Practice Problems (KEY)

Problem 1

Calculate the cutoff frequency for a circuit with:

a. \(R = 1\ \text{k}\Omega\) and \(C = 3\ \text{nF}\)

\[ f_{cutoff}=\frac{1}{2\pi RC}=\frac{1}{2\pi(1000)(3\times10^{-9})} \approx 53\ \text{kHz} \]

b. \(R = 5\ \text{k}\Omega\) and \(C = 1.5\ \mu\text{F}\)

\[ f_{cutoff}=\frac{1}{2\pi(5000)(1.5\times10^{-6})} \approx 21.22\ \text{Hz} \]

Problem 2

What should the resistor value be for the following filters:

a. \(f_{cutoff} = 1.5\ \text{kHz}\) and \(C = 500\ \text{nF}\)

\[ R=\frac{1}{2\pi(1500)(500\times10^{-9})} \approx 212\ \Omega \]

b. \(f_{cutoff} = 417\ \text{Hz}\) and \(C = 56\ \text{nF}\)

\[ R=\frac{1}{2\pi(417)(56\times10^{-9})} \approx 6.8\ \text{k}\Omega \]

c. \(f_{cutoff} = 2\ \text{kHz}\) and \(C = 500\ \mu\text{F}\)

\[ R=\frac{1}{2\pi(2000)(500\times10^{-6})} \approx 159\ \text{m}\Omega \]

Problem 3

Are the circuits below high or low pass filters? How do you know?

Evaluate the transfer function at \(f=0\) (DC) and \(f\rightarrow\infty\).

3(a): Gain \(\rightarrow 0\) at DC and \(\rightarrow 1\) at high frequency → High Pass Filter (HPF)

3(b): Gain \(\rightarrow 1\) at DC and \(\rightarrow 0\) at high frequency → Low Pass Filter (LPF)

3(c): Gain \(\rightarrow 1\) at DC and \(\rightarrow 0\) at high frequency → Low Pass Filter (LPF)

3(d): Gain \(\rightarrow 0\) at DC and \(\rightarrow 1\) at high frequency → High Pass Filter (HPF)

Problem 4

Calculate the cutoff frequency of the following systems.

a. A transmission line modeled as an R-L circuit with \(R = 4\ \Omega\) and \(L = 5\ \mu\text{H}\).

\[ f_{cutoff}=\frac{R}{2\pi L}=\frac{4}{2\pi(5\times10^{-6})} \approx 127\ \text{kHz} \]

b. An R-C low pass filter with \(R = 60\ \Omega\) and \(C = 5\ \text{nF}\).

\[ f_{cutoff}=\frac{1}{2\pi RC}=\frac{1}{2\pi(60)(5\times10^{-9})} \approx 531\ \text{kHz} \]

c. A C-R high pass filter with \(R = 100\ \Omega\) and \(C = 8\ \mu\text{F}\).

\[ f_{cutoff}=\frac{1}{2\pi RC}=\frac{1}{2\pi(100)(8\times10^{-6})} \approx 199\ \text{Hz} \]

Problem 5

Your communications radio has a lower frequency bound of \(800\ \text{kHz}\). You know it has a capacitor value of \(100\ \text{nF}\), but what is the resistor value?

\[ R=\frac{1}{2\pi f_{cutoff}C}=\frac{1}{2\pi(800000)(100\times10^{-9})} \approx 1.99\ \Omega \]

Problem 6

Design a high pass filter to get rid of a DC bias (\(0\ \text{Hz}\)) using a \(100\ \Omega\) resistor you have available.

Choose a practical cutoff, e.g. \(f_{cutoff}=10\) Hz. Solve for \(C\):

\[ C=\frac{1}{2\pi f_{cutoff}R}=\frac{1}{2\pi(10)(100)} \approx 159\ \mu\text{F} \]

Problem 7

For the circuit below, what is the magnitude of the gain, \(\left|\frac{v_{o}}{v_{in}}\right|\), at \(60\ \text{Hz}\)?

\[ \frac{v_0(t)}{v_s(t)}=\frac{1}{1+\frac{1}{j2\pi fRC}} = 0.889 + j0.314 = 0.943\angle 19.5^\circ \]

\[ \left|\frac{v_0}{v_{in}}\right| = 0.942 \]

The filter passes \(94.2\%\) of the input at 60 Hz, so 60 Hz is in the passband.