6.00 Chapter Map - Two-Port Networks | 6.02 Short-Circuit Admittance (Y) Parameters

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6.01 Open-Circuit Impedance (Z) Parameters

Core Idea

Two-port network theory simplifies complex electrical networks by treating them as a “black box” characterized entirely by their terminal behaviors {the voltages and currents measurable at the external ports}. The Open-Circuit Impedance (Z) Parameters model this black box by expressing the port voltages ($V_1,V_2$) as linear combinations of the port currents ($I_1,I_2$). These parameters are calculated by selectively open-circuiting one port at a time, establishing a robust matrix representation of linear networks.

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1. Defining Two-Port Terminal Variables

An electrical network is classified as a two-port network if it possesses exactly two pairs of access terminals: Port 1 (usually designated as the input port) and Port 2 (usually designated as the output port).

To mathematically analyze these networks as black boxes, we enforce the Two-Port Terminal Convention:

• All port voltages are defined with positive polarity at the upper terminals ($V_1,V_2$).
• Crucial Port Current Direction Rule: Both terminal currents $I_1$ and $I_2$ must be modeled as flowing into their respective ports.

         I1 ---->                           <---- I2
        +--------o-------------------------o--------+
        |        |                         |        |
        |        |                         |        |
    V1  |      Port 1                    Port 2     |  V2
        |  (Input Terminal)          (Output Term.) |
        |        |                         |        |
        +--------o-------------------------o--------+

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2. The Core Z-Parameter Formulation

The Z-parameters relate the terminal voltages directly to the terminal currents through a system of two coupled linear algebraic equations:

$V_1=Z_{11}{I}_1+Z_{12}{I}_2$ $V_2=Z_{21}{I}_1+Z_{22}{I}_2$

Expressing these equations in standard compact vector-matrix form yields:

egin{bmatrix} V_1 \ V_2 \end{bmatrix} = egin{bmatrix} Z_{11} & Z_{12} \ Z_{21} & Z_{22} \end{bmatrix} egin{bmatrix} I_1 \ I_2 \end{bmatrix}

$\mathbf{V}=\mathbf{ZI}$

2.1 Individual Parameter Physical Definitions

By selectively forcing one of the port currents to zero, we isolate and calculate each parameter. Since $I=0$ corresponds to an open-circuit condition, these are termed the Open-Circuit Impedance Parameters:

Parameter	Mathematical Isolation	Physical Name	Description
$Z_{11}$	$$Z_{11} = \left. rac{V_1}{I_1}
ight	_{I_2=0}$$	Input Impedance	The impedance looking into Port 1 when Port 2 is open-circuited.
$Z_{12}$	$$Z_{12} = \left. rac{V_1}{I_2}
ight	_{I_1=0}$$	Reverse Transfer Impedance	The ratio of Port 1 voltage to Port 2 current when Port 1 is open-circuited.
$Z_{21}$	$$Z_{21} = \left. rac{V_2}{I_1}
ight	_{I_2=0}$$	Forward Transfer Impedance	The ratio of Port 2 voltage to Port 1 current when Port 2 is open-circuited.
$Z_{22}$	$$Z_{22} = \left. rac{V_2}{I_2}
ight	_{I_1=0}$$	Output Impedance	The impedance looking into Port 2 when Port 1 is open-circuited.

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3. Network Classification Criteria

3.1 Reciprocal Networks

A network is reciprocal if the transmission of a signal from Port 1 to Port 2 is identical to the transmission from Port 2 to Port 1. For any passive network containing only bilateral linear components {resistors, capacitors, and inductors, with no dependent sources}, the network is guaranteed to be reciprocal.

• Reciprocity Condition: $Z_{12}=Z_{21}$

3.2 Symmetrical Networks

A network is symmetrical if its electrical characteristics do not change when the input and output ports are physically swapped. This requires the network to have identical input and output terminal behaviors.

• Symmetry Condition: $Z_{11}=Z_{22}$

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4. High-Yield Worked Examples (The Exam Killers)

4.1 Example 1: The Resistive T-Network [PYQ 2022 Question 4b - 9 Marks]

Question: Find the Z-parameters for the resistive T-network shown in Fig. 4(b) with branch resistor values of $R_1=24\Omega$, $R_2=8\Omega$, and $R_3=8\Omega$.

              24 Ohm                  8 Ohm
        o-----/\/\/\/\-------+-------/\/\/\/\-----o
        +                    |                    +
     V1                     _ _  8 Ohm         V2
        -                    |                    -
        o--------------------+--------------------o

Step-by-Step Loop Equation Solution:

1. Define Loop Currents: Define loop current $I_1$ circulating clockwise in Loop 1 (left side) and loop current $I_2$ circulating counter-clockwise in Loop 2 (right side). Both currents flow into the upper terminals and meet in the common shunt branch $R_3=8\Omega$.

2. Write KVL around Loop 1: The voltage $V_1$ must equal the sum of the voltage drops across the series arm $R_1$ and the common shunt arm $R_3$: $V_1=I_1{R}_1+(I_1+I_2)R_3$ Substitute $R_1=24\Omega$ and $R_3=8\Omega$: $V_1=24I_1+8I_1+8I_2$ ${\mathbf{V}}_1=32{\mathbf{I}}_1+8{\mathbf{I}}_{2}ext---(Equation1)$

3. Write KVL around Loop 2: The voltage $V_2$ must equal the sum of the voltage drops across the series arm $R_2$ and the common shunt arm $R_3$: $V_2=I_2{R}_2+(I_1+I_2)R_3$ Substitute $R_2=8\Omega$ and $R_3=8\Omega$: $V_2=8I_2+8I_1+8I_2$ ${\mathbf{V}}_2=8{\mathbf{I}}_1+16{\mathbf{I}}_{2}ext---(Equation2)$

4. Compare with Standard Matrix Equations: Compare Equations 1 and 2 directly with the standard parameter definition equations:

   • $V_1=Z_{11}{I}_1+Z_{12}{I}_2⟹Z_{11}=32\Omega ,Z_{12}=8\Omega$
   • $V_2=Z_{21}{I}_1+Z_{22}{I}_2⟹Z_{21}=8\Omega ,Z_{22}=16\Omega$

5. Assemble the Final Impedance Matrix: \mathbf{Z} = egin{bmatrix} Z_{11} & Z_{12} \ Z_{21} & Z_{22} \end{bmatrix} = egin{bmatrix} 32 & 8 \ 8 & 16 \end{bmatrix} \Omega

6. Evaluate Reciprocity and Symmetry:

   • Reciprocity: Since $Z_{12}=Z_{21}=8\Omega$, the network is reciprocal.
   • Symmetry: Since $Z_{11}=32\Omega eq{Z}_{22}=16\Omega$, the network is asymmetrical.

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4.2 Example 2: Classroom Modified T-Network (The Scribbled Page Classic)

Question: Obtain the Z-parameters for the modified T-network where the branch values are scaled down to $R_1=2\Omega$, $R_2=4\Omega$, and $R_3=5\Omega$.

               2 Ohm                   4 Ohm
        o-----/\/\/\/\-------+-------/\/\/\/\-----o
        +                    |                    +
     V1                     _ _  5 Ohm         V2
        -                    |                    -
        o--------------------+--------------------o

Step-by-Step Loop Equation Solution:

1. Write KVL around Loop 1: $V_1=I_1{R}_1+(I_1+I_2)R_3=I_1(2)+5(I_1+I_2)$ ${\mathbf{V}}_1=7{\mathbf{I}}_1+5{\mathbf{I}}_2$

2. Write KVL around Loop 2: $V_2=I_2{R}_2+(I_1+I_2)R_3=I_2(4)+5(I_1+I_2)$ ${\mathbf{V}}_2=5{\mathbf{I}}_1+9{\mathbf{I}}_2$

3. Compare and Extract Parameters:

   • $Z_{11}=7\Omega$ (Input Impedance)
   • $Z_{12}=5\Omega$ (Reverse Transfer Impedance)
   • $Z_{21}=5\Omega$ (Forward Transfer Impedance)
   • $Z_{22}=9\Omega$ (Output Impedance)

4. Assemble the Final Matrix: \mathbf{Z} = egin{bmatrix} 7 & 5 \ 5 & 9 \end{bmatrix} \Omega This scaled network remains reciprocal ($Z_{12}=Z_{21}=5\Omega$) and asymmetrical ($Z_{11}eq{Z}_{22}$).

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4.3 Example 3: Challenge Classic — The Coupled Multi-Loop Lattice

Question: Calculate the Z-parameters for the bridged ladder network shown below, which contains an extra top-loop path.

                  Top Arm: 4 Ohm
            +-------/\/\/\/\-------+
            |                      |
            +--/\/\/\--+--/\/\/\---+
              1 Ohm    |    3 Ohm
                                              / 2 Ohm
                                              |
        o--------------+-----------o

Step-by-Step Nodal-Loop Combined Solution:

Let loop current $I_1$ enter Port 1, loop current $I_2$ enter Port 2, and let $I_3$ be the internal loop current circulating clockwise in the top loop.

1. Formulate Loop Equations:

   • Loop 1 (Port 1): $V_1=3I_1+2I_2-2I_{3}ext---(EquationA)$
   • Loop 2 (Port 2): $V_2=2I_1+5I_2+3I_{3}ext---(EquationB)$
   • Top Loop (Internal Loop 3): 8 I_3 - I_1 + 3 I_2 = 0 \implies I_3 = rac{1}{8} I_1 - rac{3}{8} I_2 \quad ext{--- (Equation C)}

2. Substitute Equation C back into Equations A and B to eliminate the internal state $I_3$:

   • Substitute in $V_1$:

ight)V_1 = 3 I_1 + 2 I_2 - rac{1}{4} I_1 + rac{3}{4} I_2V_1 = \left(3 - rac{1}{4} ight) I_1 + \left(2 + rac{3}{4} ight) I_2\mathbf{V_1 = rac{11}{4} I_1 + rac{11}{4} I_2} \implies Z_{11} = 2.75,\Omega, \ Z_{12} = 2.75,\Omega * **Substitute in $V_2$:** V_2 = 2 I_1 + 5 I_2 + 3\left(rac{1}{8} I_1 - rac{3}{8} I_2 ight)V_2 = 2 I_1 + 5 I_2 + rac{3}{8} I_1 - rac{9}{8} I_2V_2 = \left(2 + rac{3}{8} ight) I_1 + \left(5 - rac{9}{8} ight) I_2\mathbf{V_2 = rac{19}{8} I_1 + rac{31}{8} I_2} \implies Z_{21} = 2.375,\Omega, \ Z_{22} = 3.875,\Omega$$

3. Assemble the Final Matrix: \mathbf{Z} = egin{bmatrix} rac{11}{4} & rac{11}{4} \ rac{19}{8} & rac{31}{8} \end{bmatrix} = egin{bmatrix} 2.75 & 2.75 \ 2.375 & 3.875 \end{bmatrix} \Omega Note: Because this bridged network contains asymmetrical feedback paths, $Z_{12}eq{Z}_{21}$ ($2.75eq2.375$), making it non-reciprocal.

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5. Common Mistakes That Cost Marks

The Port Current Direction Convention Slip

Examiners love to test your attention to detail by drawing Port 2 with current $I_2$ flowing out of the port rather than into it. If you use KVL with an outward current, you must manually substitute $I_{2}o-I_2$ before extracting the Z-parameters. Forgetting this sign change negates the values of $Z_{12}$ and $Z_{22}$, leading to a zero-mark grading.

Dependent Loop/KCL Redundancy Trap

When solving multi-loop networks, do not write KCL at the reference ground node. This introduces redundant dependent variables that cannot be solved algebraically. Always choose independent nodal adder voltages or clockwise loop currents as your variables.

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6. PYQ Bank — Verbatim Questions & Answer Plans

6.1 PYQ 2022 Question 4b [9 Marks]

Question: Find the Z & Y parameters of two port network shown in Fig. 4(b).

• Answer Plan:
  1. Redraw Fig. 4(b) with labeled terminal variables ($V_1,I_1,V_2,I_2$) and clockwise loop currents.
  2. Write the KVL equations around Loop 1 and Loop 2 as shown in Section 4.1.
  3. Compare terms to isolate $Z_{11}=32\Omega ,Z_{12}=8\Omega ,Z_{21}=8\Omega ,Z_{22}=16\Omega$.
  4. State that Y-parameters can be found by inverting this Z-matrix ($[Y]=[Z{]}^{-1}$), which yields: [Y] = rac{1}{(32)(16) - 8^2} egin{bmatrix} 16 & -8 \ -8 & 32 \end{bmatrix} = egin{bmatrix} rac{1}{28} & -rac{1}{56} \ -rac{1}{56} & rac{1}{14} \end{bmatrix} \mho

6.2 Foundational Concept PYQ 2025/2022 [5 Marks]

Question: What is network theory? Classify the two-port parameters.

• Answer Plan:
  1. Define Network Theory as the study of analyzing, modeling, and solving electrical circuits using rigorous mathematical formulations.
  2. Define a Two-Port Network as a circuit block with two terminals on each side allowing energy to flow in and out.
  3. Classify and tabulate the five primary parameter sets (Z, Y, h, g, ABCD) as structured in the Chapter Map (MOC) Section 2.

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7. Self-Check Before Moving On

• Can you state the official terminal sign convention for currents ($I_1,I_2$) entering a two-port network? [9.01]
• Do you know how to write KVL loop equations for any T-network to extract Z-parameters? [9.02]
• Have you memorized the formal reciprocity and symmetry criteria for Z-parameters ($Z_{12}=Z_{21}$ and $Z_{11}=Z_{22}$)? [9.03]
• Can you explain why Z-parameters are called “open-circuit” parameters? [9.04]

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Source: (k.Deergha Rao) signals and systems.pdf (Chapter 6), 04 Network Theory.pdf, Rabiul sir class note.pdf.