The driving-point impedance Z(s) of a network has the pole-zero locations as shown. If Z(0) = 3, then Z(s) is

To find the driving-point impedance $Z(s)$ of the network based on the given pole-zero locations and the condition $Z(0) = 3$, we can follow these steps:

1. ### Break Down the Problem

1. Identify the pole and zero locations.
2. Use the pole-zero information to formulate the impedance function.
3. Apply the condition $Z(0) = 3$ to determine any constants.

2. ### Relevant Concepts

• The general form of a driving-point impedance can be represented as: $Z(s) = K \frac{(s-z_1)(s-z_2)\ldots(s-z_m)}{(s-p_1)(s-p_2)\ldots(s-p_n)}$ where $K$ is a constant, $z_i$ are the zeros, and $p_j$ are the poles.

3. ### Analysis and Detail

Assuming the specific pole-zero locations were provided (but not explicitly stated here), we would substitute those into the formula. For example, let’s say we have:

• Zeros at $z_1$ and $z_2$
• Poles at $p_1$ and $p_2$

Then, the impedance function can be written as: $Z(s) = K \frac{(s - z_1)(s - z_2)}{(s - p_1)(s - p_2)}$

Now, we need to apply the condition $Z(0) = 3$: $Z(0) = K \frac{(-z_1)(-z_2)}{(-p_1)(-p_2)} = K \frac{z_1 z_2}{p_1 p_2}$ Setting this equal to 3: $K \frac{z_1 z_2}{p_1 p_2} = 3$

4. ### Verify and Summarize

By isolating $K$: $K = \frac{3 p_1 p_2}{z_1 z_2}$

With all values for zeros and poles identified, we could write out the final expression for $Z(s)$.

Final Answer

Thus, the final impedance function $Z(s)$ can be expressed as: $Z(s) = \left( \frac{3 p_1 p_2}{z_1 z_2} \right) \frac{(s - z_1)(s - z_2)}{(s - p_1)(s - p_2)}$

Make sure to provide the specific pole and zero values to finalize the equation.