Example - Sinusoidal Steady-state Response

Since this is DC, we can pretend that the capacitor is just a resistor:

We call the resistor $Z_C$ since we know it’s some impedance $Z_C$. With a resistor, the circuit becomes a voltage divider such that:

$$V_{out}=V_{in}\cdot \frac{R}{R+Z_C}$$

We also know that the value of our resistor is:

$$Z_C=\frac{1}{j\omega C}$$

Thus, we have:

$$\frac{V_{out}}{V_{in}}=\frac{R}{R+\frac{1}{j\omega C}}=\frac{j\omega RC}{j\omega RC+1}$$

Qualitatively, we have an open circuit at DC (remember how the capacitor symbol looks open), which means that $V_{out}$ is only connected to the resistor to ground, such that $V_{out}=0$. If we have $f\to \infty$, we would instead have $V_{out}=V_{in}$.

This means we’ve basically made a Highpass Filter – high frequencies make it through but low frequencies are blocked.

The magnitude of $V_{out}/V_{in}$ is:

$$\left|\frac{V_{out}}{V_{in}}\right|=\frac{|j\omega RC|}{|j\omega RC+1|}$$

The phase is:

$$\angle \left(\frac{V_{out}}{V_{in}}\right)=\angle (j\omega RC)-\angle (j\omega RC+1)$$