Current Gain (Math)
How Current Gain Is Defined and Computed
Current gain answers a single question: for every milliamp of signal current pushed into a device, how many milliamps come out? It is a ratio, so it carries no units. For an idealized two-port amplifier the linear current gain is simply Ai = Iout/Iin. The number becomes useful when tied to a physical device. In a bipolar transistor the base current controls a much larger collector current, and the ratio of those two is the parameter engineers call beta. A small-signal RF transistor might have a beta of 80 to 200, meaning a 50 μA base current produces a collector current of 4 to 10 mA.
Two distinct beta values appear on datasheets and must not be confused. The DC current gain, written hFE with uppercase subscript, is the large-signal ratio IC/IB measured at a fixed bias point. The small-signal current gain hfe is the slope of the IC versus IB curve at that point, formally the partial derivative ∂IC/∂IB at constant VCE. Because beta is not constant with bias current, these two figures differ: where beta is still rising with collector current (below the peak-beta point) the local slope exceeds the chord through the origin, so hfe sits above hFE; at the peak the two coincide, and above it hfe falls below hFE.
When a result is quoted in decibels, current gain uses the 20-log form because current is an amplitude quantity, not a power quantity. Many engineers reflexively reach for 10 log, which is correct only for power ratios. Confusing the two doubles or halves the apparent gain on paper, a mistake that shows up constantly in early stage budget calculations.
Governing Equations
Ai = Iout / Iin (dimensionless)
Transistor DC and small-signal beta:
β = hFE = IC / IB hfe = ∂IC / ∂IB at constant VCE
Current gain in decibels (20-log amplitude form):
Ai(dB) = 20 × log10(Iout / Iin)
Frequency roll-off (single pole):
hfe(f) ≈ hfe0 / √(1 + (f / fβ)2) fT ≈ hfe0 × fβ
Where IC = collector current, IB = base current, hfe0 = low-frequency current gain, fβ = beta cutoff frequency, fT = transition frequency where |hfe| = 1. Example: a ratio of 100 gives 20 × log10(100) = 40 dB.
Current-Ratio to Decibel Reference
| Current ratio Iout/Iin | Current gain (dB) | Equivalent context | Typical device |
|---|---|---|---|
| 1 | 0 dB | Unity (no current gain) | At fT, or ideal buffer |
| 2 | 6.02 dB | Doubling of current | Lightly loaded follower |
| 10 | 20 dB | One decade | Low-β power transistor |
| 50 | 33.98 dB | Mid-range β | RF small-signal BJT |
| 100 | 40 dB | Two decades | General-purpose BJT |
| 200 | 46.02 dB | High-β device | High-gain signal BJT (e.g. C-grade) |
Frequently Asked Questions
Is current gain calculated with 20 log or 10 log?
Current gain in decibels uses the 20-log form: Ai(dB) = 20 × log10(Iout/Iin). The factor of 20 applies to any amplitude ratio because power is proportional to the square of amplitude. A current ratio of 100 is therefore 40 dB. The 10-log form is reserved for power ratios; mixing the two is a frequent datasheet-reading error.
What is the difference between beta and h-fe?
Beta, written hFE with an uppercase subscript, is the large-signal DC current gain IC/IB at a fixed bias point. The lowercase hfe is the small-signal AC current gain, the slope ∂IC/∂IB at constant VCE. They are close but not equal: at IC = 10 mA an hFE of 150 might pair with an hfe near 170 because the local slope exceeds the chord through the origin.
How does current gain roll off with frequency?
Small-signal hfe drops at about 6 dB per octave above the beta cutoff frequency fβ, following hfe(f) = hfe0/√(1 + (f/fβ)2). Magnitude reaches unity at the transition frequency fT ≈ hfe0 × fβ. A device with hfe0 = 100 and fT = 25 GHz has a beta cutoff near 250 MHz.