Library / Circuits, electronics and noise / ELN-18
Circuits, electronics and noise
BJT small-signal transconductance
Slope of collector current against base–emitter voltage at the operating point, for the exponential junction model.
- Ref
- ELN-18
- Rev
- v1
- Field
- Circuits, electronics and noise
- Result
- gm [S]
The collector current of a bipolar transistor rises exponentially with the base–emitter voltage, Ic = Is exp(Vbe/VT). Differentiating gives the slope gm = Ic/VT. It depends only on the bias current and the temperature, not on the transistor size. At 25 mV and 1 mA it is 40 mS, or 1 mS per 25 µA.
Slope of collector current against base–emitter voltage at the operating point, for the exponential junction model. The thermal voltage is about 25 mV near room temperature.
Calculator
- Emitter resistance, 1/gm25 Ω
- Transconductance at twice the current0.08 S
- Change in Ic for a 1 mV rise in Vbe4×10−5 A
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- At 0.001 A the transconductance is 0.04 S (40 mS), and 1/gm = 25 Ω is the small-signal resistance looking into the emitter.
- The rule of thumb gm ≈ 40 × Ic holds at room temperature, here 0.04 S.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (S) | Local step | Change in result |
|---|---|---|---|---|---|---|
| Ic Quiescent collector current | 1e-5 to 0.1 A | 4e-4 | 4 | 4 | +10% | +10 % |
| VT Thermal voltage | 0.02 to 0.03 V | 0.03333 | 0.05 | 0.01667 | +10% | −9.09 % |
Sweep of Quiescent collector current
| Quiescent collector current (A) | Transconductance (S) |
|---|---|
| 1e-5 | 4e-4 |
| 3.162e-5 | 0.001265 |
| 1e-4 | 0.004 |
| 3.162e-4 | 0.01265 |
| 0.001 | 0.04 |
| 0.003162 | 0.1265 |
| 0.01 | 0.4 |
| 0.03162 | 1.265 |
| 0.1 | 4 |
Sweep one variable
Variables and how each one contributes
Transconductance
What the equation gives you. Everything else feeds it.
Quiescent collector current
The quiescent collector current, in A. Raises gm in direct proportion. Raising it by 10 % (to 0.0011) moves the result by +10 %, from 0.04 to 0.044 S.
Thermal voltage
The thermal voltage kT/q, in V, about 25.9 mV at 300 K. A higher thermal voltage means a flatter exponential, so gm falls in inverse proportion. Raising it by 10 % (to 0.0275) moves the result by −9.09 %, from 0.04 to 0.03636 S.
Worked example with the values above
- Ratio: Ic/VT = 0.001 / 0.025.
- That is 0.04 A/V.
- gm = 0.04 S.
Assumptions and limits
- An NPN transistor in the forward-active region, with a fixed base–emitter drop of about 0.7 V.
- Midband small signals, an AC-grounded supply and negligible transistor output conductance.
Common mistakes
- Forgetting to check that the collector–base junction is still reverse biased.
- Using the DC current gain where the small-signal gain is needed, or the reverse.
- Forgetting that an unbypassed emitter resistor adds (β + 1) times its value to the input resistance.
Statistics and uncertainty
gm depends on the temperature through VT, which changes by about 0.33 % per kelvin, so the gain of an amplifier with fixed bias current also changes.
Where it is used
- Setting the operating point of transistor amplifiers and switches.
- Estimating small-signal gain, input resistance and the effect of emitter degeneration.
Maths you need
- Thevenin equivalent: a source and a resistance in series
- Transistor current gain: β = Ic / Ib
- Thermal voltage: kT/q, about 25 mV at room temperature