Wye-Delta Transformations

• Resistive networks that aren’t in series or parallel may be simplified to a three-terminal equivalent network
  • Either the Wye/Tee network or the Delta/Pi network

Delta-Wye Conversion

• Superimpose Y onto $\Delta$ by adding an extra node $n$. Each resistor in Y is the product of the resistors in the two adjacent $\Delta$ branches divided by the sum of the three $\Delta$ resistors
• For the network below:

$$\begin{align} R_{1} &= \frac{R_{b}R_{c}}{R_{a}+R_{b}+R_{c}} \\ R_{2} &= \frac{R_{c}R_{a}}{R_{a}+R_{b}+R_{c}} \\ R_{3} &= \frac{R_{a}R_{b}}{R_{a}+R_{b}+R_{c}} \end{align} $$

Wye-Delta Conversion

• Each resistor in the $\Delta$ network is the sum of all products of Y resistors divided by the opposite Y resistor
• For the superimposed network:

$$\begin{align} R_{a} &= \frac{R_{1}R_{2}+R_{2}R_{3}+R_{3}R_{1}}{R_{1}} \\ R_{b} &= \frac{R_{1}R_{2}+R_{2}R_{3}+R_{3}R_{1}}{R_{2}} \\ R_{c} &= \frac{R_{1}R_{2}+R_{2}R_{3}+R_{3}R_{1}}{R_{3}} \end{align}$$

Balanced Condition

• If all the Y resistances are equal AND all the $\Delta$ resistances are equal, then:

$$R_{Y} = \frac{R_{\Delta}}{3} \ \text{or} \ R_{\Delta}= 3R_{Y}$$

Example 02.25.

(Practice Problem 2.14 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 80])

Example 02.26.

(Practice Problem 2.15 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 83])

Two-Port Networks

Overview

1. Define a matrix model for the system using the network type
2. Selectively zero out each parameter
3. Use methods of analysis to rewrite the output equations in terms of the input variables
4. Solve the output equations

Network 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}

$$

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}

$$

Hybrid

$$

\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}

$$

Inverse Hybrid

$$

\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}

$$

Transmission

$$

\begin{bmatrix}

V_1 \

I_1

\end{bmatrix}

=

\begin{bmatrix}

A & B \

C & D

\end{bmatrix}

\begin{bmatrix}

V_2 \

• I_2

\end{bmatrix}

$$

Example 10.1 (A two-port network is given below. Find the parameters $Z_{11}, Z_{12}, Z_{21}, Z_{22}$ that relate the currents and voltages of the circuit through the equations $V_1 = Z_{11}I_1 + Z_{12}I_2$ and $V_2 = Z_{21}I_1 + Z_{22}I_2$).

References

• Two-Port and Three-Port Networks (course handout)