1. Circuits in the Frequency Domain
Topic 18
Circuits in the Frequency Domain
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2. Summary P P-1 Transforms Impedances Approach
1. Transform all voltages and currents to phasors
2. Solve the circuit
3. Transform the phasors back to the time domain
Polar →Rectangular co-ordinates
Special cases a b j A θ
Rectangular → Polar co-ordinates
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Frequency Domain

3. Assessment 9.6
32mH
90Ω C vs i
(a) Find the value of C that yields a steady-state output current i with a phase angle of -105°
Basically
What we want is for the phase angle of I to be -105°
So what the problem really wants is
45°
real and imaginary parts must be the same R
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Frequency Domain R

4. (b) Find the magnitude of the steady state output current
Assessment 9.6
32mH
90Ω C vs i
(b) Find the magnitude of the steady state output current
& we just set this one =R
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Frequency Domain

5. (a) Calculate the impedance at frequencies of 2 krad/s
Assessment 9.7
20Ω
5mH 5Ω
25µF
(a) Calculate the impedance at frequencies of 2 krad/s ZN
(b) 8 krad/s is similar
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Frequency Domain

6. Standard trick to make denominator real
Assessment 9.7
20Ω
5mH 5Ω
25µF
(c) At what finite frequency does the impedance become purely resistive
Standard trick to make denominator real ZN
Need this to go to 0
(d) so so
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Frequency Domain

7. What is the maximum amplitude of the current in the inductor?
Assessment 9.9
20Ω
5mH 5Ω
25µF it
What is the maximum amplitude of the current in the inductor? iL
Given what we know already about this problem, how can we write down an expression for iL by inspection?
150 cos4000t v
We know ZN so we can easily compute the total current So
Also
7.07 mS iL
45°
.393
1.18
2.75
-.393
f=ω/2π
T=1/f=2π/ω
T=3.14 mS A
Thus
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Frequency Domain

8. Assessment 9.11 ½Vx 10Ix Vx Voc Find the Thevenin equivalent wrt a-b
20 Ω
j10 Ω
10Ix
10 Ω
-j10 Ω Ix a b
½Vx Vx
10Ix
Voc
Find the Thevenin equivalent wrt a-b
Find voc
Node voltages appears a useful approach
Note so
Not
as in text
Right node
Left node so
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Frequency Domain

9. Assessment 9.11 It 10Ix Vx Find the Thevenin equivalent wrt a-b Vt
j10 Ω Vx
10Ix
10 Ω a
Find the Thevenin equivalent wrt a-b Vt
10Ix
245° A
-j10 Ω Ix
20 Ω
Find ZTh
Turn off independent sources & apply a (phasor) test voltage at a-b b
Find the current drawn from the test source
The only solution is Ix=0
To find Ix we note so
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Frequency Domain

10. Problem 9.16 Find the steady state expression for io in the circuit ifvs 4Ω
240mH
2.5mF io
Find the steady state expression for io in the circuit if
The total impedance seen by the voltage source is 4
In polar form So
And
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Frequency Domain

11. Find and plot Vo/Vs as a function of ω (with ω on a log scale) if
Problem vs
5kΩ vo
10µF + -
Find and plot Vo/Vs as a function of ω (with ω on a log scale) if
Low Pass
Filter
Pass
region
Reject
region
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Frequency Domain

12. Repeat the plot with the capacitor and resistor exchanged
Problem part b vs
5kΩ vo
10µF + -
Repeat the plot with the capacitor and resistor exchanged
High Pass
Filter
Reject
region
Pass
region
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Frequency Domain

13. Problem 9.57 Use the node-voltage method to find Vo and Io8
250° mA
40 Ω
j20 Ω Vo 8 Io
16 Io
50 Ω
-j25 Ω - + Vx Vo
Use the node-voltage method to find Vo and Io
Label the essential nodes
Please note there is no current flowing between the two halves of the circuit because there’s no way for it to get back
i.e. this current is 0
Sum the currents at the essential nodes 2 1 3
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Frequency Domain

14. Problem 9.57 1 2 3 2 into 3 4 Now put 4 into 1 Vo Vx j20 Ω 8 Vo Vo Io
250° mA
40 Ω
j20 Ω Vo 8 Io
16 Io
50 Ω
-j25 Ω - + Vx Vo 1 2 3
2 into 3 4
Now put 4 into 1
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Frequency Domain

15. Problem 9.62
100°
50°
10° 1Ω
-j1Ω
j1Ω
Use mesh-current method to find the branch currents Ia, Ib, Ic & Id. 1 Ia Ib Ic Id I2
Mark in the mesh currents I3 I4
1 + j2 5 1 6 2 3
2 + 3 4
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Frequency Domain

16. Problem 9.12
A 1000 Hz sinusoidal voltage with a maximum amplitude of 200 volts at t=0 is applied across the terminals of an inductor. The maximum amplitude of the steady-state current in the inductor is 25 A.
(a)
What is the frequency of the inductor current? L
The same as the voltage Hz
(b)
What is the phase angle of the voltage?
2000°
Max amplitude at t=0 means it’s a cosine wave: phase angle 0°
magnitude
(c)
What is the phase angle of the current?
-90°
(d)
What is the inductive reactance of the inductor?
phase
Reactance is the imaginary part of the imedance - i.e. ωL
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Frequency Domain

17. 5/3/2019
Frequency Domain

18. End of this lecture
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Frequency Domain

19. Can this really be the end?
Ohh… Momma!
Can this really be the end?
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Frequency Domain

20. To quote Jim Morrison Yeah, Bob, This… is... the... end. 5/3/2019
Frequency Domain