C3 · BJT

The Small-Signal Model: Up Close, Even a Curve Is a Line

To a small wobble about the operating point, the transistor looks like a linear device. Shrink the signal amplitude yourself and see below what size the output cleanly resembles the input.

Shrink the signal to find a clean copy

The faint curve is the input vbe and the bold curve is the output ic. When the signal is large, the output is distorted, with sharp peaks and flat troughs. Shrink the amplitude. Below some size the output becomes the same clean sine as the input.

Signal amplitude vbevbe = 80 mV

Small-signal relation · ic = gm vbe

gm = Ic/VT ≈ 192 mA/V

rπ = β/gm ≈ 0.52 kΩ

The amplitude is comparable to the thermal voltage. The bend of the exponential curve shows through, so the output peaks tower over the troughs.

Large signal · distorted

Zoom in at the operating point and the curve is a line

The collector current bends upward exponentially with the base-emitter voltage. It is a curve, but take just a very narrow stretch around the operating point and zoom in, and any smooth curve looks like its tangent, a straight line. A small signal is exactly a wobble that stays within this narrow stretch. Inside it, input and output are in a straight-line relation, that is, linear.

The slope of the tangent, gm = Ic/VT

The slope of that tangent is the transconductance gm. Since the derivative of an exponential is proportional to itself, the slope is the operating-point collector current over the thermal voltage: gm = Ic/VT. So the larger the bias current chosen in C2, the steeper the tangent and the larger gm, and the same small input yields a larger output current. The operating point is, in effect, the handle on the gain.

The small-signal equivalent

Now the transistor can be redrawn as a single linear box. The input side looks like an input resistance rπ = β/gm, the base voltage over the base current; the output side is a current source ic = gm vbe = β ib. The unchanging DC supplies do not wobble for the signal, so for AC they are tied to ground. This equivalent circuit is the tool for computing gain in the next units, and it holds only while the signal stays within the narrow stretch.

Observeic = gm vbe

The output current is the transconductance times the input voltage.

Choosegm = Ic / ?

The transconductance is the operating current over the thermal voltage.

Fill inrπ = β / ?

The input resistance is β over the transconductance.

On your owngm rπ = ?

Multiply the two and you return to the current gain β.

Back to the first screen

When the signal was large, the output was a distorted waveform with sharp peaks and flat troughs, a direct reflection of the bend of the exponential curve. As you shrank the amplitude far below the thermal voltage, the curve looked straight within that narrow stretch and the output became a clean sine resembling the input. The slope of that line is gm, and with it the transistor becomes a linear box with only an input resistance rπ and a current source gm vbe. The large DC is the pedestal that holds the spot; only the small wobble on top is the signal we will now compute.

To a small signal about the operating point, the transistor is a linear device. Zoom into the exponential curve Ic(Vbe) at the Q-point and it looks like a line of slope gm = Ic/VT (the transconductance). So a small vbe makes ic = gm vbe. The transistor becomes a small-signal equivalent: an input resistance rπ = β/gm and a current source gm vbe (gm rπ = β). It holds only while the signal is small enough that the curve looks straight.

What comes next

You now know how to turn the transistor into a linear box. The next unit drops this box into a real circuit: the common-emitter amplifier. When the input signal wobbles the base, the current source drives current through the collector resistor to make a large output voltage. With the small-signal model you will compute that voltage gain directly for the first time.