How a transistor amplifies: biasing a common-emitter stage

What you're hearing

The base is a tap, and the collector current is the flow. A transistor has three legs: the base, the collector and the emitter. A small voltage between the base and the emitter decides how much current flows in at the collector and out at the emitter. The law is exponential: every 26 mV more at the base multiplies the current by e (2.7), so every 60 mV multiplies it by 10. Put a resistor between the supply and the collector, and that current becomes a voltage across it, by Ohm's law. Here it's 4.7 kΩ: 1 mA through it drops 4.7 V. So a few millivolts at the base change the current a little, and the 4.7 kΩ turns that into volts at the collector. More current means a bigger drop, so the collector falls as the base rises: the output is upside down.

Biasing sets where it rests, and the emitter resistor holds it there. The exponential law has no useful middle on its own. At 0.6 V between base and emitter, this transistor passes about 0.08 mA; at 0.72 V, about 8 mA, a hundred times more, while the 4.7 kΩ can pass at most 1.9 mA from 9 V. So the stage needs a steady resting point, its bias, with the sound riding on top. Two resistors make a voltage divider on the base: 100 kΩ to the 9 V supply and 22 kΩ to ground, about 1.62 V unloaded. The base draws 6 µA, which pulls it to 1.51 V. The emitter sits 0.66 V below, at 0.85 V, so 0.85 V / 1 kΩ = 0.85 mA flows. The collector rests at 9 V − 0.84 mA × 4.7 kΩ = 5.0 V. The 10 µF capacitor at the input lets the sound through but keeps the bias from leaking back into whatever drives it. The emitter resistor keeps the bias steady: the emitter has to stay about 0.66 V below the base, so the 1 kΩ sets the current, whatever the transistor itself is like. Two transistors with the same part number can differ in gain two to one or more. Halve this one's gain in the model and the current falls only 3.6%; double it and the current rises 1.9%. Bias the same transistor with just one big resistor to its base, and the same change moves the current by 25% and 21%, and the collector by about a volt. The emitter resistor is feedback: if the current rises, the emitter rises with it, and the base has less to push with.

The gain is about Rc / Re, or much more with the emitter bypassed. With the 1 kΩ in the circuit, the emitter follows the base, so the whole input appears across the 1 kΩ. Then the gain is the collector resistor over the emitter resistor: 4.7 kΩ / 1 kΩ = 4.7. The 100 kΩ load and the transistor's own 31 Ω bring it to 4.3. The 100 µF across the 1 kΩ, the bypass capacitor, holds the emitter still for audio. Now the whole input lands on the base-emitter junction itself, and the gain is its transconductance gm, current per volt, times the collector's load: gm = 0.84 mA / 25.9 mV = 32.6 mS, and 32.6 mS × 4.5 kΩ ≈ 146. The circuit gives 137 at 1 kHz, mostly because a transistor's collector isn't a perfect current source (the Early effect). Below about 50 Hz the capacitor stops holding the emitter still, so deep bass gets less: 98 at 55 Hz.

The bypassed gain follows the current, because gm does. Without the bypass it hardly moves until the transistor runs out of room: that's what feedback buys.

The bypassed stage is big but lopsided. At 5 mV in, the top of the output comes out 5% short and the bottom 4% too big, because the current grows faster going up than it shrinks going down. That's already a 2nd harmonic 27 dB under the note. At 10 mV the top is 10% short, and by about 30 mV the collector reaches its floor, about 1 V, where the transistor is fully on (saturated). At 25 mV the output swings +2.5 V and −4.1 V, and the 2nd harmonic is only 14 dB under the note. Without the bypass, the second button needs 0.15 V for the same 0.65 V out, and its 2nd harmonic is 58 dB under the note: the feedback trades gain for a cleaner stage.

Pushed, it runs out of room sooner one way, and that's even harmonics. Without the bypass, the collector can rise towards 9 V, where the transistor turns off, and fall to about 1.7 V, where it's saturated. That floor is higher than the bypassed stage's 1 V, because the emitter now rises with the current. Resting at 5.0 V, it has about 3.8 V of room up and 3.3 V down. That's why builders aim the collector near the middle, often called half the supply: the middle of the room, not of the supply. Here the floor is about 1.7 V, so the middle is a little over 5 V, not 4.5 V. At 0.7 V in, without the bypass, the centred stage is clean: 0.9% distortion. Push it further and the bottom is 10% short at 0.85 V in, the top at about 0.97 V. Off centre, one side flattens first. Turn bias up to 68 kΩ and the collector rests at 2.6 V, only 0.9 V above its floor. By about 0.23 V in, the bottom of the output is 10% short; at 0.7 V it stops at −1.0 V while the top swings 4.2 V. Turn bias down to 150 kΩ and the collector rests at 6.9 V, 2 V from the top: at 0.7 V the top flattens at +2.0 V while the bottom reaches −3.0 V. A wave flattened on one side only is no longer the same shape both ways, and that shows up as even harmonics. The 2nd is 20 to 22 dB under the note, with the 3rd close behind (11–12% distortion in all). This lopsided clipping is the sound of many simple transistor boosts and preamps pushed hard. A differential pair is two of these transistors sharing one current, which makes the curve the same both ways, so it adds mainly odd harmonics instead.

What the model covers, and what it leaves out. The 2N3904 model is the published one, with the Early effect and the base's leakage. It has no temperature, noise or internal capacitances. A real transistor's base voltage falls as it warms, and its gain changes, so the bias drifts; the emitter resistor is what holds it. SynthMaker's transistors stay at 27 °C.

Try this

• Press "Bypassed: 5 mV in", then "No bypass: 30 times the input, cleaner". The two look the same size on the Scope, but the second took 30 times the input. Turn input on the first button from 5 mV up to 25 mV and watch the bottom grow and the top lag behind.
• Press "0.7 V, bias centred", then the two other 0.7 V buttons. The same input clips at the bottom with the bias too high and at the top with it too low. On the centred button, turn bias slowly both ways: by 90 kΩ one way or 120 kΩ the other, the distortion has risen from 0.9% to 2–2.5%, and by 82 kΩ or 130 kΩ to about 5%.
• On the first button, turn bias from 220 kΩ up to 68 kΩ. The note gets louder as the current rises, the gain from 29 to 212. At 47 kΩ the transistor is saturated, its collector stuck at 1.7 V, and the gain falls to 19. Open it in SynthMaker to change the parts: change Ce e gnd 100u to Ce e x 100u and add Rx x gnd 220. Now only part of the emitter resistor is bypassed, and the gain lands in between, at 21, with the same resting point. That's how many boosts set their gain.

The parts list

circuit Stage  setting input: 0.005 V, from 0.001 V to 1 V  setting output: 10 V  setting bias: 100k, from 220k to 47k  V1 vcc gnd 9V  C1 in b 10u  R1 vcc b bias  R2 b gnd 22k  Rc vcc c 4.7k  Re e gnd 1k  Ce e gnd 100u  Q1 c b e 2N3904  C2 c out 10u  Rl out gnd 100kend

The bias knob turns R1, written from 220k down to 47k, so turning it up means a smaller resistor and more current. The four no-bypass buttons put a # in front of Ce, which takes it out.

How we checked it

Checked against ngspice, a standard circuit simulator: this is the circuit of SynthMaker's common-emitter check. With its emitter bypassed and a 5 mV, 1 kHz sine in, it rests within 0.19 µV of ngspice and plays within 0.01 mV rms of a 0.71 V peak at the 2× rate it runs at. The bias knob, the no-bypass buttons and the clipping aren't part of that check. Its pushed buttons, clipping at the top or the bottom, are within 0.02% of ngspice across the audible band.

Build it step by step

• The manual: Circuit demos. A one-transistor germanium treble booster among the example sounds is this same stage, emitter bypassed.
• The circuit format: Start a circuit, for writing transistors in a parts list.

Next and previous

• This is the first page of chapter 5, transistors and valves.

All circuits

Further reading

• The stage itself, its gain and the bypass capacitor: Wikipedia: Common emitter.
• Divider bias, emitter feedback and why the resting point drifts: Wikipedia: Bipolar transistor biasing.
• The transistor's exponential law and saturation: Wikipedia: Bipolar junction transistor.
• Why the collector isn't a perfect current source: Wikipedia: Early effect.
• The transistor model SynthMaker and ngspice use: Wikipedia: Gummel–Poon model.
• How distortion is measured as harmonics: Wikipedia: Total harmonic distortion.
• D. T. Yeh, "Digital Implementation of Musical Distortion Circuits by Analysis and Simulation", PhD thesis, Stanford University, 2009 (the DK method). PDF
• M. Holters and U. Zölzer, "A Generalized Method for the Derivation of Non-Linear State-Space Models from Circuit Schematics", EUSIPCO 2015. PDF