The bipolar transistor as an amplifier: small and large signal
A common emitter without degeneration is linear only for a few millivolts: with 10 mV at the base it already produces almost 10 % of second harmonic. The scale of the bipolar transistor is 25.85 mV.
01Small signal and large signal
The bipolar transistor is an exponential device: the collector current follows the base-emitter voltage as \( i_C = I_S\, e^{v_{BE}/V_T} \). No exponential is a straight line, so no amplifier built with transistors is exactly linear. What you do is work around a quiescent point and separate two regimes:
- Small signal: the swing is so small that the exponential looks like its tangent. The linear hybrid-π model from the unit on the bipolar transistor applies, and the gain does not depend on the amplitude.
- Large signal: the swing covers an appreciable part of the curve. Distortion appears, the gain changes with the amplitude, and if the signal is large the output is clipped against saturation or cutoff.
The boundary between the two is not a fixed number: it depends on how much distortion is tolerated. This page calculates it.
02The quiescent point
Before amplifying you have to bias: set a stable collector current and a collector voltage that leaves room to swing both up and down. Biasing with a voltage divider at the base and an emitter resistor sets the current almost independently of β, because \( I_C \approx (V_B - V_{BE})/R_E \). For the maximum symmetrical swing, the quiescent collector voltage is set near the midpoint between saturation and the supply. The technician program develops these circuits in small-signal amplifiers.
03The three small-signal configurations
Depending on which terminal is common to the input and the output, the transistor gives three amplifiers with very different properties. With \( g_m = I_C / V_T \), \( r_\pi = \beta / g_m \) and an unbypassed emitter resistance \( R_E \):
| Configuration | Voltage gain | Input resistance | Output resistance | Use |
|---|---|---|---|---|
| Common emitter | \( -\dfrac{g_m R_C}{1 + g_m R_E} \) | \( r_\pi + (\beta + 1) R_E \) | \( \approx R_C \) | Voltage gain |
| Common collector (follower) | \( \approx 1 \) | \( r_\pi + (\beta + 1) R_L \), high | \( \approx 1/g_m + R_s/(\beta+1) \), low | Buffer, driving low-resistance loads |
| Common base | \( g_m R_C \) | \( \approx 1/g_m \), very low | \( \approx R_C \) | High frequency, current input |
The unbypassed emitter resistor is the central tool of the design: it reduces the gain by the factor \( 1 + g_m R_E \), but in exchange it makes the gain depend on resistors and not on the transistor (\( A_v \approx -R_C / R_E \) if \( g_m R_E \gg 1 \)), raises the input resistance and, as you will see shortly, greatly reduces the distortion. It is local negative feedback.
04Large signal: where the distortion comes from
With an input voltage \( v_{be} = \hat V \sin \omega t \) at the base, the current is \( I_C\, e^{x \sin \omega t} \) with \( x = \hat V / V_T \). Expanding the exponential gives terms in \( \sin^2 \), \( \sin^3 \)…, which are the second and third harmonics. For small amplitudes, the second harmonic relative to the fundamental is
That is why a common emitter without degeneration is only “linear” for input signals of a few millivolts: \( V_T = 25.85 \) mV is the natural scale of the bipolar transistor. The exact expression uses Bessel functions, and the lab calculates it numerically, sample by sample.
If the amplitude keeps growing, the collector voltage hits its limits: it cannot rise higher than the supply (the transistor cuts off) nor fall below about 0.2 V above the emitter (it saturates). The output is clipped, and the distortion jumps.
A common emitter powered from 12 V. The collector current is calculated with the full exponential for each instant (without linearizing), along with the output and its harmonic content. The dashed line is what the small-signal model predicts.
05Where it appears in electronics
| Case | Which regime matters |
|---|---|
| Microphone preamplifier | Small signal: noise and gain; distortion barely appears with millivolts |
| Output stage of an audio amplifier | Large signal: clipping and distortion define the usable power |
| Radio receiver with a strong interfering signal | The nonlinearity mixes signals and generates intermodulation products |
| Mixer and multiplier | The exponential nonlinearity is used on purpose |
06In the lab
Build a common emitter with a 2N3904 at 1 mA and \( R_C = 4.7 \) kΩ, with the emitter bypassed. Measure the gain with 1 mV at the input and compare it with \( g_m R_C \).
Raise the input to 10 and 20 mV and observe the asymmetry of the output on the oscilloscope. With the oscilloscope's FFT function, measure the second harmonic and compare it with \( \hat V / 4 V_T \).
Remove the bypass capacitor to leave 100 Ω of unbypassed emitter resistance. Adjust the input to obtain the same output as before and compare the distortion.
07Common mistakes
- Applying the small-signal model at any amplitude. It holds for a few millivolts at the base.
- Biasing without room to swing. A quiescent collector near the supply clips right away on one half-cycle.
- Designing the gain with β. With degeneration, the resistors set it.
- Thinking that more gain is always better. More gain without degeneration means more distortion.
08Self-assessment
What is the gain of a common emitter at 2 mA with \( R_C = 2.2 \) kΩ and the emitter bypassed?
\( g_m R_C = (2/25.85) \times 2200 \approx 170 \).
What second harmonic is expected with a 4 mV peak at the base, without degeneration?
\( 4 / (4 \times 25.85) \approx 3.9 \) %.
What advantages does the unbypassed emitter resistor have?
Gain set by resistors, higher input resistance and less distortion, at the cost of less gain.
Which configuration suits driving a low-resistance load?
The common collector, because of its low output resistance.
09Further reading
- Adel S. Sedra and Kenneth C. Smith. Microelectronic Circuits. 7th ed., Oxford University Press, 2014. The small-signal bipolar configurations, with all the parameters derived.
- 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 chapter on distortion, with the series expansion of the exponential and the effect of feedback.
- Douglas Self. Audio Power Amplifier Design. 6th ed., Focal Press, 2013. The distortion of bipolar stages measured in real amplifiers.