6.00 Two-Port Network Theory Compact Review

Chapter 6: Two-Port Network Theory - Compact Review

Overview

Two-port network theory treats linear electrical circuits as “black boxes,” modeling their behavior strictly through terminal variables: input/output voltages ($V_1,V_2$) and input/output currents ($I_1,I_2$). This compact note compiles all terminal equations, parameter definitions, reciprocity/symmetry boundaries, cascading rules, and parameter-to-parameter conversion matrices required for rapid pre-exam revision.

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6.01 Two-Port Network Terminal Variables & Z-Parameters

*(Target: Theory Descriptive / Formulas & Numerical Solving)*

• Two-Port Terminal Convention: Voltages are positive at upper terminals ($V_1,V_2$). Port currents ($I_1,I_2$) are modeled as flowing into the network.
• Z-Parameters (Open-Circuit Impedance): Relate port voltages directly to port currents: $V_1=Z_{11}{I}_1+Z_{12}{I}_2$ $V_2=Z_{21}{I}_1+Z_{22}{I}_2$ In matrix form: $\left[\begin{array}{c}{V}_1\\ V_2\end{array}\right]=\left[\begin{array}{cc}{Z}_{11}& Z_{12}\\ Z_{21}& Z_{22}\end{array}\right]\left[\begin{array}{c}{I}_1\\ I_2\end{array}\right]⟹\mathbf{V}=\mathbf{ZI}$
• Physical Parameters Definitions (Isolated by open-circuiting ports):
  • $Z_{11}$ (Input Impedance): $Z_{11}={\frac{V_1}{I_1}|}_{I_2=0}[\Omega ]$
  • $Z_{12}$ (Reverse Transfer Impedance): $Z_{12}={\frac{V_1}{I_2}|}_{I_1=0}[\Omega ]$
  • $Z_{21}$ (Forward Transfer Impedance): $Z_{21}={\frac{V_2}{I_1}|}_{I_2=0}[\Omega ]$
  • $Z_{22}$ (Output Impedance): $Z_{22}={\frac{V_2}{I_2}|}_{I_1=0}[\Omega ]$
• Z-Parameter Network Symmetries:
  • Reciprocity condition: $Z_{12}=Z_{21}$.
  • Symmetry condition: $Z_{11}=Z_{22}$.

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6.02 Short-Circuit Admittance (Y) Parameters

*(Target: Theory Descriptive / Formulas & Numerical Solving)*

• Y-Parameters (Short-Circuit Admittance): Relate port currents directly to port voltages: $I_1=Y_{11}{V}_1+Y_{12}{V}_2$ $I_2=Y_{21}{V}_1+Y_{22}{V}_2$ In matrix form: $\left[\begin{array}{c}{I}_1\\ I_2\end{array}\right]=\left[\begin{array}{cc}{Y}_{11}& Y_{12}\\ Y_{21}& Y_{22}\end{array}\right]\left[\begin{array}{c}{V}_1\\ V_2\end{array}\right]⟹\mathbf{I}=\mathbf{YV}$
• Physical Parameters Definitions (Isolated by short-circuiting ports):
  • $Y_{11}$ (Short-Circuit Input Admittance): $Y_{11}={\frac{I_1}{V_1}|}_{V_2=0}[\text{S}]$
  • $Y_{12}$ (Short-Circuit Reverse Transfer Admittance): $Y_{12}={\frac{I_1}{V_2}|}_{V_1=0}[\text{S}]$
  • $Y_{21}$ (Short-Circuit Forward Transfer Admittance): $Y_{21}={\frac{I_2}{V_1}|}_{V_2=0}[\text{S}]$
  • $Y_{22}$ (Short-Circuit Output Admittance): $Y_{22}={\frac{I_2}{V_2}|}_{V_1=0}[\text{S}]$
• Y-Parameter Network Symmetries:
  • Reciprocity condition: $Y_{12}=Y_{21}$.
  • Symmetry condition: $Y_{11}=Y_{22}$.
• Pi-to-Y Parametric Mapping (No derivation): For a resistive $\Pi$-network with series feedback branch $R_C$ and shunt branches $R_A$ (Port 1), $R_B$ (Port 2): $Y_{11}=\frac{1}{R_A}+\frac{1}{R_C},Y_{12}=Y_{21}=-\frac{1}{R_C},Y_{22}=\frac{1}{R_B}+\frac{1}{R_C}$

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6.03 The Fundamental Z-to-Y Matrix Inverse Identity

*(Target: Mathematical Proof / Parameter Conversion)*

• Philosophical Identity: Admittance and impedance matrices are exact mathematical inverses of each other: $\mathbf{Y}={\mathbf{Z}}^{-1}\text{and}\mathbf{Z}={\mathbf{Y}}^{-1}\text{(derivation)}$
• Algebraic Conversion Formulas: Let determinant ${\Delta }_Z=Z_{11}{Z}_{22}-Z_{12}{Z}_{21}$: $Y_{11}=\frac{Z_{22}}{{\Delta }_Z},Y_{12}=\frac{-Z_{12}}{{\Delta }_Z},Y_{21}=\frac{-Z_{21}}{{\Delta }_Z},Y_{22}=\frac{Z_{11}}{{\Delta }_Z}\text{(derivation)}$
• Let determinant ${\Delta }_Y=Y_{11}{Y}_{22}-Y_{12}{Y}_{21}$: $Z_{11}=\frac{Y_{22}}{{\Delta }_Y},Z_{12}=\frac{-Y_{12}}{{\Delta }_Y},Z_{21}=\frac{-Y_{21}}{{\Delta }_Y},Z_{22}=\frac{Y_{11}}{{\Delta }_Y}\text{(derivation)}$

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6.04 Transmission (ABCD) Parameters & Cascaded Networks

*(Target: Theory Descriptive / Formulas & Cascade Multiplying)*

• Negative Current Convention: Current $I_2$ is modeled as flowing out of Port 2 (represented as $-I_2$ in terminal equations) to simplify cascaded network series connections.
• ABCD-Parameters (Transmission Matrix): Relate sending-end variables ($V_1,I_1$) to receiving-end variables ($V_2,-I_2$): $V_1=A{V}_2-B{I}_2$ $I_1=C{V}_2-D{I}_2$ In matrix form: $\left[\begin{array}{c}{V}_1\\ I_1\end{array}\right]=\left[\begin{array}{cc}A& B\\ C& D\end{array}\right]\left[\begin{array}{c}{V}_2\\ -I_2\end{array}\right]⟹{\mathbf{X}}_{\text{send}}=\mathbf{T}{\mathbf{X}}_{\text{rec}}$
• Physical Parameters Definitions (Open-Circuit or Short-Circuit conditions):
  • $A$ (Open-Circuit Voltage Ratio, unitless): $A={\frac{V_1}{V_2}|}_{I_2=0}$
  • $B$ (Short-Circuit Transfer Impedance, $[\Omega ]$): $B={\frac{V_1}{-I_2}|}_{V_2=0}$
  • $C$ (Open-Circuit Transfer Admittance, $[\text{S}]$): $C={\frac{I_1}{V_2}|}_{I_2=0}$
  • $D$ (Short-Circuit Current Ratio, unitless): $D={\frac{I_1}{-I_2}|}_{V_2=0}$
• ABCD Network Symmetries:
  • Reciprocity condition: ${\Delta }_T=AD-BC=1$.
  • Symmetry condition: $A=D$.
• T-to-ABCD Parametric Mapping (No derivation): For a resistive T-network with series arms $R_1$, $R_2$ and common shunt arm $R_3$: $A=1+\frac{R_1}{R_3},B=R_1+R_2+\frac{R_1{R}_2}{R_3},C=\frac{1}{R_3},D=1+\frac{R_2}{R_3}\text{(derivation)}$
• Cascaded Networks Theorem: If two systems with transmission matrices ${\mathbf{T}}_1$ and ${\mathbf{T}}_2$ are connected in cascade (series), the overall transmission matrix ${\mathbf{T}}_{\text{total}}$ is the direct matrix product of individual matrices: ${\mathbf{T}}_{\text{total}}={\mathbf{T}}_1\cdot {\mathbf{T}}_2=\left[\begin{array}{cc}{A}_1& B_1\\ C_1& D_1\end{array}\right]\left[\begin{array}{cc}{A}_2& B_2\\ C_2& D_2\end{array}\right]\text{(derivation)}$

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6.05 Hybrid (h) & Inverse Hybrid (g) Parameter Formulations

*(Target: Theory Descriptive / Transistor Modeling Context)*

• h-Parameters (Hybrid): Combine series input voltage and parallel output current, selecting input current $I_1$ and output voltage $V_2$ as independent variables. Heavily used for Bipolar Junction Transistor (BJT) modeling: $V_1=h_{11}{I}_1+h_{12}{V}_2$ $I_2=h_{21}{I}_1+h_{22}{V}_2$ In matrix form: $\left[\begin{array}{c}{V}_1\\ I_2\end{array}\right]=\left[\begin{array}{cc}{h}_{11}& h_{12}\\ h_{21}& h_{22}\end{array}\right]\left[\begin{array}{c}{I}_1\\ V_2\end{array}\right]⟹{\mathbf{X}}_h=\mathbf{H}{\mathbf{U}}_h$
• Physical h-Parameter Definitions:
  • $h_{11}$ (Short-Circuit Input Impedance, $[\Omega ]$): $h_{11}={\frac{V_1}{I_1}|}_{V_2=0}$
  • $h_{12}$ (Open-Circuit Reverse Voltage Gain, unitless): $h_{12}={\frac{V_1}{V_2}|}_{I_1=0}$
  • $h_{21}$ (Short-Circuit Forward Current Gain, unitless): $h_{21}={\frac{I_2}{I_1}|}_{V_2=0}$
  • $h_{22}$ (Open-Circuit Output Admittance, $[\text{S}]$): $h_{22}={\frac{I_2}{V_2}|}_{I_1=0}$
• g-Parameters (Inverse Hybrid): The mathematical dual of h-parameters, selecting input voltage $V_1$ and output current $I_2$ as independent variables: $I_1=g_{11}{V}_1+g_{12}{I}_2$ $V_2=g_{21}{V}_1+g_{22}{I}_2$ In matrix form: $\left[\begin{array}{c}{I}_1\\ V_2\end{array}\right]=\left[\begin{array}{cc}{g}_{11}& g_{12}\\ g_{21}& g_{22}\end{array}\right]\left[\begin{array}{c}{V}_1\\ I_2\end{array}\right]⟹{\mathbf{X}}_g=\mathbf{G}{\mathbf{U}}_g$
• Physical g-Parameter Definitions:
  • $g_{11}$ (Open-Circuit Input Admittance, $[\text{S}]$): $g_{11}={\frac{I_1}{V_1}|}_{I_2=0}$
  • $g_{12}$ (Short-Circuit Reverse Current Gain, unitless): $g_{12}={\frac{I_1}{I_2}|}_{V_1=0}$
  • $g_{21}$ (Open-Circuit Forward Voltage Gain, unitless): $g_{21}={\frac{V_2}{V_1}|}_{I_2=0}$
  • $g_{22}$ (Short-Circuit Output Impedance, $[\Omega ]$): $g_{22}={\frac{V_2}{I_2}|}_{V_1=0}$
• g-to-h Matrix Inverse Identity: $\mathbf{G}={\mathbf{H}}^{-1}=\frac{1}{{\Delta }_h}\left[\begin{array}{cc}{h}_{22}& -h_{12}\\ -h_{21}& h_{11}\end{array}\right]\text{where }{\Delta }_h=h_{11}{h}_{22}-h_{12}{h}_{21}\text{(derivation)}$
• Hybrid Symmetries (Reciprocity & Symmetry Rules):
  • Reciprocity: $h_{12}=-h_{21}$ and $g_{12}=-g_{21}$.
  • Symmetry: ${\Delta }_h=h_{11}{h}_{22}-h_{12}{h}_{21}=1$ and ${\Delta }_g=g_{11}{g}_{22}-g_{12}{g}_{21}=1$.
• Transistor CE Equivalent Notations: Input resistance $h_{ie}=h_{11}$, reverse gain feedback $h_{re}=h_{12}$, active current amplification factor (beta) $h_{fe}=h_{21}$, and output channel admittance $h_{oe}=h_{22}$.

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6.06 Master Parameter Symmetries & Conversions

*(Target: Summary Comparisons & Conversion Matrices)*

Table 6.1 Two-Port Network Symmetries & Boundary Criteria

This comparison table compiles the terminal equations, independent variables, and identical symmetry constraints across all parameter models:

Parameter Set	Governing Matrix Equations	Independent Variables	Dependent Variables	Reciprocity Condition	Symmetry Condition
Impedance (Z)	$\mathbf{V}=\mathbf{ZI}$	$I_1,I_2$	$V_1,V_2$	$Z_{12}=Z_{21}$	$Z_{11}=Z_{22}$
Admittance (Y)	$\mathbf{I}=\mathbf{YV}$	$V_1,V_2$	$I_1,I_2$	$Y_{12}=Y_{21}$	$Y_{11}=Y_{22}$
Transmission (T)	${\mathbf{X}}_{\text{send}}=\mathbf{T}{\mathbf{X}}_{\text{rec}}$	$V_2,-I_2$	$V_1,I_1$	$AD-BC=1$	$A=D$
Hybrid (h)	${\mathbf{X}}_h=\mathbf{H}{\mathbf{U}}_h$	$I_1,V_2$	$V_1,I_2$	$h_{12}=-h_{21}$	${\Delta }_h=1$
Inverse Hybrid (g)	${\mathbf{X}}_g=\mathbf{G}{\mathbf{U}}_g$	$V_1,I_2$	$I_1,V_2$	$g_{12}=-g_{21}$	${\Delta }_g=1$

Table 6.2 Matrix Conversion Reference Matrix

Use this table to seamlessly convert any parameter matrix directly into another:

Matrix	Z-Parameters	Y-Parameters	ABCD-Parameters	h-Parameters
$[\mathbf{Z}]$	$\left[\begin{array}{cc}{Z}_{11}& Z_{12}\\ Z_{21}& Z_{22}\end{array}\right]$	$\frac{1}{{\Delta }_Y}\left[\begin{array}{cc}{Y}_{22}& -Y_{12}\\ -Y_{21}& Y_{11}\end{array}\right]$	$\left[\begin{array}{cc}\frac{A}{C}& \frac{{\Delta }_T}{C}\\ \frac{1}{C}& \frac{D}{C}\end{array}\right]$	$\left[\begin{array}{cc}\frac{{\Delta }_h}{h_{22}}& \frac{h_{12}}{h_{22}}\\ -\frac{h_{21}}{h_{22}}& \frac{1}{h_{22}}\end{array}\right]$
$[\mathbf{Y}]$	$\frac{1}{{\Delta }_Z}\left[\begin{array}{cc}{Z}_{22}& -Z_{12}\\ -Z_{21}& Z_{11}\end{array}\right]$	$\left[\begin{array}{cc}{Y}_{11}& Y_{12}\\ Y_{21}& Y_{22}\end{array}\right]$	$\left[\begin{array}{cc}\frac{D}{B}& -\frac{{\Delta }_T}{B}\\ -\frac{1}{B}& \frac{A}{B}\end{array}\right]$	$\left[\begin{array}{cc}\frac{1}{h_{11}}& -\frac{h_{12}}{h_{11}}\\ \frac{h_{21}}{h_{11}}& \frac{{\Delta }_h}{h_{11}}\end{array}\right]$
$[\mathbf{T}]$	$\left[\begin{array}{cc}\frac{Z_{11}}{Z_{21}}& \frac{{\Delta }_Z}{Z_{21}}\\ \frac{1}{Z_{21}}& \frac{Z_{22}}{Z_{21}}\end{array}\right]$	$\left[\begin{array}{cc}-\frac{Y_{22}}{Y_{21}}& -\frac{1}{Y_{21}}\\ -\frac{{\Delta }_Y}{Y_{21}}& -\frac{Y_{11}}{Y_{21}}\end{array}\right]$	$\left[\begin{array}{cc}A& B\\ C& D\end{array}\right]$	$\left[\begin{array}{cc}-\frac{{\Delta }_h}{h_{21}}& -\frac{h_{11}}{h_{21}}\\ -\frac{h_{22}}{h_{21}}& -\frac{1}{h_{21}}\end{array}\right]$
$[\mathbf{H}]$	$\left[\begin{array}{cc}\frac{{\Delta }_Z}{Z_{22}}& \frac{Z_{12}}{Z_{22}}\\ -\frac{Z_{21}}{Z_{22}}& \frac{1}{Z_{22}}\end{array}\right]$	$\left[\begin{array}{cc}\frac{1}{Y_{11}}& -\frac{Y_{12}}{Y_{11}}\\ \frac{Y_{21}}{Y_{11}}& \frac{{\Delta }_Y}{Y_{11}}\end{array}\right]$	$\left[\begin{array}{cc}\frac{B}{D}& \frac{{\Delta }_T}{D}\\ -\frac{1}{D}& \frac{C}{D}\end{array}\right]$	$\left[\begin{array}{cc}{h}_{11}& h_{12}\\ h_{21}& h_{22}\end{array}\right]$

Determinants: ${\Delta }_Z=Z_{11}{Z}_{22}-Z_{12}{Z}_{21}$, ${\Delta }_Y=Y_{11}{Y}_{22}-Y_{12}{Y}_{21}$, ${\Delta }_T=AD-BC$, ${\Delta }_h=h_{11}{h}_{22}-h_{12}{h}_{21}$.

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A. Common Mistakes That Cost Marks

Exam Pitfalls & Marks-Savers Checklist

1. The Negative Output Current Trap ($I_2^{\prime }$ vs. $I_2$): Standard two-port directions specify current entering both Port 1 and Port 2. However, ABCD parameters use $-I_2$ (leaving Port 2). Always check if your nodal loop currents have a negative sign on Port 2 terms when mapping to transmission forms.
2. Feedback Element Admittance Minus Sign: In resistive Pi-networks, the transfer admittance terms $Y_{12}$ and $Y_{21}$ are negative (i.e., $Y_{12}=Y_{21}=-1/R_C$). Forgetting the negative sign on transfer elements is a 3-mark penalty trap.
3. The Reciprocal Hybrid Sign Swap: Remember that in hybrid reciprocity, $h_{12}=-h_{21}$ (NOT $h_{12}=h_{21}$). A positive sign here indicates a non-reciprocal system (like an active transistor amplifier).

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B. PYQ Bank — Verbatim Questions & Answer Plans

1. Network Theory Classification [KUET 2025, 2022 - 4 to 5 Marks]

• Question: What is network theory? Classify the two-port network.
• Answer Plan:
  1. Define Network Theory (Section 6.00 Overview: study of analyzing, modeling, and solving electrical circuits using mathematical methods).
  2. Define Two-Port Network (Section 6.01: access terminals with input Port 1 and output Port 2).
  3. List and classify parameter matrix models: Open-circuit Impedance (Z), Short-circuit Admittance (Y), Hybrid (h), Inverse Hybrid (g), and Transmission (ABCD) parameters, mapping each to its specific application use-case.

2. General T-Network Parameter Extraction [KUET 2024, 2022 - 8 to 9 Marks]

• Question: Calculate the Z and Y parameters for a resistive T-network with resistor values of $$\begin{gathered} R_1=24 \\ ,\Omega \end{gathered}$$, $$\begin{gathered} R_2=8 \\ ,\Omega \end{gathered}$$, and $$\begin{gathered} R_3=8 \\ ,\Omega \end{gathered}$$.
• Answer Plan:
  1. Apply loop equations (KVL) around Port 1 and Port 2 to find Z-parameters: $V_1=32I_1+8I_2$, $V_2=8I_1+16I_2$.
  2. Write final Impedance matrix: $\mathbf{Z}=\left[\begin{array}{cc}32& 8\\ 8& 16\end{array}\right]\Omega$.
  3. Apply matrix inversion $[Y]=[Z{]}^{-1}$ to derive Y-parameters: ${\Delta }_Z=32\times 16-8\times 8=448⟹\mathbf{Y}=\frac{1}{448}\left[\begin{array}{cc}16& -8\\ -8& 32\end{array}\right]=\left[\begin{array}{cc}\frac{1}{28}& -\frac{1}{56}\\ -\frac{1}{56}& \frac{1}{14}\end{array}\right]\text{S}$

3. Cascaded Transmission ABCD Calculation [KUET 2025, 2024 - 12 Marks]

• Question: Obtain ABCD parameters for the network with series arms $$\begin{gathered} R_1=1 \\ ,\Omega \end{gathered}$$, $$\begin{gathered} R_2=2 \\ ,\Omega \end{gathered}$$, and shunt $$\begin{gathered} R_3=5 \\ ,\Omega \end{gathered}$$.
• Answer Plan:
  1. Substitute values into standard T-to-ABCD formulas:
     • $A=1+R_1/R_3=1+1/5=1.2$
     • $$\begin{gathered} B=R_1+R_2+R_1{R}_2/R_3=1+2+2/5=3.4 \\ ,\Omega \end{gathered}$$
     • $C=1/R_3=1/5=0.2\text{ S}$
     • $D=1+R_2/R_3=1+2/5=1.4$
  2. Assemble standard Transmission matrix: $\mathbf{T}=\left[\begin{array}{cc}1.2& 3.4\\ 0.2& 1.4\end{array}\right]$.

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C. Self-Check Before Moving On

• Can you write down all 5 sets of parameter matrix equations from memory?
• Do you know the exact reciprocity and symmetry conditions for Z, Y, ABCD, and h parameters?
• Can you convert a Z-parameter matrix to a Y-parameter matrix using the matrix inversion identity?
• Do you understand why ABCD parameters use a negative current convention for Port 2?

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Source: (k.Deergha Rao) signals and systems.pdf, 04 Network Theory.pdf, Rabiul sir class note.pdf