Bipolar transistor: DC, small signal and switching
All the bipolar transistors in the world have the same transconductance at 1 mA: 38.7 mS. Their β, on the other hand, varies from 100 to 300 between two samples of the same type.
01Two junctions very close together
An NPN bipolar transistor is two PN junctions back to back, but it is not the same as two diodes: its trick is that the middle base is much thinner than the diffusion length of the electrons. With the base-emitter junction forward biased, the emitter injects electrons into the base; almost none of them recombine, and on reaching the edge of the base-collector junction, which is reverse biased, the field of that junction sweeps them into the collector. A small base current controls a large collector current.
The geometry and the doping levels are designed for that: the emitter heavily doped, so that the current of the base-emitter junction is carried almost entirely by electrons (in the PN junction the current is dominated by the less doped side), the base thin and lightly doped, and the collector lightly doped and wide, to withstand voltage. The Electronics Technician program page on bipolar transistors covers their use in circuits; here we look inside.
02Where the gain comes from
The emitter current has two components: electrons that go to the base, which are the useful ones, and holes that the base returns to the emitter, which are lost. Of the electrons, in addition, some recombine in the base. Two factors measure those losses:
With a 0.5 µm base and a diffusion length of 20 µm, \( \alpha_T = 0.9997 \): the base alone would allow a \( \beta \) of more than 3000. What limits the actual gain, 100 to 300 in a small-signal transistor, is the injection efficiency, which is why the emitter is doped a hundred times more than the base. Since \( \beta \) comes from a ratio of doping levels and thicknesses, it varies a lot between transistors of the same type: the 2N3904 datasheet allows 100 to 300 at 10 mA. A well-designed circuit does not depend on the exact value of β.
03Operating modes and the Ebers–Moll model
| Mode | Base-emitter | Base-collector | Use |
|---|---|---|---|
| Forward active | Forward | Reverse | Amplification |
| Saturation | Forward | Forward | Closed switch |
| Cutoff | Reverse | Reverse | Open switch |
| Reverse active | Reverse | Forward | Almost never: \( \beta_R \) is less than 1 |
The Ebers–Moll model describes all four modes with a single pair of equations. In its transport form, with \( V_{BC} = V_{BE} - V_{CE} \):
The Early effect completes the picture: as \( V_{CE} \) rises, the depletion region of the base-collector junction eats into the base, the effective base narrows and \( I_C \) grows. It is modeled with a factor \( (1 + V_{CE}/V_A) \), where the Early voltage \( V_A \) ranges from 50 to 150 V in small-signal transistors.
The transistor has Ebers–Moll parameters of the order of a 2N3904 (\( I_S = 6.7 \) fA, \( \beta_F = 200 \), \( \beta_R = 0.74 \), \( V_A = 74 \) V). The curves are \( I_C \) versus \( V_{CE} \) for base currents from 20 to 100 µA. You choose the supply, the collector resistor and the base current: the operating point lies where the load line crosses the curve for that current.
04Small-signal model
Around the operating point in forward active mode, the transistor is replaced by the hybrid-π model. All of its parameters come from the collector current:
The transconductance depends only on the current and the temperature, not on the transistor: all bipolar transistors in the world have \( g_m = 38.7 \) mS at 1 mA. And the product \( g_m r_o = V_A / V_T \) is the intrinsic gain, the maximum voltage gain that a single transistor can give: with \( V_A = 74 \) V, about 2900 times, almost 70 dB. The transition frequency \( f_T \), where the current gain falls to 1, is 300 MHz in the 2N3904 at 10 mA.
05The transistor as a switch
In saturation the transistor is a closed switch with a drop of 0.1 to 0.3 V: the 2N3904 guarantees \( V_{CE(sat)} \leq 0.2 \) V with 10 mA of collector current and 1 mA of base current. To ensure saturation the base is overdriven, giving it considerably more current than \( I_C / \beta \); the ratio \( I_C / I_B \) in saturation is called forced β, and it is designed to be between 10 and 20.
The cost of overdriving is the stored charge: with both junctions forward biased, the base fills with excess minority carriers, and when the base current is cut off the transistor keeps conducting until that charge is removed. This is the storage time, which in the 2N3904 reaches 200 ns compared with a 35 ns rise time. It is fought with:
- A Schottky diode between base and collector (the “Baker clamp”): it diverts the excess base current and keeps the base-collector junction from becoming forward biased. It is the S in 74LS logic.
- A speed-up capacitor in parallel with the base resistor, which gives a current spike on turn-on and turn-off.
- Not saturating: ECL logic always works in the active region, and for decades it was the fastest family.
06The limits of the device
| Limit | What causes it |
|---|---|
| \( BV_{CEO} \) lower than \( BV_{CBO} \) | With the base open, the carriers of the incipient avalanche act as base current and the transistor amplifies them: in the 2N3904, 40 V versus 60 V |
| Maximum power | The junction temperature: \( P = V_{CE} I_C \) and the thermal resistance to ambient |
| Second breakdown | A hotter spot on the junction conducts more, heats up more, and the current concentrates until it melts it; that is why the safe operating area is cut back at high voltage |
| β drop at high current | High injection in the base (the Webster effect) and the expansion of the base into the collector (the Kirk effect) |
07Where it appears in electronics
| Case | Why a bipolar |
|---|---|
| Precision operational amplifiers | Better \( V_{BE} \) matching than the MOS threshold matching, and less low-frequency noise |
| Bandgap voltage references | The sum of a \( V_{BE} \), which falls with temperature, and a \( V_{BE} \) difference, which rises, gives a stable 1.2 V |
| Radio frequency (SiGe) | Silicon-germanium transistors with \( f_T \) of hundreds of GHz |
| Audio output stages | Complementary NPN-PNP pairs in class AB |
| Driving relays and loads | A cheap switch controlled from a microcontroller |
08In the lab
With a 2N3904, a source for the base through 100 kΩ and another for the collector, measure \( I_C \) versus \( V_{CE} \) for three base currents. Extend the lines of the active region until they cross the voltage axis and estimate \( V_A \).
With \( V_{CE} \) fixed at 5 V, measure \( V_{BE} \) for collector currents from 10 µA to 10 mA and verify that each decade raises the voltage by about 60 mV. Calculate \( g_m \) at each point.
Drive the transistor as a switch with a square wave, first with a forced β of 10 and then of 50, and use the oscilloscope to measure the delay between the falling edge of the base and the rise of the collector. Add a Schottky diode between base and collector and repeat.
09Common mistakes
- Thinking of the transistor as two diodes. Two diodes soldered together do not amplify: the base has to be thin.
- Designing with an exact value of β. It varies from 100 to 300 between samples.
- Assuming that in saturation \( I_C = \beta I_B \). In saturation the current is set by the external circuit.
- Using \( BV_{CBO} \) as the maximum voltage. With the base open or at high resistance, the lower one applies, \( BV_{CEO} \).
- Forgetting the stored charge. A heavily saturated transistor is slow to turn off.
10Self-assessment
Why is the emitter doped much more heavily than the base?
So that the current of the base-emitter junction is carried almost entirely by the electrons injected into the base, which are the ones that reach the collector: the injection efficiency goes up and with it β.
What is \( g_m \) at 2 mA?
\( 2 \text{ mA} / 25.85 \text{ mV} \approx 77 \) mS, whatever the transistor.
With \( \alpha = 0.995 \), what is β?
\( \beta = 0.995 / 0.005 = 199 \).
What is forced β and why is it designed to be low?
It is \( I_C / I_B \) with the transistor saturated. It is made much lower than the minimum β of the transistor to guarantee saturation with any sample.
Why is \( BV_{CEO} \) lower than \( BV_{CBO} \)?
Because with the base open the current of the incipient avalanche acts as base current and the transistor amplifies it, bringing the breakdown forward.
11Further reading
- Adel S. Sedra and Kenneth C. Smith. Microelectronic Circuits. 7th ed., Oxford University Press, 2014. The hybrid-π model and the use of the bipolar transistor in circuits, with just the right amount of physics.
- Donald A. Neamen. Semiconductor Physics and Devices. 4th ed., McGraw-Hill, 2012. Injection efficiency, the transport factor and the Ebers–Moll model derived from the physics.
- Paul R. Gray, Paul J. Hurst, Stephen H. Lewis and Robert G. Meyer. Analysis and Design of Analog Integrated Circuits. 5th ed., Wiley, 2009. The device models chapter is the reference for the Early effect, the capacitances and high-current effects.