Concept Flashcards

1) What three passive components are used to create analog filters?

Resistors, inductors, and capacitors (R, L, C).

2) What is the impedance of a capacitor in steady-state AC analysis?

\[ Z_C = -\frac{j}{\omega C}\ \Omega \]

3) What is the impedance of an inductor in steady-state AC analysis?

\[ Z_L = j\omega L\ \Omega \]

4) What is the relationship between angular frequency and frequency?

\[ \omega = 2\pi f \]

5) For a series R–C circuit with output across the capacitor, what is the gain

\(\frac{V_C}{V_S}\)?

\[ \frac{V_C}{V_S}=\frac{1}{j\omega RC + 1}=\frac{1}{j2\pi fRC + 1} \]

6) How do you determine if a 2-element filter is low pass or high pass using limits?

Evaluate the gain at:

\(f \rightarrow 0\) (DC / low frequency)

\(f \rightarrow \infty\) (very high frequency)

If gain \(\rightarrow 1\) at low frequency and \(\rightarrow 0\) at high frequency → Low Pass Filter (LPF).
If gain \(\rightarrow 0\) at low frequency and \(\rightarrow 1\) at high frequency → High Pass Filter (HPF).

7) Why is an R–C circuit (output across the capacitor) a low pass filter?

Because:

\[ \lim_{f\rightarrow 0}\frac{V_C}{V_S}=1 \quad\text{and}\quad \lim_{f\rightarrow\infty}\frac{V_C}{V_S}=0 \]

Low frequencies pass, high frequencies are attenuated.

8) What is the “3 dB point” and why does it define cutoff frequency?

The cutoff frequency is commonly defined at the **3 dB point**, where: - Output power is half the input power:

\(P_{out}=\frac{P_{in}}{2}\)

Because power is proportional to voltage squared, this corresponds to:

\[ \left|\frac{V_{out}}{V_{in}}\right| = \frac{1}{\sqrt{2}} \approx 0.707 \]

9) What is the magnitude of a complex number

\(A = \text{Re} + j\text{Im}\)?

\[ |A| = \sqrt{(\text{Re})^2 + (\text{Im})^2} \]

10) What is the magnitude form of the R–C (output across C) gain?

Starting with:

\[ \frac{V_C}{V_S}=\frac{1}{1+j2\pi fRC} \]

Magnitude:

\[ \left|\frac{V_C}{V_S}\right| = \frac{1}{\sqrt{1^2+(2\pi fRC)^2}} \]

11) What is the cutoff frequency for an R–C filter?

\[ f_{cutoff}=\frac{1}{2\pi RC} \]

12) For a series C–R circuit with output across the resistor, what is the gain

\(\frac{V_R}{V_S}\)?

\[ \frac{V_R}{V_S} = \frac{R}{R - j\frac{1}{\omega C}} = \frac{1}{1 - j\frac{1}{2\pi fRC}} \]

13) Why is a C–R circuit (output across the resistor) a high pass filter?

Because:

\[ \lim_{f\rightarrow 0}\frac{V_R}{V_S}=0 \quad\text{and}\quad \lim_{f\rightarrow\infty}\frac{V_R}{V_S}=1 \]

Low frequencies are blocked, high frequencies pass.

14) What is the cutoff frequency for an R–L filter (both LPF and HPF cases)?

\[ f_{cutoff}=\frac{R}{2\pi L} \]

15) What does “rolloff” mean and why does it matter in real filters?

Rolloff is the gradual reduction in output amplitude outside the pass band (real filters do not drop to zero instantly).
It matters because non-ideal rolloff can cause nearby frequency channels to “bleed” into each other if the rolloff is not steep enough, so systems must use:

1. Steeper rolloff when possible, and

2. Adequate channel spacing to prevent interference.