Library / Circuits, electronics and noise / ELN-19
Circuits, electronics and noise
Hybrid-pi base input resistance
Small-signal resistance looking into the intrinsic base of a transistor with the emitter at AC ground, from the small-signal current gain and the transconductance.
- Ref
- ELN-19
- Rev
- v1
- Field
- Circuits, electronics and noise
- Result
- r_π [Ω]
In the hybrid-pi model the base current is gm·vbe/β, so the small-signal resistance looking into the intrinsic base is rπ = vbe/ib = β/gm. It is large for a low bias current and falls as the current rises. External bias resistors and the base spreading resistance come in parallel or series with it and are not included.
Small-signal resistance looking into the intrinsic base of a transistor with the emitter at AC ground, from the small-signal current gain and the transconductance. External bias resistors are not included.
Calculator
- Base current per collector current, 1/β0.008333
- Base current at 1 mA collector current, 1 mA/β8.333×10−6 A
- Input resistance with Re = 100 Ω unbypassed, rπ + (β + 1)·10015,100 Ω
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.
- With β = 120 and gm = 0.04 S the base input resistance is 3,000 Ω.
- Because gm rises with bias current, doubling the collector current halves rπ and doubles the load the stage puts on its source.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (Ω) | Local step | Change in result |
|---|---|---|---|---|---|---|
| gm Transconductance | 0.001 to 0.4 S | 300 | 120,000 | 119,700 | +10% | −9.09 % |
| β Small-signal current gain | 20 to 500 | 500 | 12,500 | 12,000 | +10% | +10 % |
Sweep of Transconductance
| Transconductance (S) | Base input resistance (Ω) |
|---|---|
| 0.001 | 120,000 |
| 0.05088 | 2,359 |
| 0.1008 | 1,191 |
| 0.1506 | 796.7 |
| 0.2005 | 598.5 |
| 0.2504 | 479.3 |
| 0.3003 | 399.7 |
| 0.3501 | 342.7 |
| 0.4 | 300 |
Sweep one variable
Variables and how each one contributes
Base input resistance
What the equation gives you. Everything else feeds it.
Small-signal current gain
The small-signal current gain β, no units. Raises rπ in direct proportion. Raising it by 10 % (to 132) moves the result by +10 %, from 3,000 to 3,300 Ω.
Transconductance
The transconductance, in S. A larger gm means more collector current for each base volt, so rπ falls in inverse proportion. Raising it by 10 % (to 0.044) moves the result by −9.09 %, from 3,000 to 2,727 Ω.
Worked example with the values above
- Ratio: β/gm = 120 / 0.04.
- That is 3,000 Ω.
- rπ = 3,000 Ω.
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
Transistor current gain β varies by a factor of two or more between parts and with temperature, so a stable design depends on emitter feedback, not on β. Thermal voltage changes by about 0.33 % per kelvin.
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