Download RC and RL circuits. Given the following circuit with Vin = 10V sin(ωt

RC and RL circuits.
Given the following circuit with Vin = 10V sin(ωt), C = 100μF and R = 10kΩ, answer the
following questions.
a) Calculate the reactance of the capacitor
and the total impedance of the circuit as a
function of ω.
b) Plot the log(reactance) of the capacitor
against the log(ω) for ω = 10 to 109 rad/s.
c) If the voltage across the capacitor is
used as the output signal, Plot the gain of
the circuit (Vc / Vin) against ω. The choice
of log or linear is up to you, defend your
choice based on the information it shows
clearly.
d) same question as c) but for Vr used as the output.
e) Describe the circuit with the output choices above as a high-pass or low pass filter. A high
pass filter gives higher gain at high frequencies than at low frequencies (where the gain should
be less than 1).
f) If you wanted a circuit which would cut off signals with frequencies higher than 10MHz, can
you figure out what values of C and R to use and where to take the output signal from? [Look at
the impedence formula and how it depends on ω.]
Given the following circuit with Vin = 10V sin(ωt), L = 10mH and R = 10kΩ, answer the
following questions.
a) Calculate the reactance of the inductor and
the total impedance of the circuit as a function of
ω.
b) Plot the log(reactance) of the inductor against
the log(ω) for ω = 10 to 109 rad/s.
c) If the voltage across the inductor is used as
the output signal, Plot the gain of the circuit (Vc /
Vin) against ω. The choice of log or linear is up
to you, defend your choice based on the
information it shows clearly.
d) Describe the circuit with the output choice above as a high-pass or low pass filter.
3.a) With R = 8Ω and C = .25μF, calculate the frequency ω, with units, that will give R = XC .
[4 points]
b) Using ωo for the frequency you calculated in 3, calculate the total impedance, Z, for the
following frequencies and plot the results on the log plot provided. [10 points]
ω/ ωo 10-4
10-3
10-2
10-1
1
10
102
103
104
Xc
Z
Impedance
7
6
log(Z)
5
4
3
2
1
0
2
3
4
5
6
7
log(omega)
8
9
10
c) Locate your answer for part (2) on the plot. What feature of the plot corresponds to that
answer? [4 points]
d) Describe how to quickly sketch this log-log plot for an RC circuit with only one calculation,
based on your answer to (6). [5 points]
e) Using the same resistance of 8Ω, what value for the inductance in an LR circuit would give
XL = 8Ω at the same ωo value calculated in 3 ? [4 points]
f) Using ωo for the frequency you calculated in 3, calculate the total impedance, Z, of the LR
circuit for the following frequencies and plot the results on the log plot provided. [10 points]
ω/ ωo 10-4
10-3
10-2
10-1
1
10
102
103
104
XL
Z
Impedance
7
6
log(Z)
5
4
3
2
1
0
2
3
4
5
6
7
log(omega)
8
9
10
4a). Given a circuit with a resistor, R, a capacitor, C, and an inductor, L, in series, write the
expression for the impedance of the circuit. [4 points]
b). Find an expression for the frequency at which the capacitive reactance and the inductive
reactance are of equal magnitude. [6 points]
c). How does the impedance of the circuit at this frequency compare with the impedance at
smaller and greater frequencies? Support your answer with mathematics. [8 points]
d). Given your answers about the impedance above, what happens to the current amplitude in
the circuit as a function of frequency? [5 points]
e). What happens to a driven oscillator when the driving frequency equals the resonance
frequency? [4 points]
f). What is the resonance frequency for an LC oscillator with L = 0.1mH and C = 47μF?
[4 points]
g). Using Ohm’s law modified for AC circuits with reactance, write the expression for the
amplitude of the current in an LRC circuit driven by a signal with amplitude Vin. [4 points]
h). Plot the current versus frequency ω for the oscillator for the values in question 6 and resistor
values of 100Ω to 1kΩ (2 curves, one for each R). Use a range of frequencies from 10-3 to 103
times your answer to (6), do both curves on one graph and use another piece of paper: graph
paper or Excel printout. (These are like your plots done in lab) [12 points] {I would use Excel to
do the calculations and plotting!}
i). What is the effect of increasing the resistance on the resonance frequency and the curve
shape? [ 8 points]
Bonus: Can you think of a way to make a filter which would remove frequencies below 100Hz
and remove frequencies above 50MHz ?
Support your answer with a circuit diagram and a graph of the gain along with pertinent
calculations and descriptions.