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Two Port Networks

Z, Y, h, g, ABCD parameters, interconnections, symmetry & reciprocity conditions.

Introduction & Variables

2-Port N/W Port 1 Vā‚ + - Iā‚ Port 2 Vā‚‚ + - Iā‚‚

4 Variables: Vā‚, Iā‚, Vā‚‚, Iā‚‚

Choose 2 as dependent, 2 as independent.
Possible combinations: ${}^4C_2 = 6$

Explore the 6 Parameter Sets:

Independent: Iā‚, Iā‚‚

Dependent: Vā‚, Vā‚‚

Z-Parameters (Impedance)

$$ \begin{bmatrix} V_1 \\ V_2 \end{bmatrix} = \begin{bmatrix} Z_{11} & Z_{12} \\ Z_{21} & Z_{22} \end{bmatrix} \begin{bmatrix} I_1 \\ I_2 \end{bmatrix} $$
(Measured with ports open circuited)
$Z_{11}$
Input Impedance
$(I_2=0)$
$Z_{12}$
Reverse Transfer
$(I_1=0)$
$Z_{21}$
Forward Transfer
$(I_2=0)$
$Z_{22}$
Output Impedance
$(I_1=0)$
Iā‚ [Z] Vā‚‚ Port 2 Open (Iā‚‚=0)

Y-Parameters (Admittance)

$$ \begin{bmatrix} I_1 \\ I_2 \end{bmatrix} = \begin{bmatrix} Y_{11} & Y_{12} \\ Y_{21} & Y_{22} \end{bmatrix} \begin{bmatrix} V_1 \\ V_2 \end{bmatrix} $$
(Measured with ports short circuited)
$Y_{11}$
Input Admittance
$(V_2=0)$
$Y_{12}$
Reverse Transfer
$(V_1=0)$
$Y_{21}$
Forward Transfer
$(V_2=0)$
$Y_{22}$
Output Admittance
$(V_1=0)$
Vā‚ [Y] Iā‚‚ Port 2 Shorted (Vā‚‚=0)

Hybrid Parameters

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

Mixed units: $h_{11}(\Omega), h_{12}(\text{pure}), h_{21}(\text{pure}), h_{22}(S)$

BJT Small Signal Model h_ie h_re*Vce h_fe*Ib 1/h_oe
$h_{11}$: Input Imp. ($\Omega$)
$h_{21}$: Current Gain (pure)
$$ \begin{bmatrix} I_1 \\ V_2 \end{bmatrix} = \begin{bmatrix} g_{11} & g_{12} \\ g_{21} & g_{22} \end{bmatrix} \begin{bmatrix} V_1 \\ I_2 \end{bmatrix} $$

Inverse pattern: $[g] = [h]^{-1}$

FET Model Concept Gate Cgs Drain gm*Vgs ro
$g_{21}$: Transconductance
$g_{22}$: Output Imp. ($\Omega$)

ABCD (Transmission) Parameters

$$ \begin{bmatrix} V_1 \\ I_1 \end{bmatrix} = \begin{bmatrix} A & B \\ C & D \end{bmatrix} \begin{bmatrix} V_2 \\ -I_2 \end{bmatrix} $$

Note the $-I_2$ (current leaving port 2)

$A = \frac{V_1}{V_2}\big|_{I_2=0}$
(pure)
$B = \frac{V_1}{-I_2}\big|_{V_2=0}$
($\Omega$)
$C = \frac{I_1}{V_2}\big|_{I_2=0}$
(S)
$D = \frac{I_1}{-I_2}\big|_{V_2=0}$
(pure)
$$ \text{Reciprocal: } AD - BC = 1 $$
$$ \text{Symmetrical: } A = D $$

abcd (Inverse Transmission)

$$ \begin{bmatrix} V_2 \\ I_2 \end{bmatrix} = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} V_1 \\ -I_1 \end{bmatrix} $$

Relationship with ABCD:

$$ a = D, \quad b = B $$
$$ c = C, \quad d = A $$

(Assuming Reciprocal, $\Delta_T = 1$)

Conditions for Symmetry & Reciprocity

Parameter Reciprocal Condition Symmetrical Condition
[Z] $Z_{12} = Z_{21}$ $Z_{11} = Z_{22}$
[Y] $Y_{12} = Y_{21}$ $Y_{11} = Y_{22}$
[h] $h_{12} = -h_{21}$ $h_{11}h_{22} - h_{12}h_{21} = 1$
[g] $g_{12} = -g_{21}$ $g_{11}g_{22} - g_{12}g_{21} = 1$
[ABCD] $AD - BC = 1$ $A = D$
Live Condition Checker
Reciprocal: YES Symmetrical: YES

Connections of Two-Port Networks

$$ [Z_{total}] = [Z_A] + [Z_B] $$

Impedance matrices simply add together.

[ZA] [ZB]

Transformer as Two-Port Network

V1 V2 n = 2

Turns ratio $n = \frac{N_2}{N_1}$

$$ \begin{bmatrix} A & B \\ C & D \end{bmatrix} = \begin{bmatrix} 1/n & 0 \\ 0 & n \end{bmatrix} $$
A = 0.5, B = 0
C = 0, D = 2.0

T & Ļ€ Network Conversions

Z1 Z2 Z3
T-Network Parameters
$$ [Z] = \begin{bmatrix} Z_1+Z_3 & Z_3 \\ Z_3 & Z_2+Z_3 \end{bmatrix} $$
Z1:
Z2:
Z3:
[40, 30]
[30, 50]

Parameter Interconversion Calculator

Input any complete parameter set to compute all others.

[Z]
[ 10, 5 ]
[ 5, 20 ]
[Y]
...
[ABCD]
...
[h]
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