to 0 V are the same component.

That is also why every board has decoupling capacitors scattered across it. They are what makes the rail a signal ground. Leave them out and the rail moves, stages start talking to each other through the supply, and things oscillate.

The ritual

1. Draw the DC circuit. Capacitors become open circuits — they pass no steady current. The signal source disappears. Work out the Q point: I_C, V_C, V_E.

2. Draw the AC circuit. Capacitors become wires — at signal frequencies their impedance is negligible. Every DC supply becomes ground. Work out the gain and the impedances.

3. Only then do algebra, and always on one drawing at a time.

Use the toggle on the bench to flip between the three views of one circuit. Components appear and vanish, and the point is that this is not a simplification — each drawing is exactly, completely true about its own question.

Notation, which is doing real work here

Sedra & Smith are strict about this, and it is worth adopting on day one, because the letter you write records which drawing you are in:

The rule underneath: the case of the letter says whether it varies; the case of the subscript says whether it includes DC. Which makes i_C = I_C + i_c the two-drawing method written as an equation.

“Capacitors block DC and pass AC” is the sentence everyone learns, and it hides the mechanism so completely that people end up believing a capacitor is a kind of one-way valve for wiggles.

Here is the honest version. Charge flows onto one plate and off the other continuously, as long as the voltage keeps changing. Stop changing it and the flow stops. A capacitor does not block current. It blocks steady current.

X_C = 1 / (2πfC)

Put f = 0 into that and you get infinity, which is “blocks DC”. Put f = 1 kHz and C = 10 µF into it and you get 15.9 Ω, which is “passes AC”. Two famous claims, one formula, no magic.

Which makes the coupling capacitor a filter

A coupling capacitor is not a gate that is open or shut. It is in series with whatever resistance follows it, so the two form a divider — a high-pass filter. At high frequencies X_C is tiny and everything gets through. At low frequencies X_C grows until it is comparable with the resistance, and the signal starts to disappear.

The turnover is where X_C equals the resistance:

f_−3dB = 1 / (2πRC)

Drag the capacitor value on the bench and watch the whole response curve slide left and right. Make it too small and you have built a circuit that works fine on a test tone at 1 kHz and has no bass at all.

And it does a second job

The coupling capacitor is also what makes bias possible at all. The previous stage might sit at 2 V and yours might need 3 V. Wire them directly and the two bias networks fight. Put a capacitor between them and each side keeps its own DC level while the signal walks straight across. That is why the two ideas — bias and coupling — always turn up together.

Chapter 2, lesson 5 told you to put a resistor in the emitter, because it makes the bias independent of β and stops thermal runaway. It is not optional.

It also costs you almost all of your gain. A common-emitter stage with an emitter resistor has a voltage gain of roughly

A_v = −R_C / (R_E + r_e)

With R_C = 4.7 kΩ and R_E = 1.2 kΩ, that is a gain of about −3.8. You have a transistor capable of nearly two hundred, and a resistor is throwing away almost all of it.

Why it works, in the language of lesson 3

1. In the DC drawing the capacitor is an open circuit. The emitter resistor is fully present, setting the current and holding the bias steady against β and temperature. Everything chapter 2 promised still holds.

2. In the AC drawing the capacitor is a wire. It shorts the emitter resistor out completely. The emitter is at signal ground, and the gain becomes −R_C/r_e.

The same component is present in one drawing and absent from the other. If you needed one demonstration that the two-drawing method is describing something real rather than being a study aid, this is it.

The catch, which is why this is not always done

Gain of −R_C/r_e sounds wonderful until you notice what r_e is made of. It is 25 mV divided by a current, and that 25 mV is kT/q — it is proportional to absolute temperature. So the gain of a fully bypassed stage drifts with temperature, varies from device to device, and is wildly non-linear for large signals, because r_e changes as the current swings.

Leave a small resistor unbypassed — say 100 Ω of the 1.2 kΩ — and the gain becomes −R_C/(100 + r_e). You have given up most of the gain you just gained, and bought back predictability and low distortion. Almost every real amplifier does exactly this. Try the partial-bypass setting on the bench and watch the numbers trade off.

That trade — gain for linearity, by feeding some of the output back — is negative feedback again, the same idea as chapter 3, lesson 4. You will meet it a third time as the thing op-amps are made of.

Everything in this chapter, used once, in the order a designer actually uses it. Follow along on the bench — it recalculates the whole circuit as you move each decision, and warns you when a choice breaks something you settled earlier.

The six decisions

1. Pick the quiescent current. Higher gives more gain and more drive; lower wastes less power. 1 mA is a sensible small-signal default and the number this bench starts at.

2. Pick R_C so the collector sits near half the supply. R_C = (V_CC/2) / I_C. On 12 V at 1 mA that is 6 kΩ, so 5.6 kΩ from the shelf. That was lesson 2.

3. Pick R_E for a stable bias. Put about a tenth of the supply across it — 1.2 V at 1 mA means 1.2 kΩ. Enough that the ±60 mV of Vbe variation between devices is a small fraction of it.

4. Pick the divider. The base must sit at V_E + 0.65 ≈ 1.85 V. And its current should be about ten times the base current, so the divider holds firm regardless of β. That was chapter 2, lesson 5.

5. Pick the bypass. Fully bypassed for maximum gain, or leave 100 Ω unbypassed for a gain you can predict. Lesson 5.

6. Pick the capacitors so every turnover sits well below your lowest frequency. Lesson 4.

The three ways it goes wrong

Push the sliders until the bench complains, then work out which decision you would undo. That is what design actually is — not solving equations, but noticing which constraint you broke.