BJT Small Signal Model

The BJT small signal model replaces a transistor with linear circuit elements valid for small AC signals around the Q-point. Every BJT amplifier calculation for gain, impedance, and bandwidth uses this model. It is the bridge between the transistor physics and circuit algebra.

Core Concept

When an AC signal rides on the DC bias of a BJT, the transistor behaves like a linear voltage-controlled current source for small signals. The transconductance gm relates the small-signal collector current to the base-emitter voltage: ic = gm * vbe. For a 2N3904 at IC = 1 mA, gm = 1mA/26mV = 38.5 mA/V.

The hybrid-pi model captures this with three elements. The resistor rπ between base and emitter models the base current path. The current source gm*vπ from collector to emitter models transistor action. The resistor rO from collector to emitter models the Early effect (output resistance finite because of channel-length-like modulation in BJTs).

A second small signal model, the T-model, uses an emitter resistor re = 1/gm and a current-controlled current source alpha*ie at the collector. Both models give identical results for gain and impedance. The hybrid-pi is preferred for GATE and most amplifier analysis because it naturally shows input and output ports.

Key Equations

Transconductance: gm = IC / VT where VT = 26 mV at 300K. IC is the Q-point collector current in amperes.

Base-emitter resistance: rπ = beta / gm = beta * VT / IC. This is also written as hfe/gm.

Output resistance: rO = VA / IC where VA is the Early voltage (50-200V for typical BJTs).

Emitter resistance in T-model: re = VT / IC = 1 / gm. Note: re = rπ / beta.

Given:
  2N3904 BJT in common emitter configuration
  Q-point: IC = 2 mA, VCE = 5V
  beta = 150, VA = 100V, VT = 26 mV

Why this formula:
  Compute all small signal parameters from Q-point.

Formula:
  gm = IC / VT
  rπ = beta / gm
  rO = VA / IC

Substitution:
  gm = 2 mA / 26 mV = 2 / 26 (in mA/mV = A/V = Siemens)

Calculation:
  gm = 0.0769 A/V = 76.9 mA/V

  rπ = beta / gm = 150 / 0.0769 = 1950 Ω ≈ 1.95 kΩ

  rO = VA / IC = 100 / 0.002 = 50000 Ω = 50 kΩ

  re = 1/gm = 1/0.0769 = 13 Ω

Final Answer:
  gm = 76.9 mA/V
  rπ = 1.95 kΩ
  rO = 50 kΩ
  re = 13 Ω

Exam Tip: GATE frequently gives IC in mA and asks for gm. Use gm = IC/VT directly with VT = 26 mV, giving gm in mA/mV = A/V (Siemens). Then rπ = beta/gm. The most common mistake is using VT = 25 mV or 0.026 V incorrectly with mA units and getting rπ off by a factor of 1000. Keep units consistent: if IC is in mA, VT must be in mV to get gm in mA/mV = S.

Key Properties

• gm = IC/VT: proportional to Q-point current. Doubling IC doubles gm and halves rπ.
• rπ = beta/gm: the small-signal input resistance looking into the base. At IC=1mA, beta=100, rπ = 2.6 kΩ.
• rO = VA/IC: the small-signal output resistance. With VA=100V and IC=1mA, rO = 100 kΩ.
• The T-model uses re = 1/gm in the emitter branch and a current source alpha*ie at the collector. It is easier to use for common-base analysis.
• For the Early effect, the collector current expression becomes IC = IS * exp(VBE/VT) * (1 + VCE/VA). A larger VA means a flatter IC vs VCE curve.
• Both hybrid-pi and T-model give the same voltage gain, current gain, and impedance results.

Quick Revision

• gm = IC/VT (VT = 26 mV at 300K).
• rπ = beta/gm: base-emitter small signal resistance.
• rO = VA/IC: output resistance, models Early effect.
• re = 1/gm = VT/IC: used in T-model and common-base analysis.
• Hybrid-pi: rπ in series with base, gm*vπ current source at collector.
• T-model: re in emitter branch, alpha*ie current source at collector.
• Both models are equivalent; choose based on which configuration is being analyzed.
• Exam trap: Students mix up rπ and rO. rπ is at the input (base-emitter), rO is at the output (collector-emitter). Using rO in the gain formula where rπ belongs gives a completely wrong answer.