Appearance
2.3 — The Bipolar Junction Transistor
In December 1947, John Bardeen and Walter Brattain at Bell Labs pressed two gold contacts against a piece of germanium and found that a small current in one contact controlled a much larger current in the other. William Shockley, annoyed at being left off the discovery, worked out the theory of the junction transistor over the following weeks and published the design that actually got manufactured. All three shared the 1956 Nobel Prize, and Bell Labs' patent attorneys chose the name from "transresistance", because the device transfers current across a resistance.
The one fact that makes it work: a small base current controls a large collector current. Everything else in this chapter is the consequence.
This chapter stands entirely on Chapter 2.1's PN junction. A transistor is two of them sharing a middle layer, and every behaviour below — the 0.7 V turn-on, the temperature drift, the one-way action — is a junction property you have already met, now appearing twice.
The letters this chapter uses, said out loud
Transistor datasheets are dense with subscripted symbols, and the subscripts follow one rule that makes all of them readable.
The subscript rule. Two letters means a voltage between those two terminals, in that order: V_{BE} is the voltage at the base minus the voltage at the emitter. One letter means a current into that terminal: I_C is the current into the collector. Learn that and most of a datasheet becomes readable without a key.
\beta — "beta", the current gain, also written h_{FE} on datasheets. It is the ratio of collector current to base current, a pure number typically between 50 and 400. It is not a design parameter — it varies by a factor of three between two transistors of the same part number and drifts with temperature, which is why section 5's biasing schemes are all built to work regardless of its value.
\alpha — "alpha", the fraction of emitter current that reaches the collector rather than leaking out of the base. It is just under 1, typically 0.99, and \beta = \alpha/(1-\alpha) links the two. That relationship is why a small change in \alpha produces a large change in \beta, and therefore why \beta is so unreliable.
V_{BE} — the base-emitter voltage, about 0.7 V for silicon when conducting. This is a PN junction's forward drop from Chapter 2.2, and it falls by roughly 2 mV for every degree the transistor warms, which is the cause of the thermal runaway discussed in section 5.
V_{CE(sat)} — the collector-emitter voltage when the transistor is switched fully on, around 0.2 V. It sets the loss when the device is used as a switch.
I_C, I_B, I_E — collector, base and emitter currents, related by I_E = I_C + I_B, which is just Kirchhoff's current law from Chapter 1.2 applied to the device as a node.
V_T — the thermal voltage, about 26 mV at room temperature. It is kT/q: Boltzmann's constant times absolute temperature divided by the electron's charge. It is not a property of the transistor at all but of physics at that temperature, and it sets the steepness of every exponential in this chapter.
1. The structure, and the counter-intuitive bit
A bipolar junction transistor is three doped regions in a row: emitter, base, collector. An NPN has n-type emitter, p-type base, n-type collector. A PNP is the mirror image. NPN is far more common, because electrons are three times more mobile than holes (Chapter 2.1), so an NPN is faster and can carry more current for the same size.
Two junctions, so surely it is just two diodes back to back? No, and the reason is the reason the device exists. Wire two separate diodes together and you get nothing useful. The transistor works because of two deliberate asymmetries in how the three regions are built:
- The base is extremely thin — under a micrometre in modern devices — and lightly doped.
- The emitter is very heavily doped, typically a hundred times more than the base.
Those two choices are the invention. Here is what they buy.
Walk through the operation
Forward bias the base-emitter junction, so it conducts like an ordinary diode from Chapter 2.2 and needs about 0.7 V.
Step 1. Because the emitter is heavily doped and the base lightly doped, the current across that junction is overwhelmingly electrons flowing from emitter into base, rather than holes flowing the other way. Doping asymmetry decides which carrier dominates, and the design deliberately makes it electrons.
Step 2. Those electrons are now in p-type material, where they are minority carriers surrounded by holes. Normally they would recombine within a microsecond or so.
Step 3. But the base is under a micrometre thick, so an electron crosses it in a fraction of that time. Only about one in a hundred finds a hole. Ninety-nine percent arrive at the far edge of the base still free.
Step 4. The base-collector junction is reverse biased, so its field points in the direction that sweeps electrons from the base into the collector. For a minority electron in the base, that field is an escalator going the right way. Every electron that reaches it is collected.
The result. I_C \approx 99\% of the emitter current; I_B \approx 1\%. And crucially, the collector current is set by how many electrons the emitter injects, which is set by V_{BE} — and it does not depend on the collector voltage at all, as long as that junction stays reverse biased. That last clause is why the collector behaves like a current source, which is exactly what a good amplifier needs.
2. The current relationships
By KCL, all current entering leaves:
I_E = I_C + I_B
Define the two gain parameters:
\alpha = \frac{I_C}{I_E} \approx 0.99 \ \text{(rarely used directly)}
\boxed{\beta = \frac{I_C}{I_B}} \qquad\text{typically 50 to 400}
\beta (beta) is also written h_{FE} on datasheets. They relate as
\beta = \frac{\alpha}{1-\alpha}, \qquad \alpha = \frac{\beta}{\beta+1}
Notice how brutal that relationship is. \alpha = 0.99 gives \beta = 99; \alpha = 0.995 gives \beta = 199. A half-percent change in \alpha doubles \beta, which is why \beta varies enormously between individual transistors of the same part number — a 2N3904's datasheet quotes a range of 100 to 300, and temperature moves it further.
The design rule that follows: never build a circuit whose behaviour depends on \beta. Section 5 is entirely about how to obey that rule.
Also useful:
I_E = (\beta+1)I_B, \qquad I_C = \beta I_B
3. The three operating regions
The two junctions can each be forward or reverse biased, giving four combinations. Three of them matter.
| Region | B-E junction | B-C junction | Behaviour | Used for |
|---|---|---|---|---|
| Cutoff | reverse | reverse | no current, open switch | logic 0 |
| Active | forward | reverse | I_C = \beta I_B, current source | amplifying |
| Saturation | forward | forward | V_{CE} ≈ 0.2 V, closed switch | logic 1 |
Cutoff. V_{BE} below about 0.5 V. Nothing conducts, I_C \approx 0, and the collector sits at the supply voltage. The transistor is an open switch.
Active. V_{BE} \approx 0.7 V and V_{CE} comfortably above V_{BE}. The collector current is controlled by the base and is nearly independent of the collector voltage. This is the amplifying region, and every analog circuit in this chapter lives here.
Saturation. You have supplied more base current than the collector circuit can use. The collector voltage drops until the base-collector junction goes forward biased too, and then V_{CE} bottoms out at V_{CE(sat)}, about 0.2 V. The transistor is now a closed switch and I_C is set entirely by the external circuit, not by \beta.
There is a fourth region, reverse active — both junctions swapped — where the device works backwards with a \beta of about 2 or 3 because the asymmetries are all the wrong way round. It is useless, but knowing it exists explains why a transistor's collector and emitter cannot be interchanged.
The output characteristic
Plot I_C against V_{CE} for several fixed base currents and you get the family of curves every textbook shows. Each curve rises steeply from the origin (saturation), then bends over and runs almost flat (active). The flat region is where the transistor is a current source; its slight upward slope is the Early effect, caused by the collector's depletion region widening into the base as V_{CE} rises, making the base effectively thinner and \beta slightly higher. It is quantified by the Early voltage V_A, typically 50 to 150 V, and it sets a ceiling on how much gain a single stage can produce.
4. The transistor as a switch
Start here, because it is easier than amplifying and it is what most transistors in the world actually do.
The design procedure, in order:
- Decide the load current I_C you need. A relay coil, an LED, a motor.
- Choose a base current with deliberate excess — a factor of 5 to 10 more than I_C/\beta. This is called overdriving the base, and it forces saturation regardless of which end of the \beta range your particular transistor sits at.
- Compute the base resistor from the driving voltage.
Worked example. Switch a 12 V relay drawing 80 mA, driven from a 3.3 V microcontroller pin. Transistor \beta_{min} = 100.
Minimum base current for the job:
I_{B(min)} = \frac{80\ \text{mA}}{100} = 0.8\ \text{mA}
Overdrive by 5:
I_B = 4\ \text{mA}
Base resistor — the microcontroller pin is at 3.3 V and V_{BE} is 0.7 V, so 2.6 V is across the resistor:
R_B = \frac{3.3-0.7}{0.004} = 650\ \Omega
Use 560 Ω (the nearest standard value below, giving a bit more drive). Check the microcontroller can supply it: (3.3-0.7)/560 = 4.6 mA, which is within a typical 20 mA pin limit ✓.
Power in the transistor when on: V_{CE(sat)}\times I_C = 0.2\times0.08 = 16 mW. Negligible — this is why saturation is used for switching. In the active region the same transistor might be dropping 6 V at 80 mA, which is 480 mW and needs a heatsink.
And fit the flyback diode across the relay coil, cathode to the positive supply, for the reason Chapter 1.5 worked out in numbers. Without it the transistor will not survive many operations.
Speed, and why saturation costs you
A saturated transistor has excess minority carriers stored in its base. Turning it off means sweeping them out, which takes a storage time of hundreds of nanoseconds. For a relay at 10 Hz this is irrelevant. For a switching supply at 100 kHz it is fatal, and the fix is either a Schottky clamp diode from base to collector that prevents full saturation, or using a MOSFET instead (Chapter 2.4), which has no minority carrier storage at all.
5. Biasing for amplification
To amplify, the transistor must sit in the active region at rest, so that the signal can swing both up and down without leaving it. Setting up that resting condition is biasing, and the resting values of I_C and V_{CE} are the quiescent point or Q-point.
The wrong way: fixed bias
One resistor from the supply to the base:
I_B = \frac{V_{CC}-0.7}{R_B}, \qquad I_C = \beta I_B
Simple, and unusable. I_C is directly proportional to \beta, which varies 3:1 between samples and rises with temperature. Build a hundred of these and they will bias anywhere from cutoff to saturation.
Worse, there is a runaway loop: temperature rises, so \beta rises and V_{BE} falls by 2 mV per °C, so I_C rises, so power dissipation rises, so temperature rises further. Thermal runaway, and germanium devices used to destroy themselves this way routinely.
The right way: voltage-divider bias
Four resistors, and it is the standard for a reason.
How it achieves stability, in four steps:
- R1 and R2 form a divider holding the base at a fixed voltage V_B, essentially independent of the transistor.
- V_E = V_B - 0.7, also fixed.
- I_E = V_E/R_E — fixed by two resistors and nothing else. \beta does not appear.
- Since I_C \approx I_E, the collector current is stable.
And it self-corrects: if I_C tries to rise with temperature, V_E rises, so V_{BE} = V_B - V_E falls, which reduces I_C again. That is negative feedback, appearing here for the first time in the volume, and Part 6 is about nothing else.
The design rule: make the divider current about ten times I_B, so that base current does not disturb the divider. Then the approximation in step 1 is honest.
Worked bias design
Target: I_C = 2 mA, V_{CC} = 12 V, \beta \approx 150.
Step 1 — allocate the supply voltage. A good default is V_E \approx 1 V (enough to swamp V_{BE}'s temperature drift), V_{CE} \approx 6 V (mid-rail, so the output can swing both ways), leaving 5 V across R_C.
Step 2 — emitter resistor.
R_E = \frac{1\ \text{V}}{2\ \text{mA}} = 500\ \Omega \;\to\; \text{use } 470\ \Omega
Step 3 — collector resistor.
R_C = \frac{5\ \text{V}}{2\ \text{mA}} = 2.5\ \text{k}\Omega \;\to\; \text{use } 2.4\ \text{k}\Omega
Step 4 — base voltage needed.
V_B = V_E + 0.7 = 1.7\ \text{V}
Step 5 — divider. Base current is $2\ \text{mA}/150 = 13.3\ \mu$A, so make the divider carry about 133 µA:
R_2 = \frac{1.7}{133\ \mu\text{A}} = 12.8\ \text{k}\Omega \;\to\; 12\ \text{k}\Omega
R_1 = \frac{12-1.7}{133\ \mu\text{A}} = 77\ \text{k}\Omega \;\to\; 75\ \text{k}\Omega
Step 6 — verify. With 75 k and 12 k:
V_B = 12\cdot\frac{12}{87} = 1.655\ \text{V}, \quad V_E = 0.955\ \text{V}, \quad I_E = \frac{0.955}{470} = 2.03\ \text{mA}
V_C = 12 - (2.03\ \text{mA}\times2400) = 12-4.87 = 7.13\ \text{V}, \quad V_{CE} = 7.13-0.955 = 6.17\ \text{V}
Comfortably in the active region with room to swing ✓.
Now test the design's robustness. Repeat with \beta = 50 instead of 150. Base current becomes 40 µA, which pulls the divider down slightly, giving V_B \approx 1.61 V and I_C \approx 1.94 mA — a 4% change for a threefold change in \beta. That is why this circuit is the standard.
6. Small-signal analysis
Once biased, treat the signal separately from the DC. This is superposition (Chapter 1.4) applied to a device that is only approximately linear, and it is legitimate as long as the signal is small enough that the curve looks straight over its range.
The key parameter is the transconductance, how much collector current changes per volt of base-emitter change. Differentiating the exponential relationship I_C = I_Se^{V_{BE}/V_T}:
g_m = \frac{dI_C}{dV_{BE}} = \frac{I_C}{V_T}
\boxed{g_m = \frac{I_C}{26\ \text{mV}}}
At 1 mA, g_m = 38.5 mA/V. Notice what this says: transconductance depends only on the bias current, not on the transistor. Two completely different transistors biased at the same current have the same g_m. That is a remarkable and useful fact, and it is why BJT circuits are designed around currents.
The input resistance looking into the base:
r_\pi = \frac{\beta}{g_m} = \frac{\beta V_T}{I_C}
and the emitter resistance seen looking into the emitter:
r_e = \frac{V_T}{I_E} \approx \frac{26\ \text{mV}}{I_E}
At 1 mA, r_e = 26\ \Omega. This little number is the single most useful quantity in BJT design.
Common-emitter gain
With the emitter resistor bypassed by C_E (so it is a short circuit for signals):
A_v = -g_mR_C = -\frac{R_C}{r_e}
The minus sign means inversion — the output is upside down relative to the input, because more base current pulls the collector down towards ground.
For the worked design above, I_C = 2 mA gives r_e = 13\ \Omega, and R_C = 2.4 kΩ:
A_v = -\frac{2400}{13} = -185
A gain of 185. But it depends on V_T, which drifts with temperature, and on the exact bias current. Leave the emitter resistor unbypassed and the gain becomes
A_v = -\frac{R_C}{r_e+R_E} \approx -\frac{R_C}{R_E} = -\frac{2400}{470} = -5.1
Far less gain, but now set by two resistors and therefore stable, predictable and much more linear. That trade — gain for predictability — is the fundamental bargain of feedback, and Part 6 formalises it. A common compromise is to bypass only part of the emitter resistance, splitting it into two.
7. The three configurations
Any of the three terminals can be the common reference, giving three circuits with sharply different personalities.
| Common emitter | Common collector | Common base | |
|---|---|---|---|
| Voltage gain | high, inverting | ≈ 1 | high, non-inverting |
| Current gain | high (β) | high (β+1) | ≈ 1 |
| Input impedance | medium (~1 kΩ) | high (~100 kΩ) | very low (~26 Ω) |
| Output impedance | high | very low (~ohms) | very high |
| Typical use | general amplifier | buffer, output stage | RF front end |
Common emitter
The workhorse. Signal in at the base, out at the collector, emitter grounded (or grounded through the bypass capacitor). Both voltage and current gain, and it inverts.
Common collector — the emitter follower
Signal in at the base, out at the emitter, collector tied to the supply. The emitter voltage simply follows the base voltage, 0.7 V below it. Gain of one — so why bother?
Because of the impedances. Looking into the base you see
R_{in} = \beta(r_e + R_E) \approx \beta R_E
which for \beta = 150 and R_E = 1 kΩ is 150 kΩ. Looking back into the emitter you see
R_{out} = r_e + \frac{R_{source}}{\beta}
which is typically a few tens of ohms.
So the follower takes a signal from a source that cannot supply current, and hands it on to a load that demands current. It has a current gain of \beta and a power gain to match, even though the voltage gain is one. Every audio output stage, every voltage regulator's pass element, and every place a high-impedance signal meets a low-impedance load has one of these.
That is also the reason for its name: the emitter follows the base, and now you know exactly why — the base-emitter junction is a diode holding a near-constant 0.7 V, so whatever the base does, the emitter does 0.7 V lower.
Common base
Signal in at the emitter, out at the collector, base grounded. Input impedance is just r_e, a few tens of ohms, which sounds useless until you want to receive a signal from a 50 Ω coaxial cable (Chapter 7.7) — then it is exactly right. It has no inversion and excellent high-frequency behaviour, because it avoids the Miller effect, in which the collector-base capacitance of a common-emitter stage is multiplied by the voltage gain and destroys the bandwidth.
The cascode, a common-emitter stage feeding a common-base stage, gets the common-emitter's gain and the common-base's bandwidth together, and it is the standard high-frequency amplifier building block.
8. The current mirror, and the Darlington
Two configurations worth knowing because they are everywhere inside integrated circuits.
The current mirror. Connect one transistor's base to its own collector (a "diode-connected" transistor) and tie its base to a second transistor's base. Both have the same V_{BE}, so both pass the same collector current. Set the current in the first with a resistor and the second copies it, regardless of what voltage its collector sits at. That is a near-ideal current source, and it is how every op-amp inside Chapter 2.5 biases itself. It works only because the two transistors are on the same chip at the same temperature and were made in the same process — which is precisely what an integrated circuit provides and a bag of discrete parts does not.
The Darlington pair. Two transistors with the first's emitter feeding the second's base. The overall gain is the product:
\beta_{total} \approx \beta_1\beta_2
so two devices with \beta = 100 give 10,000. The price is two base-emitter drops in series, so the pair needs 1.4 V to turn on, and V_{CE(sat)} rises to about 0.9 V because the first transistor's saturation voltage sits above the second's junction. Used where enormous current gain matters more than voltage headroom — power drivers, and the classic TIP120.
9. What actually destroys transistors
Four failure modes, in rough order of how often they catch people.
Exceeding the power rating. The limit is P = V_{CE}I_C, and datasheets give a maximum with a derating curve — the rating is quoted at 25 °C case temperature and falls to zero at the maximum junction temperature. A "1 W" transistor in free air with no heatsink can usually dissipate about 0.4 W before it cooks.
Exceeding V_{CEO}. The collector-emitter breakdown voltage. Inductive loads are the usual culprit, for exactly the reason Chapter 1.5 calculated.
Thermal runaway, described in section 5. The emitter resistor is the standard cure; in power stages, small emitter resistors in each device also force current to share between parallel transistors.
Reverse base-emitter breakdown. The base-emitter junction breaks down at only about 6 V in reverse and is permanently degraded by it — \beta drops and never recovers. This is easy to do accidentally with an AC-coupled input, and the standard protection is a diode across the junction.
The BJT is controlled by base current, which means the driving circuit must supply real power. The next chapter's device is controlled by a voltage on an insulated gate that draws essentially no current at all, and that single difference is why it, not the BJT, is what a modern processor is made of.
Every formula above, built from scratch
None of the results in this chapter are worth memorising, because each one can be rebuilt in under a minute from something simpler. What follows is that rebuilding, one result at a time, so the formula and the reason for it sit on the same page as the explanation that needed them.
The BJT
I_E = I_C+I_B, \qquad \alpha = \frac{I_C}{I_E}, \qquad \beta = \frac{I_C}{I_B}
\beta = \frac{\alpha}{1-\alpha}, \qquad \alpha = \frac{\beta}{\beta+1}, \qquad I_E = (\beta+1)I_B
Derivation of the \beta–\alpha link: divide I_E = I_C+I_B through by I_C to get 1/\alpha = 1 + 1/\beta, then rearrange.
Collector current from the junction physics:
I_C = I_Se^{V_{BE}/V_T}
— the same exponential as the diode, because the base-emitter junction is a diode.
Small-signal parameters
g_m = \frac{dI_C}{dV_{BE}} = \frac{I_C}{V_T}
r_\pi = \frac{\beta}{g_m} = \frac{\beta V_T}{I_C}, \qquad r_e = \frac{V_T}{I_E} \approx \frac{26\ \text{mV}}{I_E}, \qquad r_o = \frac{V_A}{I_C}
V_A is the Early voltage, 50 to 150 V typically.
Bias design
Voltage-divider bias:
V_B = V_{CC}\frac{R_2}{R_1+R_2}, \qquad V_E = V_B-0.7, \qquad I_E = \frac{V_E}{R_E}, \qquad V_{CE} = V_{CC}-I_C(R_C+R_E)
Why \beta is absent: the divider is designed so that its current is roughly ten times I_B, so the base draws a negligible share and V_B is set by resistors alone.
Rule of thumb targets: V_E \approx 0.1V_{CC}, V_{CE} \approx 0.5V_{CC}, divider current \approx 10I_B.
Gains by configuration
A_{v(CE)} = -\frac{R_C}{r_e} \ \text{(emitter bypassed)}, \qquad -\frac{R_C}{r_e+R_E} \ \text{(unbypassed)}
A_{v(CC)} = \frac{R_E}{r_e+R_E} \approx 1, \qquad A_{v(CB)} = +\frac{R_C}{r_e}
Impedances:
R_{in(CE)} = R_1\|R_2\|r_\pi, \qquad R_{in(CC)} = \beta(r_e+R_E), \qquad R_{in(CB)} = r_e
R_{out(CC)} = r_e + \frac{R_{source}}{\beta}
Switching and multi-device
I_{B} \ge 5\ \text{to}\ 10\times\frac{I_C}{\beta_{min}}, \qquad R_B = \frac{V_{drive}-0.7}{I_B}
\beta_{Darlington} \approx \beta_1\beta_2, \qquad V_{BE(total)} = 1.4\ \text{V}, \qquad V_{CE(sat)} \approx 0.9\ \text{V}