In this problem, we need to determine the input capacitance, $C_{\text{in}}$, seen by the source in the given circuit. We can use the Miller theorem to express this capacitance in terms of the amplifier gain $A$ and the capacitance $C$ connected between the amplifier's input and output.

Miller's Theorem

Miller's theorem allows us to split a feedback capacitance (or impedance) connected between the input and output of an amplifier into two separate capacitances at the input and output, based on the gain of the amplifier. This is especially useful for analyzing circuits where feedback components influence the input and output impedance.

For a capacitance $C$ between the input and output, the Miller equivalent input capacitance $C_{\text{in}}$ is given by: $C_{\text{in}} = C (1 - A)$ where:

• $C$ is the feedback capacitance (in this case, connected between the input and output).
• $A$ is the gain of the amplifier.

Applying to the Problem

In the given circuit:

1. The amplifier has a voltage gain of $A$.
2. The capacitance $C$ is connected between the input and output.

Using the Miller theorem, we can rewrite the capacitance $C$ as an equivalent input capacitance: $C_{\text{in}} = C (1 - A)$

Final Answer

Thus, the input capacitance $C_{\text{in}}$ in terms of $A$ and $C$ is: $C_{\text{in}} = C (1 - A)$

Would you like further details on how Miller’s theorem was applied, or have any additional questions?

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Here are 5 related questions to deepen understanding:

1. How does the Miller effect impact the bandwidth of an amplifier circuit?
2. What happens to $C_{\text{in}}$ when the gain $A$ is negative?
3. How would you find the output capacitance $C_{\text{out}}$ in terms of $A$ and $C$?
4. If the gain $A$ is unity (i.e., $A = 1$), what will be the input capacitance $C_{\text{in}}$?
5. Why is it beneficial to treat the amplifier as ideal in this analysis?

Tip: When analyzing circuits with feedback elements, always check if the Miller theorem can simplify the equivalent input and output impedance.