What are h (hybrid) parameters and why are they used for transistors?

The hybrid or h parameters mix the two forms: V1 = h11 I1 + h12 V2 and I2 = h21 I1 + h22 V2, so the four parameters have mixed units. They are popular for transistors because they map onto the common-emitter small-signal model: h11 is the input resistance (h-i-e), h21 is the forward current gain (h-f-e or beta), h12 is the reverse voltage ratio and h22 is the output admittance (h-o-e).

What is the difference between a two-port and a four-terminal network?

Every two-port has four terminals, but a four-terminal network is only a two-port if the current entering one terminal of each port equals the current leaving the other terminal of the same port, the so-called port condition. When that condition holds, the four terminals form two proper ports and the two-port parameters apply.

Two-Port Networks: Z, Y, h & ABCD Parameters Explained

Q: What is a two-port network?

A two-port network is a circuit treated as a black box with two pairs of terminals: an input port and an output port. It is described by four variables, the port voltages V1 and V2 and the port currents I1 and I2. Two equations relate them, and the coefficients are the network's parameters. Amplifiers, filters, transmission lines and transformers are all modelled as two-ports.

Q: What are the Z (impedance) parameters?

The Z or impedance parameters express the port voltages in terms of the port currents: V1 = z11 I1 + z12 I2 and V2 = z21 I1 + z22 I2. They have units of ohms and are called open-circuit parameters because each is found by open-circuiting a port so that one current is zero, for example z11 = V1/I1 with I2 = 0.

Q: What are the Y (admittance) parameters?

The Y or admittance parameters express the port currents in terms of the port voltages: I1 = y11 V1 + y12 V2 and I2 = y21 V1 + y22 V2. They have units of siemens and are called short-circuit parameters because each is found by short-circuiting a port so that one voltage is zero, for example y11 = I1/V1 with V2 = 0.

Q: What are the ABCD (transmission) parameters?

The transmission or ABCD parameters give the input in terms of the output: V1 = A V2 - B I2 and I1 = C V2 - D I2, with I2 taken as leaving the output port. Their big advantage is cascading: when two-ports are connected in a chain, the overall ABCD matrix is simply the product of the individual ABCD matrices, which makes them ideal for transmission lines and cascaded stages.

Q: Why are there so many parameter sets?

Because different problems make different variables convenient. Z and Y suit networks combined in series or parallel; h parameters suit transistor amplifiers; ABCD parameters suit cascaded stages and transmission lines. All the sets describe the same network and can be converted from one to another, so you choose whichever makes the algebra easiest.

Q: When are the Z parameters not defined?

A parameter set fails to exist when the required matrix cannot be inverted. For example, an ideal series impedance has no Y parameters, and an ideal shunt admittance has no Z parameters, because the corresponding open- or short-circuit measurement gives an infinite or indeterminate value. In such cases you use a different parameter set that is well defined for that network.

Q: What is a reciprocal two-port network?

A two-port is reciprocal if it contains only passive, bilateral elements such as resistors, capacitors and inductors, with no dependent sources. For a reciprocal network the transfer parameters are equal: z12 = z21, y12 = y21, h12 = -h21, and the ABCD determinant AD - BC = 1. Networks with transistors or other dependent sources are generally not reciprocal.

Q: What is a symmetrical two-port network?

A two-port is symmetrical if its two ports can be swapped without changing its behaviour, which happens when the network looks the same from either port. Then the driving-point parameters are equal, z11 = z22 and y11 = y22, and for the ABCD parameters A = D. A symmetrical network is also always reciprocal.

Q: How do you combine two-port networks?

The best parameter set depends on the connection. For two-ports in series (their input and output ports in series) the Z matrices add. For two-ports in parallel the Y matrices add. For two-ports in cascade (output of one into the input of the next) the ABCD matrices multiply. Choosing the matching parameter set turns the combination into simple matrix addition or multiplication.

What Is a Two-Port Network?

A two-port network is any circuit treated as a black box with two pairs of terminals — an input port and an output port. You do not care what is inside; you only care how the port voltages and currents relate.

Each port has a voltage and a current, giving four variables: V₁, I₁ at the input and V₂, I₂ at the output. Two equations link them, so if you know any two variables the other two follow. The coefficients in those equations are the two-port parameters — a compact fingerprint of the whole circuit.

A two-port network drawn as a black box N with an input port carrying V1 and I1 and an output port carrying V2 and I2

A two-port network N: an input port (V₁, I₁) and an output port (V₂, I₂). The relationship between these four variables is the network's fingerprint.

2 ports

Input & output

4 vars

V₁, I₁, V₂, I₂

2 eqns

Relate them

Z Y h T

Parameter sets

Why it's powerful

A complicated amplifier or filter collapses to just four numbers. You can analyse, cascade and design with those numbers without ever re-solving the internal circuit.

The Four Variables & Sign Convention

Everything rests on a clear convention: both port currents are taken to flow into the network at the top terminal, and each port voltage is measured + at the top.

The four variables of a two-port network and the standard convention with both currents flowing into the network

Four variables, two equations, and the parameter sets that follow. Each set simply chooses which two variables are the causes and which two are the effects.

Two of the four variables are treated as independent (the causes) and two as dependent (the effects). Which two you pick decides the parameter set: currents as causes gives the Z parameters, voltages as causes gives the Y parameters, and mixed choices give the h and ABCD parameters.

Impedance (Z) Parameters

The Z parameters treat the currents as causes and give the voltages: they are open-circuit parameters, measured in ohms.

Z parameters express the port voltages from the port currents. Each is found by open-circuiting a port so one current is zero.

V₁ = z₁₁I₁ + z₁₂I₂ · V₂ = z₂₁I₁ + z₂₂I₂

open-circuit test: z₁₁ = V₁/I₁ and z₂₁ = V₂/I₁ with I₂ = 0 (output open)

z₁₁ and z₂₂ are the driving-point (input/output) impedances; z₁₂ and z₂₁ are the transfer impedances. Z parameters are handy when two-ports are connected in series, because then the Z matrices simply add.

Admittance (Y) Parameters

The Y parameters are the mirror image: voltages are the causes and currents the effects. They are short-circuit parameters, measured in siemens.

Y parameters express the port currents from the port voltages. Each is found by short-circuiting a port so one voltage is zero.

I₁ = y₁₁V₁ + y₁₂V₂ · I₂ = y₂₁V₁ + y₂₂V₂

short-circuit test: y₁₁ = I₁/V₁ and y₂₁ = I₂/V₁ with V₂ = 0 (output short)

The Y matrix is the inverse of the Z matrix ([Y] = [Z]⁻¹). Y parameters are the natural choice when two-ports are connected in parallel, because then the Y matrices add.

Hybrid (h) Parameters & the Transistor Model

The h parameters mix the two views — input current and output voltage are the causes — which is exactly what suits a transistor.

Hybrid h parameters and the common-emitter small-signal transistor model with hie, hfe and hoe

The h parameters map straight onto the common-emitter small-signal model of a BJT: h_ie input resistance, h_fe current gain, h_oe output admittance.

V₁ = h₁₁I₁ + h₁₂V₂ · I₂ = h₂₁I₁ + h₂₂V₂

h₁₁ = h_ie (input R, Ω), h₂₁ = h_fe (current gain β), h₁₂ = h_re, h₂₂ = h_oe (output S)

Because the four h parameters have mixed units (an ohm, a siemens and two dimensionless ratios), they are called "hybrid". Datasheets quote a transistor's h_fe (its current gain, β) and h_ie directly.

Transmission (ABCD) Parameters & Cascade

The ABCD (transmission or chain) parameters write the input in terms of the output — the one form that multiplies neatly when networks are chained.

ABCD transmission parameters V1 = A V2 - B I2, I1 = C V2 - D I2, and the cascade rule that the ABCD matrices multiply

ABCD parameters relate input to output. Cascade two-ports and the overall matrix is just the product [T] = [T_a]·[T_b].

V₁ = A·V₂ − B·I₂ · I₁ = C·V₂ − D·I₂

I₂ is taken as leaving the output port; for a reciprocal network AD − BC = 1

Here A is a voltage ratio, D a current ratio, B an impedance and C an admittance. Cascading is the killer feature: chain several stages and you just multiply their ABCD matrices in order — the reason they dominate transmission-line and filter analysis.

Choosing & Combining Parameter Sets

Every set describes the same network, so pick the one that makes your problem easiest — especially when interconnecting two-ports.

Parameters	Cause → Effect	Units	Best for
Z (impedance)	I → V (open-circuit)	Ω	Two-ports in series (Z matrices add)
Y (admittance)	V → I (short-circuit)	S	Two-ports in parallel (Y matrices add)
h (hybrid)	I₁, V₂ → V₁, I₂	mixed	Transistor amplifiers
ABCD (transmission)	output → input	mixed	Two-ports in cascade (matrices multiply)

Reciprocity & symmetry

A network of only passive, bilateral parts is reciprocal: z₁₂ = z₂₁, y₁₂ = y₂₁, AD − BC = 1. If it also looks the same from either port it is symmetrical: z₁₁ = z₂₂ and A = D.

Applications

Two-port modelling is a workhorse of analog, RF and power engineering.

Transistor amplifiers

h and y parameters model BJTs and FETs for gain and impedance.

Cascaded stages

ABCD matrices multiply through multi-stage amplifiers and filter chains.

Transmission lines

Lines and cables are modelled as ABCD two-ports for power flow and matching.

Filter design

π and T (delta/wye) sections are two-ports combined by Z, Y or ABCD.

Transformers

A transformer is a classic two-port with simple ABCD parameters.

RF & microwave

S-parameters extend the two-port idea to high frequencies.

Key Terms at a Glance

The essential two-port vocabulary students and engineers search for.

Port

A terminal pair with equal in/out current.

Z parameters

Open-circuit impedances (Ω).

Y parameters

Short-circuit admittances (S).

h parameters

Hybrid, mixed units; transistor model.

ABCD

Transmission; multiply to cascade.

Reciprocal

z₁₂ = z₂₁, AD−BC = 1.

Frequently Asked Questions

Quick, clear answers to the questions people ask most about two-port networks.

What is a two-port network?

A circuit treated as a black box with two terminal pairs — an input port and an output port — described by four variables: V₁, I₁, V₂, I₂. Two equations relate them, and the coefficients are the network's parameters. Amplifiers, filters, lines and transformers are all modelled this way.

What are the Z (impedance) parameters?

They give the voltages from the currents: V₁ = z₁₁I₁ + z₁₂I₂, V₂ = z₂₁I₁ + z₂₂I₂. In ohms, they are open-circuit parameters — each found with a port open so one current is zero (e.g. z₁₁ = V₁/I₁ with I₂ = 0).

What are the Y (admittance) parameters?

They give the currents from the voltages: I₁ = y₁₁V₁ + y₁₂V₂, I₂ = y₂₁V₁ + y₂₂V₂. In siemens, they are short-circuit parameters (a port shorted so one voltage is zero). The Y matrix is the inverse of the Z matrix.

What are h parameters and why are they used for transistors?

The hybrid set: V₁ = h₁₁I₁ + h₁₂V₂, I₂ = h₂₁I₁ + h₂₂V₂, with mixed units. They map onto the BJT small-signal model: h_ie input resistance, h_fe current gain (β), h_oe output admittance — the values datasheets quote.

What are the ABCD (transmission) parameters?

They give input from output: V₁ = A V₂ − B I₂, I₁ = C V₂ − D I₂ (I₂ leaving the output). Their advantage is cascading: chain two-ports and the overall ABCD matrix is the product of the individual matrices.

Why are there so many parameter sets?

Different problems make different variables convenient: Z and Y for series/parallel combinations, h for transistor amplifiers, ABCD for cascaded stages and lines. All describe the same network and convert to each other, so you use whichever gives the simplest algebra.

When are the Z parameters not defined?

When the required matrix can't be inverted. An ideal series impedance has no Y parameters, and an ideal shunt admittance has no Z parameters, because the open- or short-circuit measurement becomes infinite or indeterminate. Then you switch to a parameter set that is well defined for that network.

What is a reciprocal two-port network?

One built only from passive, bilateral elements (R, L, C) with no dependent sources. For a reciprocal network z₁₂ = z₂₁, y₁₂ = y₂₁, h₁₂ = −h₂₁ and AD − BC = 1. Transistor networks are generally not reciprocal.

What is a symmetrical two-port network?

One that looks the same from either port, so the ports can be swapped without changing its behaviour. Then z₁₁ = z₂₂, y₁₁ = y₂₂ and A = D. A symmetrical network is always reciprocal.

How do you combine two-port networks?

Match the parameter set to the connection: series → add the Z matrices, parallel → add the Y matrices, cascade → multiply the ABCD matrices. The right choice turns the combination into simple matrix addition or multiplication.

Conclusion & Key Takeaways

A two-port network reduces any circuit to four numbers relating its input and output — and the Z, Y, h and ABCD sets are just four ways to write them.

Black box, 2 ports

Four variables V₁, I₁, V₂, I₂.

Z: open-circuit

V from I, in Ω; add in series.

Y: short-circuit

I from V, in S; add in parallel.

h: hybrid

Transistor small-signal model.

ABCD: cascade

Multiply to chain stages.

Reciprocal / symmetric

z₁₂=z₂₁; z₁₁=z₂₂.

Two-Port Networks

Complete Learning Path — Two-Port Networks