9.4 Small-signal models and parameters

Small-signal model overview

Small-signal models let you analyze how a BJT circuit responds to tiny AC signals riding on top of a DC bias. Instead of dealing with the transistor's full nonlinear I-V characteristics, you linearize around the DC operating point (the Q-point) and replace the transistor with a simple equivalent circuit of resistors, capacitors, and dependent sources. This makes it possible to calculate voltage gain, current gain, and input/output impedances using standard linear circuit techniques.

Why small-signal models matter

• They turn a nonlinear device into a linear circuit you can solve with KVL, KCL, and superposition.
• They let you determine voltage gain, current gain, input impedance, and output impedance for amplifier stages.
• They're the foundation for designing amplifiers, oscillators, and active filters with predictable performance.
• Without them, you'd need full numerical simulation for every design iteration.

Linear vs. nonlinear models

Nonlinear models (like the Ebers-Moll or Gummel-Poon models) capture transistor behavior across the full range of voltages and currents. They're necessary for large-signal analysis, such as switching circuits or clipping behavior.

Small-signal (linear) models are a first-order Taylor expansion of those nonlinear equations around the Q-point. They're only valid when the AC signal amplitudes are small enough that the device stays approximately linear. A common rule of thumb for BJTs: the AC base-emitter voltage swing should satisfy $v_{be} \ll V_T$, where $V_T \approx 26 \text{ mV}$ at room temperature.

Operating point selection

The Q-point is the DC bias condition ($I_C$, $V_{CE}$) around which you linearize. Every small-signal parameter depends on this point, so getting the bias right is critical.

• The transistor must be biased in the forward-active region for amplifier operation ($V_{BE} \approx 0.7 \text{ V}$, $V_{CE} > V_{CE,sat}$).
• DC bias circuits (voltage divider bias, current mirror bias, etc.) are designed to establish and stabilize the Q-point against temperature and $\beta$ variations.
• If the Q-point drifts, all your small-signal parameters change with it.

Small-signal equivalent circuits

Once you've set the Q-point, you replace the BJT with a network of linear elements whose values are determined by the DC bias conditions and device physics.

The hybrid-π model for BJTs

This is the most widely used BJT small-signal model. For a common-emitter configuration, it contains:

• Transconductance $g_m = \frac{I_C}{V_T}$, where $I_C$ is the DC collector current and $V_T \approx 26 \text{ mV}$ at room temperature. This is the gain of the dependent current source: $i_c = g_m v_{be}$.
• Base-emitter resistance $r_\pi = \frac{\beta}{g_m} = \frac{V_T}{I_B}$, which represents the small-signal resistance looking into the base.
• Output resistance $r_o = \frac{V_A}{I_C}$, where $V_A$ is the Early voltage. This accounts for the finite slope of the output characteristics in the active region.
• Base-collector capacitance $C_\mu$ (the Miller capacitance) and base-emitter capacitance $C_\pi$, which become important at higher frequencies.

For example, if $I_C = 1 \text{ mA}$, $\beta = 100$, and $V_A = 100 \text{ V}$:

• $g_m = \frac{1 \text{ mA}}{26 \text{ mV}} \approx 38.5 \text{ mA/V}$
• $r_\pi = \frac{100}{38.5 \text{ mA/V}} \approx 2.6 \text{ k}\Omega$
• $r_o = \frac{100 \text{ V}}{1 \text{ mA}} = 100 \text{ k}\Omega$

Resistor and capacitor models

In the small-signal equivalent circuit, resistors remain ideal resistances. DC voltage sources become short circuits (zero AC impedance), and DC current sources become open circuits (infinite AC impedance). Capacitors introduce frequency-dependent impedance:

$Z_C = \frac{1}{j\omega C}$

At low frequencies, coupling and bypass capacitors may not be effective shorts, which limits the low-frequency response of amplifier stages.

Dependent sources

The core of any transistor small-signal model is its dependent source. In the hybrid-π model, a voltage-controlled current source (VCCS) produces collector current $i_c = g_m v_{be}$ controlled by the base-emitter voltage. The dependent source is what gives the transistor its amplifying ability in the linear model.

Admittance parameters (y-parameters)

Y-parameters describe a two-port network by relating port currents to port voltages. They're measured under short-circuit conditions and are especially convenient for analyzing parallel-connected networks, since y-parameter matrices simply add for parallel two-ports.

Y-parameter definition and matrix

$\begin{bmatrix} I_1 \\ I_2 \end{bmatrix} = \begin{bmatrix} y_{11} & y_{12} \\ y_{21} & y_{22} \end{bmatrix} \begin{bmatrix} V_1 \\ V_2 \end{bmatrix}$

• $y_{11}$: short-circuit input admittance (set $V_2 = 0$, measure $I_1/V_1$)
• $y_{12}$: short-circuit reverse transfer admittance (set $V_1 = 0$, measure $I_1/V_2$)
• $y_{21}$: short-circuit forward transfer admittance (set $V_2 = 0$, measure $I_2/V_1$)
• $y_{22}$: short-circuit output admittance (set $V_1 = 0$, measure $I_2/V_2$)

Deriving y-parameters from an equivalent circuit

1. Draw the small-signal equivalent circuit of the two-port.
2. Short-circuit the output port ($V_2 = 0$). Apply a test voltage $V_1$ at the input. Solve for $I_1$ and $I_2$ to get $y_{11}$ and $y_{21}$.
3. Short-circuit the input port ($V_1 = 0$). Apply a test voltage $V_2$ at the output. Solve for $I_1$ and $I_2$ to get $y_{12}$ and $y_{22}$.

Y-parameter measurement

Y-parameters can be measured with a vector network analyzer (VNA) or impedance analyzer. The VNA typically measures s-parameters, which are then converted to y-parameters mathematically. Direct measurement requires applying short-circuit terminations at each port in turn.

Impedance parameters (z-parameters)

Z-parameters are the dual of y-parameters. They relate port voltages to port currents under open-circuit conditions and are convenient for series-connected networks, since z-parameter matrices add directly for series two-ports.

Z-parameter definition and matrix

$\begin{bmatrix} V_1 \\ V_2 \end{bmatrix} = \begin{bmatrix} z_{11} & z_{12} \\ z_{21} & z_{22} \end{bmatrix} \begin{bmatrix} I_1 \\ I_2 \end{bmatrix}$

• $z_{11}$: open-circuit input impedance (set $I_2 = 0$, measure $V_1/I_1$)
• $z_{12}$: open-circuit reverse transfer impedance (set $I_1 = 0$, measure $V_1/I_2$)
• $z_{21}$: open-circuit forward transfer impedance (set $I_2 = 0$, measure $V_2/I_1$)
• $z_{22}$: open-circuit output impedance (set $I_1 = 0$, measure $V_2/I_2$)

Deriving z-parameters from an equivalent circuit

1. Draw the small-signal equivalent circuit.
2. Open-circuit the output port ($I_2 = 0$). Apply a test current $I_1$ at the input. Solve for $V_1$ and $V_2$ to get $z_{11}$ and $z_{21}$.
3. Open-circuit the input port ($I_1 = 0$). Apply a test current $I_2$ at the output. Solve for $V_1$ and $V_2$ to get $z_{12}$ and $z_{22}$.

Z-parameter measurement

Like y-parameters, z-parameters are most often obtained by converting from s-parameter measurements on a VNA. Direct measurement requires open-circuit terminations at each port, which can be impractical at high frequencies where stray capacitance makes a true open circuit difficult to achieve.

Hybrid parameters (h-parameters)

H-parameters use a mix of short-circuit and open-circuit conditions, which makes them especially natural for BJT analysis. Transistor datasheets commonly specify h-parameters ($h_{fe}$, $h_{ie}$, etc.) because they map directly onto measurable transistor properties.

H-parameter definition and matrix

$\begin{bmatrix} V_1 \\ I_2 \end{bmatrix} = \begin{bmatrix} h_{11} & h_{12} \\ h_{21} & h_{22} \end{bmatrix} \begin{bmatrix} I_1 \\ V_2 \end{bmatrix}$

Notice the mixed variables: the input equation relates $V_1$ to $I_1$ and $V_2$, while the output equation relates $I_2$ to $I_1$ and $V_2$.

• $h_{11}$ (or $h_{ie}$): short-circuit input impedance ($V_2 = 0$). For a BJT, this corresponds to $r_\pi$.
• $h_{12}$ (or $h_{re}$): open-circuit reverse voltage ratio ($I_1 = 0$). Typically very small and often neglected.
• $h_{21}$ (or $h_{fe}$): short-circuit forward current gain ($V_2 = 0$). This is the small-signal $\beta$.
• $h_{22}$ (or $h_{oe}$): open-circuit output admittance ($I_1 = 0$). This is approximately $1/r_o$.

Deriving h-parameters from an equivalent circuit

1. Short-circuit the output ($V_2 = 0$). Drive the input with a test current $I_1$. Solve for $V_1/I_1 = h_{11}$ and $I_2/I_1 = h_{21}$.
2. Open-circuit the input ($I_1 = 0$). Apply a test voltage $V_2$ at the output. Solve for $V_1/V_2 = h_{12}$ and $I_2/V_2 = h_{22}$.

H-parameter measurement

H-parameters can be measured with a VNA (via s-parameter conversion) or with dedicated test fixtures. At low frequencies, you can measure them directly: apply a known AC current at the base with the collector AC-shorted, then measure the resulting base voltage and collector current to extract $h_{11}$ and $h_{21}$.

Scattering parameters (s-parameters)

At high frequencies (typically above a few hundred MHz), it becomes impractical to create true short or open circuits at the device ports. S-parameters solve this by characterizing the device in terms of incident and reflected traveling waves, referenced to a characteristic impedance (usually $Z_0 = 50 \text{ }\Omega$).

S-parameter definition and matrix

$\begin{bmatrix} b_1 \\ b_2 \end{bmatrix} = \begin{bmatrix} s_{11} & s_{12} \\ s_{21} & s_{22} \end{bmatrix} \begin{bmatrix} a_1 \\ a_2 \end{bmatrix}$

Here $a_1, a_2$ are incident wave amplitudes and $b_1, b_2$ are reflected wave amplitudes, normalized to $\sqrt{Z_0}$.

• $s_{11}$: input reflection coefficient (with output matched to $Z_0$)
• $s_{21}$: forward transmission coefficient (forward gain)
• $s_{12}$: reverse transmission coefficient (reverse isolation)
• $s_{22}$: output reflection coefficient (with input matched to $Z_0$)

Why s-parameters dominate at RF

Unlike y, z, or h-parameters, s-parameters don't require short or open circuits, which are nearly impossible to realize cleanly at microwave frequencies. A VNA directly measures s-parameters by sending a known incident wave and measuring what comes back (reflected) and what passes through (transmitted).

S-parameter measurement

1. Connect the device under test (DUT) between the two ports of a calibrated VNA.
2. The VNA sends an incident wave from port 1 while port 2 is terminated in $Z_0$. It measures $s_{11}$ and $s_{21}$.
3. The VNA then sends from port 2 with port 1 terminated, measuring $s_{22}$ and $s_{12}$.
4. Calibration (SOLT, TRL, etc.) removes systematic errors from cables and connectors.

Parameter conversions

You'll often need to convert between parameter types. The key relationships:

Y ↔ Z conversion

Y-parameters and z-parameters are matrix inverses of each other:

$[Z] = [Y]^{-1} \qquad \text{and} \qquad [Y] = [Z]^{-1}$

For a 2×2 matrix, the inverse is straightforward. If $\Delta_y = y_{11}y_{22} - y_{12}y_{21}$, then:

$z_{11} = \frac{y_{22}}{\Delta_y}, \quad z_{12} = \frac{-y_{12}}{\Delta_y}, \quad z_{21} = \frac{-y_{21}}{\Delta_y}, \quad z_{22} = \frac{y_{11}}{\Delta_y}$

H-parameters to Y/Z conversion

These conversions involve algebraic manipulation of the individual parameters. For example, converting h-parameters to y-parameters:

$y_{11} = \frac{1}{h_{11}}, \quad y_{12} = \frac{-h_{12}}{h_{11}}, \quad y_{21} = \frac{h_{21}}{h_{11}}, \quad y_{22} = \frac{\Delta_h}{h_{11}}$

where $\Delta_h = h_{11}h_{22} - h_{12}h_{21}$.

S-parameters to other parameter sets

Converting s-parameters to y, z, or h-parameters requires the reference impedance $Z_0$. For example, converting to z-parameters:

$[Z] = Z_0 ([I] - [S])^{-1} ([I] + [S])$

where $[I]$ is the identity matrix. These conversions are typically handled by VNA software or CAD tools, but understanding the relationship matters for interpreting results correctly.

Small-signal parameter applications

Amplifier design

Small-signal parameters directly give you the quantities you need for amplifier design:

• Voltage gain of a common-emitter stage: $A_v = -g_m (R_C \| r_o)$
• Input impedance: determined by $r_\pi$ (and any feedback or degeneration resistors)
• Output impedance: determined by $r_o$ (modified by feedback topology)

H-parameters are standard for low-frequency BJT amplifier analysis. S-parameters take over for RF amplifier design, where stability analysis and impedance matching are done directly from the s-parameter data.

Oscillator design

The Barkhausen criterion for oscillation requires that the loop gain equal unity with zero phase shift. Small-signal parameters of the active device, combined with the feedback network, determine whether this condition is met at the desired frequency. S-parameters are preferred for RF oscillator design (e.g., VCOs, crystal oscillators at UHF).

Filter design

Active filters use transistors or op-amps as gain elements. The small-signal parameters of these devices affect the filter's frequency response, Q-factor, and dynamic range. Y-parameters and z-parameters are convenient for passive filter network synthesis, while s-parameters are used for distributed (microstrip, stripline) filter design at microwave frequencies.

Limitations of small-signal models

Frequency range limitations

The simple hybrid-π model works well at low-to-moderate frequencies. As frequency increases, parasitic capacitances ($C_\pi$, $C_\mu$) and lead inductances become significant. The transistor's gain rolls off, and the model must include these reactive elements to remain accurate. Beyond the device's $f_T$ (unity-gain frequency), the small-signal model predicts less than unity current gain, and the device can no longer amplify.

Amplitude range limitations

The linearization is only valid for small excursions around the Q-point. For a BJT, the exponential relationship $I_C = I_S e^{V_{BE}/V_T}$ means that even modest swings in $v_{be}$ (say, more than about 5-10 mV) introduce noticeable distortion. If your signal is large enough to push the transistor toward saturation or cutoff, you need large-signal (nonlinear) models instead.

Model accuracy considerations

• Process variations: Two transistors from the same batch can have different $\beta$ values, so $r_\pi$ and $g_m$ will differ.
• Temperature dependence: $V_T = kT/q$ changes with temperature, shifting all parameters. $I_C$ itself is temperature-sensitive.
• Device aging: Parameters can drift over the lifetime of a device, especially under stress conditions.

Always validate small-signal models against measured data when precision matters. Treat the model as an approximation that gets you close, then refine with simulation or measurement.