g Parameters

Two-port network analysis requires flexibility in choosing which port variables are independent and which are dependent. When the h parameter set uses input current and output voltage as independent variables, the g parameter set does exactly the opposite: it takes input voltage and output current as the independent variables. This complementary relationship makes g parameters the natural inverse of h parameters and useful in specific circuit configurations.

Core Concept of g Parameters

The g parameters, also called inverse hybrid parameters, describe a two-port network by expressing the input current I1 and the output voltage V2 as functions of the input voltage V1 and the output current I2. The name inverse hybrid comes from the fact that the role of voltages and currents is exactly swapped compared to h parameters: where h used I1 as independent, g uses V1; where h used V2 as independent, g uses I2.

The defining equations for the g parameter model are: I1 = g11 * V1 + g12 * I2 and V2 = g21 * V1 + g22 * I2. Each parameter is determined by setting either V1 = 0 (short-circuiting the input port) or I2 = 0 (open-circuiting the output port). The conditions are the complement of those used in h parameter measurement.

The parameter g11 is the input admittance with the output open-circuited (I2 = 0). The parameter g12 is the reverse current ratio with the input short-circuited (V1 = 0). The parameter g21 is the forward voltage gain with the output open-circuited, which corresponds to the open-circuit voltage amplification. The parameter g22 is the output impedance with the input short-circuited.

Mathematical Expression

The g parameter matrix equation in compact form is: [I1 / V2] = [g11 g12 / g21 g22] * [V1 / I2]. The units of g11 are siemens (admittance), g12 is dimensionless, g21 is dimensionless (voltage gain), and g22 is ohms (impedance). This unit distribution is the direct complement of h parameters: where h11 is in ohms, g11 is in siemens, and so on.

The relationship between the g matrix and the h matrix is an inversion: [g] = [h]^(-1). This means that if you know all four h parameters of a network, the g parameters can be computed from the inverse of the 2x2 h matrix using the standard formula for 2x2 matrix inversion. Specifically, g11 = h22 / (h11*h22 - h12*h21), g12 = -h12 / (h11*h22 - h12*h21), and similarly for g21 and g22.

Practical Understanding

The g parameter set is less commonly used in transistor circuit analysis compared to h parameters, but it appears in specific configurations. Common-base and common-gate transistor configurations are more naturally described using g parameters because in those configurations, the input tends to behave as a voltage-controlled device where input voltage is the natural excitation variable.

In FET (Field Effect Transistor) small-signal modeling, the transconductance gm is essentially g21 of the device. The forward voltage gain from gate to drain in open-circuit output condition maps directly to the g21 parameter. Understanding g parameters therefore connects to understanding FET models, which is relevant for both GATE Electronics and analog design courses.

Given:
h11 = 2 kΩ, h12 = 1 × 10⁻³, h21 = 40, h22 = 50 µS

Why this formula applies:
g matrix = [h matrix]⁻¹ using 2×2 inversion formula
g = (1 / det[h]) * [h22 -h12 / -h21 h11]

Formula:
det[h] = h11 * h22 - h12 * h21

Substitution:
det[h] = (2000)(50 × 10⁻⁶) - (1 × 10⁻³)(40)
det[h] = 0.1 - 0.04 = 0.06

Calculation:
g11 = h22 / det[h] = 50 × 10⁻⁶ / 0.06 = 8.33 × 10⁻⁴ S
g12 = -h12 / det[h] = -1 × 10⁻³ / 0.06 = -0.01667
g21 = -h21 / det[h] = -40 / 0.06 = -666.7
g22 = h11 / det[h] = 2000 / 0.06 = 33.33 kΩ

Final Answer:
g11 = 833 µS, g12 = -0.0167, g21 = -666.7, g22 = 33.33 kΩ

Exam Tip: The g matrix is the inverse of the h matrix. For GATE, if h parameters are given and g parameters are asked, use [g] = [h]^(-1) via the 2x2 inverse formula. Also remember: g11 is admittance (siemens), g22 is impedance (ohms), and g12, g21 are dimensionless, which is the exact reverse of h parameter units.

Mechanism: g Parameter Measurement Conditions

• g11 is measured as I1/V1 with I2 = 0 (output open), giving input admittance. Its units are siemens.
• g21 is measured as V2/V1 with I2 = 0 (output open), giving forward open-circuit voltage gain. It is dimensionless.
• g12 is measured as I1/I2 with V1 = 0 (input short), giving reverse short-circuit current ratio. It is dimensionless.
• g22 is measured as V2/I2 with V1 = 0 (input short), giving output impedance. Its units are ohms.
• The g matrix equals the inverse of the h matrix. If h parameters are given, g parameters follow by 2x2 matrix inversion using det[h] = h11*h22 - h12*h21.

Quick Revision

• g equations: I1 = g11*V1 + g12*I2 and V2 = g21*V1 + g22*I2. Independent variables are V1 and I2.
• g11 (siemens) = I1/V1 at I2=0; g12 (unitless) = I1/I2 at V1=0; g21 (unitless) = V2/V1 at I2=0; g22 (ohms) = V2/I2 at V1=0.
• [g] = [h]^(-1). Use det[h] = h11*h22 - h12*h21 for matrix inversion.
• g parameter units are complementary to h: where h11 is ohms, g11 is siemens; where h22 is siemens, g22 is ohms.
• g21 represents open-circuit voltage gain, directly related to transconductance gm in FET models.
• Exam trap: g11 is admittance, not impedance. Do not confuse input admittance g11 with input impedance h11.
• For a reciprocal network: g12 = -g21, same structural condition as h parameters.