Example

Let's find the impedance and hybrid parameter equivalent circuits of the circuit shown in Figure 58.

**Figure 58:** Example circuit.
$\begin{figure} \begin{center} \epsfig{file=images/tpnimg5.eps}\end{center}\end{figure}$

Impedance. Inspection of (67) indicates that we can find z₁₁ and z₂₁ by open-circuiting the right port (i₂=0), and injecting a current i₁ in the left port and determining the two voltages, Figure 59.

**Figure 59:** Example circuit with experiment for determining z₁₁ and z₂₁.
$\begin{figure} \begin{center} \epsfig{file=images/tpnimg6.eps}\end{center}\end{figure}$

In this experiment there will be no voltage drop across the 2 k $\Omega$ resistor, and the current i₁ flows through both 1 k $\Omega$ resistors. Equation (67) says that

$\begin{displaymath}\begin{array}{rl} v_1 & = z_{11}i_1 , \\ \\ v_2 & = z_{21} i_1 . \end{array}\end{displaymath}$

The circuit of Figure 59 says that

$\begin{displaymath}\begin{array}{rl} v_1 & = (1k+1k) i_1 \\ \\ v_2 & = 1k i_1 . \end{array}\end{displaymath}$

Therefore we get

$\begin{displaymath}z_{11} = 2 \ k\Omega, \ \ z_{21} = 1 \ k\Omega . \end{displaymath}$

To find z₁₂ and z₂₂ we need a second experiment, Figure 60.

**Figure 60:** Example circuit with experiment for determining z₁₂ and z₂₂.
$\begin{figure} \begin{center} \epsfig{file=images/tpnimg7.eps}\end{center}\end{figure}$

Equation (67) says that

$\begin{displaymath}\begin{array}{rl} v_1 & = z_{12}i_2 , \\ \\ v_2 & = z_{22} i_2 . \end{array}\end{displaymath}$

The circuit of Figure 60 says that

$\begin{displaymath}\begin{array}{rl} v_1 & = 1k i_1 \\ \\ v_2 & = (2k + 1k) i_1 . \end{array}\end{displaymath}$

Therefore we get

$\begin{displaymath}z_{12} = 1 \ k\Omega, \ \ z_{22} = 3k \ \Omega . \end{displaymath}$

Now that the impedance parameters have been found, you can draw the impedance parameter equivalent circuit model (exercise).

Hybrid. Inspection of (68) indicates that we can find h₁₁ and h₂₁ by short circuiting the right port, injecting a current i₁ in the left port and determining v₁ and i₂, Figure 61.

**Figure 61:** Example circuit with experiment for determining h₁₁ and h₂₁.
$\begin{figure} \begin{center} \epsfig{file=images/tpnimg8.eps}\end{center}\end{figure}$

Equation (68) says that

$\begin{displaymath}\begin{array}{rl} v_1 & = h_{11}i_1 , \\ \\ i_2 & = h_{21} i_1 . \end{array}\end{displaymath}$

From 61, we have

$\begin{displaymath}v_1 = (1k + 1k \parallel 2k) i_1 , \end{displaymath}$

and also by current division

$\begin{displaymath}-i_2 = \frac{1k}{1k+2k} i_1 . \end{displaymath}$

Therefore

$\begin{displaymath}h_{11} = \frac{5}{3} \ k\Omega, \ \ h_{21} = -\frac{1}{3} . \end{displaymath}$

Exercise. Show that h₁₂ = 1/3, $h_{22} = 1/3 \times 10^{-3}$ $\Omega^{-1}$ and draw the hybrid parameter equivalent circuit. Hint. Open circuit the left port and apply a voltage v₂ to the right port.

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