Small Signal Model

When a diode operates with a small AC signal superimposed on a DC bias, it does not behave as a simple on-off switch. Around the DC operating point, the diode can be represented by an equivalent linear circuit. This linearized representation is called the small signal model of a diode, and it introduces two key parameters: dynamic resistance and diffusion conductance. These parameters are essential for analyzing diode behavior in amplifier bias circuits, RF detectors, and communication circuit design.

Core Concept Explanation

The diode current-voltage relationship is nonlinear: I = IS × (exp(V/VT) - 1). However, for small AC signals around a fixed DC bias point, the curve can be approximated as locally linear. The slope of this curve at the operating point gives the dynamic conductance (gm), and its reciprocal is the dynamic resistance (rd). These replace the diode in small signal analysis.

The physical meaning of rd is the incremental opposition the diode presents to a small change in voltage at the bias point. A high DC current bias means the exponential curve is steep at that point, so even a small voltage change produces a large current change, meaning rd is small. Conversely, at low bias current, the curve is nearly flat and rd is large. This dependency on the operating point is a key difference from a resistor.

The complete small signal model also includes bulk resistance (rs), which accounts for the ohmic resistance of the semiconductor material and the metal-semiconductor contacts outside the junction. At high currents, the voltage drop across rs becomes significant and must be included. The model also includes both capacitances CD and CT discussed in the junction capacitance article, making this a complete high-frequency model.

Mathematical Expression

Taking the derivative of the Shockley diode equation with respect to V gives the dynamic conductance. At the Q-point bias current ID, this derivative evaluated at V = VQ yields gm = dI/dV = IS/VT × exp(VQ/VT). Since ID = IS × exp(VQ/VT) for VQ >> VT, this simplifies to:

gm = ID / VT, which means rd = VT / ID. At room temperature, VT = kT/q = 26 mV, so rd = 26 mV / ID. If ID = 1 mA, rd = 26 ohm. If ID = 2 mA, rd = 13 ohm. The dynamic resistance decreases with increasing forward current because the diode becomes more conducting. This result is fundamental and appears directly in GATE problems.

Practical Understanding

In practical circuits, the small signal model allows engineers to analyze how a diode behaves for audio, RF, or mixed-signal inputs. In a half-wave rectifier with a small ripple superimposed, the effective output resistance seen by the filter is approximately rd + rs. In emitter-follower and common-emitter BJT circuits, the forward-biased emitter-base junction contributes a dynamic resistance that appears in voltage gain expressions.

At high frequencies, the capacitances CD and CT cannot be ignored. CD dominates in forward bias and limits how fast the device can respond to AC variations. CT dominates in reverse bias and sets the frequency limit for varactor tuning applications. The total small signal impedance of the diode is a complex function of frequency, bias current, and device geometry.

Given:
Diode DC bias current ID = 2 mA
Thermal voltage VT = 26 mV at T = 300 K
Bulk resistance rs = 5 Ω

Why this formula applies:
Small signal analysis around a forward bias Q-point; dynamic resistance is slope of I-V at operating point.

Formula:
rd = VT / ID
gm = 1 / rd = ID / VT

Substitution:
rd = 26 mV / 2 mA = 26e-3 / 2e-3

Calculation:
rd = 13 Ω
gm = 1 / 13 ≈ 76.9 mS
Total small signal resistance = rd + rs = 13 + 5 = 18 Ω

Final Answer:
rd = 13 Ω,  gm ≈ 76.9 mS,  Total resistance = 18 Ω

Exam Tip: rd = 26/ID (mA) gives rd in ohms directly. If ID = 1 mA, rd = 26 Ω. If ID = 2 mA, rd = 13 Ω. This shortcut solves GATE small signal problems in seconds. Always add rs if bulk resistance is given.

• Small signal model linearizes diode behavior around a DC Q-point for AC analysis.
• Dynamic resistance rd = VT/ID and conductance gm = ID/VT are derived from the derivative of the Shockley equation.
• rd decreases with increasing forward current; gm increases with forward current.
• Bulk resistance rs accounts for ohmic contact and semiconductor resistance outside the depletion region.
• At high frequencies, diffusion capacitance CD (forward bias) and transition capacitance CT (reverse bias) must be included in the model.

Quick Revision

• rd = VT / ID = 26 mV / ID; at ID = 1 mA, rd = 26 Ω.
• gm = ID / VT = 1 / rd; conductance increases with bias current.
• Small signal model components: rd, rs, CD, CT.
• rs is the bulk/series resistance; does not vary with bias current.
• GATE trap: Always check whether rs is given separately; total resistance is rd + rs, not just rd.
• CD dominates in forward bias; CT dominates in reverse bias for frequency considerations.
• VT = 26 mV at 300 K; VT = kT/q increases with temperature.