8. Putting Devices in Circuits: Small-Signal Models

A transistor in a real amplifier is biased at some DC operating point and then asked to amplify a small AC signal superimposed on the DC. To analyze this, we use small-signal models — linearized approximations that capture the AC behavior around the Q-point.

8.1 The big idea

Imagine the transistor's full I-V curve. At any operating point, you can replace the curve locally with a tangent line. The tangent's slope is $g_m$ at that point. So as long as the AC swings are small enough that you do not stray far from the tangent, the transistor behaves as a linear device — and we can use all of network analysis (Thevenin, Norton, Bode plots, etc.) on it.

The "small-signal" approximation breaks when you swing too far. Push the input too hard and the output clips. But in a properly designed amplifier, the swings stay in the linear region.

8.2 BJT hybrid-π and h-parameter models

The most common small-signal model for the BJT is the hybrid-π model:

   B ─────[r_π]─────┬──────────── C
                    │
                  [g_m·v_be]  ← controlled current source
                    │
   E ───────────────┴──────────── E

• $r_\pi = V_T / I_B = \beta / g_m$ — base-emitter input resistance.
• $g_m = I_C / V_T$ — transconductance.
• $r_o$ (output resistance, models Early effect) — added in parallel with the current source.
• For high-frequency analysis: add capacitances $C_\pi$ (base-emitter) and $C_\mu$ (base-collector). We will get to those in Chapter 5.

Equivalent model: h-parameters, which present the BJT as a two-port network with hybrid mixed inputs/outputs:

$V_{BE} = h_{ie} \cdot I_B + h_{re} \cdot V_{CE}$ $I_C = h_{fe} \cdot I_B + h_{oe} \cdot V_{CE}$

with $h_{ie} \approx r_\pi$, $h_{fe} \approx \beta$, $h_{re}$ small (typically 10⁻⁴), $h_{oe} \approx 1/r_o$ small. h-parameters are what older textbooks and many exam questions use; they are equivalent to hybrid-π but with different naming.

8.3 FET small-signal model

Simpler than BJT because the gate draws no current:

   G ──────────┬────[g_m·v_gs]────── D
              (∞)         │          │
                          │         [r_o]
                          │          │
   S ────────────────────┴──────────┴── S

No $r_\pi$ — the gate input impedance is essentially infinite. Just the controlled current source and output resistance. Add $C_{gs}$ and $C_{gd}$ for high-frequency.

8.4 Three configurations, side by side

For each transistor and each configuration (CB, CE, CC for BJT; CG, CS, CD for FET), the small-signal model gives specific input impedance, output impedance, voltage gain, and current gain. The standard summary:

• CE/CS: general-purpose gain stage. The "default" amplifier.
• CB/CG: high-frequency RF and current amplifier.
• CC/CD (emitter/source follower): buffer. Use after a high-impedance node to drive a low-impedance load.

8.5 Worked example: a simple CE amplifier

Take the self-bias CE amplifier from section 6, with $R_C = 2.7$ kΩ, $R_E = 1$ kΩ, $V_{CC} = 12$ V, biased at $I_C = 1$ mA.

• $g_m = I_C/V_T = 1/26 = 38$ mS.
• $r_\pi = \beta/g_m = 100/0.038 \approx 2.6$ kΩ (assuming $\beta = 100$).
• Voltage gain (with $R_E$ bypassed by a capacitor for AC, so AC sees $R_E = 0$): $A_v = -g_m R_C = -0.038 \times 2700 = -103$. So gain is about 100, with 180° phase inversion.
• Input impedance: $r_\pi$ in parallel with the bias divider — typically a few kΩ.
• Output impedance: $R_C$ in parallel with $r_o$ — typically a few kΩ.

If we don't bypass $R_E$, the gain becomes $A_v \approx -R_C / R_E = -2.7$ — much smaller, but more linear and with higher input impedance and more bandwidth (the $R_E$ provides local feedback). Whether to bypass or not is a classic design choice.