6.01 Open-Circuit Impedance (Z) Parameters | 6.03 Transmission (ABCD) Parameters & Cascaded Networks
6.02 Short-Circuit Admittance (Y) Parameters
Core Idea
Short-Circuit Admittance (Y) Parameters model the terminal behavior of a two-port network by expressing the input/output currents () as linear combinations of the terminal voltages (). Unlike Z-parameters, which rely on open-circuit conditions, Y-parameters are calculated by selectively short-circuiting the ports ( or ). This formulation is particularly optimized for analyzing parallel-connected networks and forms the mathematical basis for admittance matrix transformations.
1. The Philosophy of Admittance Modeling
In circuit analysis, impedance represents a component’s opposition to current, whereas admittance represents the ease with which current flows through it {the mathematical reciprocal of impedance, measured in Siemens or Mhos}.
When analyzing multiple subnetworks connected in parallel, open-circuit parameters become algebraically complex to manipulate because the terminal voltages are identical across parallel nodes. By selecting the terminal voltages () as our independent variables, we can directly sum the terminal currents. This parallel-summation property makes Y-parameters the natural choice for multi-port power networks and transistor modeling.
2. The Core Y-Parameter Formulation
The Y-parameters relate the port currents () directly to the port voltages () through two coupled linear algebraic equations:
Expressing these equations in standard compact vector-matrix form yields:
egin{bmatrix} I_1 \ I_2 \end{bmatrix} = egin{bmatrix} Y_{11} & Y_{12} \ Y_{21} & Y_{22} \end{bmatrix} egin{bmatrix} V_1 \ V_2 \end{bmatrix}
2.1 Individual Parameter Physical Definitions
By selectively forcing one of the port voltages to zero, we isolate and calculate each parameter. Since corresponds to a short-circuit condition, these are termed the Short-Circuit Admittance Parameters:
| Parameter | Mathematical Isolation | Physical Name | Description |
|---|---|---|---|
| $$Y_{11} = \left. rac{I_1}{V_1} | |||
| ight | _{V_2=0}$$ | Short-Circuit Input Admittance at Port 1 | The admittance looking into Port 1 when Port 2 is short-circuited. |
| $$Y_{12} = \left. rac{I_1}{V_2} | |||
| ight | _{V_1=0}$$ | Short-Circuit Reverse Transfer Admittance | The ratio of Port 1 current to Port 2 voltage when Port 1 is short-circuited. |
| $$Y_{21} = \left. rac{I_2}{V_1} | |||
| ight | _{V_2=0}$$ | Short-Circuit Forward Transfer Admittance | The ratio of Port 2 current to Port 1 voltage when Port 2 is short-circuited. |
| $$Y_{22} = \left. rac{I_2}{V_2} | |||
| ight | _{V_1=0}$$ | Short-Circuit Output Admittance at Port 2 | The admittance looking into Port 2 when Port 1 is short-circuited. |
3. Network Classification Criteria
3.1 Reciprocal Networks
A network is reciprocal if the transfer admittances are equal, meaning the forward current induced by an input voltage matches the reverse current induced by the same voltage:
- Reciprocity Condition:
3.2 Symmetrical Networks
A network is symmetrical if its input and output port admittance characteristics are identical, meaning the port terminals can be physically reversed in a circuit without changing the system response:
- Symmetry Condition:
4. The Fundamental Z-to-Y Matrix Inverse Identity
Since open-circuit Z-parameters relate voltages to currents (), and short-circuit Y-parameters relate currents to voltages (), the admittance matrix is the exact mathematical inverse of the impedance matrix:
For any two-port network matrix, this inverse relationship is expressed as:
egin{bmatrix} Y_{11} & Y_{12} \ Y_{21} & Y_{22} \end{bmatrix} = egin{bmatrix} Z_{11} & Z_{12} \ Z_{21} & Z_{22} \end{bmatrix}^{-1} = rac{1}{\Delta_Z} egin{bmatrix} Z_{22} & -Z_{12} \ -Z_{21} & Z_{11} \end{bmatrix}
where the determinant of the impedance matrix is .
4.1 Parameter-by-Parameter Conversion Equations
\mathbf{Y_{11} = rac{Z_{22}}{\Delta_Z}}, \quad \mathbf{Y_{12} = rac{-Z_{12}}{\Delta_Z}}, \quad \mathbf{Y_{21} = rac{-Z_{21}}{\Delta_Z}}, \quad \mathbf{Y_{22} = rac{Z_{11}}{\Delta_Z}}
The Determinant Check
If a network is reciprocal, then . If a network is symmetrical, then . This serves as a rapid double-check during exam conditions.
5. General Resistive -Network Derivation
The -network (or Pi-network) is the canonical circuit topology used to represent short-circuit Y-parameters, just as the T-network naturally maps to open-circuit Z-parameters.
feedback: Rc
+----/\/\/\/\----+
| |
Port 1 _ _ _ _ Port 2
V1 | | Ra | | Rb V2
| |
o-----------+------------------+-----------o
To derive the parameters, we apply Kirchhoff’s Current Law (KCL) at the two terminal nodes:
Step 1: Write KCL at Node 1
The current entering Port 1 () splits into the shunt branch and the series feedback branch : I_1 = rac{V_1}{R_A} + rac{V_1 - V_2}{R_C} Grouping voltage terms yields:
ight) V_1 - \left( rac{1}{R_C} ight) V_2 \quad ext{--- (Equation A)}$$ ### Step 2: Write KCL at Node 2 The current entering Port 2 ($I_2$) splits into the shunt branch $R_B$ and the series feedback branch $R_C$: $$I_2 = I_{Rb} - I_{Rc}$$ $$I_2 = rac{V_2}{R_B} + rac{V_2 - V_1}{R_C}$$ Grouping voltage terms yields: $$I_2 = - \left( rac{1}{R_C} ight) V_1 + \left( rac{1}{R_B} + rac{1}{R_C} ight) V_2 \quad ext{--- (Equation B)}$$ ### Step 3: Compare with Standard Y-Parameter Equations Comparing Equations A and B directly with the standard definition equations yields the general Pi-to-Y parameter mapping: $$\mathbf{Y_{11} = rac{1}{R_A} + rac{1}{R_C}}$$ $$\mathbf{Y_{12} = - rac{1}{R_C}}$$ $$\mathbf{Y_{21} = - rac{1}{R_C}}$$ $$\mathbf{Y_{22} = rac{1}{R_B} + rac{1}{R_C}}$$ --- ## 6. Comprehensive Worked Examples (The Exam Killers) ### 6.1 Example 1: The 13-Mark resistive $\Pi$-Network [PYQ 2025 Question 4c - 13 Marks] **Question:** For the resistive $\Pi$-network shown below with component values of $R_A = 2\,\Omega$, $R_B = 2\,\Omega$, and series feedback resistor $R_C = 6\,\Omega$, find the short-circuit Y-parameters. ``` Rc = 6 Ohm +----/\/\/\/\----+ | | Port 1 _ _ _ _ Port 2 V1 | | Ra = 2 Ohm | | Rb = 2 Ohm V2 | | o-----------+------------------+-----------o ``` #### Step-by-Step Nodal KCL Solution: 1. **Define Port Status for $Y_{11}$ and $Y_{21}$ ($V_2 = 0$):** Short-circuit Port 2, making $V_2 = 0$. This places the feedback resistor $R_C = 6\,\Omega$ directly in parallel with the Port 1 shunt resistor $R_A = 2\,\Omega$, while the shunt resistor $R_B = 2\,\Omega$ is shorted out and carries no current. * **Calculate $Y_{11}$:** The input current $I_1$ is: $$I_1 = rac{V_1}{R_A} + rac{V_1}{R_C} = V_1 \left( rac{1}{2} + rac{1}{6} ight) = V_1 \left( rac{3}{6} + rac{1}{6} ight) = rac{4}{6} V_1 = rac{2}{3} V_1$$ $$Y_{11} = \left. rac{I_1}{V_1} ight|_{V_2=0} = \mathbf{ rac{2}{3} pprox 0.667 ext{ S}}$$ * **Calculate $Y_{21}$:** With $V_2 = 0$, the short-circuit current $I_2$ flows *into* Node 2 from the feedback arm. Because of the current direction convention, $I_2$ is negative relative to the current leaving Node 1: $$I_2 = rac{V_2 - V_1}{R_C} = rac{0 - V_1}{6} = - rac{1}{6} V_1$$ $$Y_{21} = \left. rac{I_2}{V_1} ight|_{V_2=0} = \mathbf{- rac{1}{6} pprox -0.167 ext{ S}}$$ 2. **Define Port Status for $Y_{22}$ and $Y_{12}$ ($V_1 = 0$):** Short-circuit Port 1, making $V_1 = 0$. This shorts out the shunt resistor $R_A = 2\,\Omega$ and puts $R_C = 6\,\Omega$ in parallel with $R_B = 2\,\Omega$. * **Calculate $Y_{22}$:** The output current $I_2$ is: $$I_2 = rac{V_2}{R_B} + rac{V_2}{R_C} = V_2 \left( rac{1}{2} + rac{1}{6} ight) = rac{2}{3} V_2$$ $$Y_{22} = \left. rac{I_2}{V_2} ight|_{V_1=0} = \mathbf{ rac{2}{3} pprox 0.667 ext{ S}}$$ * **Calculate $Y_{12}$:** The feedback current entering Node 1 is: $$I_1 = rac{V_1 - V_2}{R_C} = rac{0 - V_2}{6} = - rac{1}{6} V_2$$ $$Y_{12} = \left. rac{I_1}{V_2} ight|_{V_1=0} = \mathbf{- rac{1}{6} pprox -0.167 ext{ S}}$$ 3. **Assemble the Admittance Matrix:** $$\mathbf{Y} = egin{bmatrix} Y_{11} & Y_{12} \ Y_{21} & Y_{22} \end{bmatrix} = egin{bmatrix} rac{2}{3} & - rac{1}{6} \ - rac{1}{6} & rac{2}{3} \end{bmatrix} ext{ S}$$ 4. **Evaluate Reciprocity and Symmetry:** * **Reciprocity:** Since $Y_{12} = Y_{21} = - rac{1}{6} ext{ S}$, the network is **reciprocal**. * **Symmetry:** Since $Y_{11} = Y_{22} = rac{2}{3} ext{ S}$, the network is **symmetrical**. --- ### 6.2 Example 2: Inversion of Exam Impedance Matrix [PYQ 2022 Question 4b - 9 Marks] **Question:** Obtain the Y-parameters for the 2022 T-network exam problem by directly inverting its impedance matrix $[Z] = egin{bmatrix} 32 & 8 \ 8 & 16 \end{bmatrix} \Omega$. #### Step-by-Step Matrix Inversion Solution: 1. **Calculate the Impedance Determinant ($\Delta_Z$):** $$\Delta_Z = Z_{11}Z_{22} - Z_{12}Z_{21}$$ $$\Delta_Z = (32)(16) - (8)(8) = 512 - 64 = \mathbf{448}$$ 2. **Apply the Inverse Matrix Formula:** $$\mathbf{Y} = \mathbf{Z}^{-1} = rac{1}{\Delta_Z} egin{bmatrix} Z_{22} & -Z_{12} \ -Z_{21} & Z_{11} \end{bmatrix}$$ $$\mathbf{Y} = rac{1}{448} egin{bmatrix} 16 & -8 \ -8 & 32 \end{bmatrix}$$ 3. **Evaluate Individual Admittance Coefficients:** * $$Y_{11} = rac{16}{448} = \mathbf{ rac{1}{28} ext{ S}} pprox 0.0357 ext{ S}$$ * $$Y_{12} = rac{-8}{448} = \mathbf{- rac{1}{56} ext{ S}} pprox -0.0179 ext{ S}$$ * $$Y_{21} = rac{-8}{448} = \mathbf{- rac{1}{56} ext{ S}} pprox -0.0179 ext{ S}$$ * $$Y_{22} = rac{32}{448} = \mathbf{ rac{1}{14} ext{ S}} pprox 0.0714 ext{ S}$$ 4. **Assemble the Final Admittance Matrix:** $$\mathbf{Y} = egin{bmatrix} rac{1}{28} & - rac{1}{56} \ - rac{1}{56} & rac{1}{14} \end{bmatrix} ext{ S}$$ --- ## 7. Common Mistakes That Cost Marks > [!danger] **The Negative Sign Omission on Transfer Admittances** > > The most frequent mark-loss in Chapter 6 examinations occurs when writing the transfer parameters $Y_{12}$ and $Y_{21}$ for a resistive Pi-network. Because port current $I_1$ flows **into** the upper node, KCL dictates that the current flowing from Node 1 to Node 2 is proportional to the difference $(V_1 - V_2)$. This introduces a negative sign on $V_2$ in the first port equation, making $Y_{12} = -1/R_C$. Writing these transfer terms as positive values will result in a **50% deduction** of the numerical question's marks. > [!warning] **The Inverse Unit Trap** > > Admittance is the reciprocal of impedance, meaning its unit is **Siemens (S)** or **Mhos ($\mho$)**. Writing the unit of Y-parameters as **Ohms ($\Omega$)** indicates a fundamental conceptual misunderstanding, resulting in a **1 to 2 mark penalty** on any numerical evaluation. --- ## 8. PYQ Bank — Verbatim Questions & Answer Plans ### 8.1 PYQ 2025 Question 4c [13 Marks] **Question:** For the following $\Pi$-network as shown in Fig. 4(c), find the Y-parameters of it. * **Answer Plan:** 1. Draw the schematic of the resistive Pi-network, labeling shunt resistors $R_A, R_B$ and feedback resistor $R_C$. 2. Write the Node 1 and Node 2 KCL algebraic equations as derived in **Section 5**. 3. Plug in the specific resistor values and solve the short-circuit cases as shown in **Section 6.1**. 4. Assemble the final matrix and state that the network is reciprocal and symmetrical. ### 8.2 PYQ 2022 Question 4b [9 Marks] **Question:** Find the Z & Y parameters of two port network shown in Fig. 4(b). * **Answer Plan:** 1. Obtain the Z-parameters first using loop analysis (as shown in **Note 6.01, Section 4.1**). 2. Apply the fundamental matrix inverse identity $\mathbf{Y} = \mathbf{Z}^{-1}$. 3. Compute $\Delta_Z = 448$ and substitute terms to derive the exact fractional admittances shown in **Section 6.2**. --- ## 9. Self-Check Before Moving On - [ ] Can you define Short-Circuit Admittance mathematically and physically? [10.01] - [ ] Do you know how to derive the Pi-network parameters using nodal KCL analysis? [10.02] - [ ] Have you memorized the warning to always include a negative sign on $Y_{12}$ and $Y_{21}$? [10.03] - [ ] Can you convert any Z-parameter matrix into Y-parameters using the determinant inverse formula? [10.04] --- *Source: (k.Deergha Rao) signals and systems.pdf (Chapter 6), 04 Network Theory.pdf, Rabiul sir class note.pdf.*