1. Topic 17
Phasors
( )

2. Steady State Response t=0 R i Same frequency as the source L vs
Amplitude depends on the network and the source
Switch closes at t=0
KVL
DE’s are at the heart of much engineering analysis
This is a differential equation
Actually, phase does as well
You don’t know how to solve them!
Transient solution
Steady-state solution
Response to the sudden change (switch closing)
Response to the steady drive of the source
Dies out in time
Persists as long as the source does
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Phasors

3. Steady State Response are Familiar
Temperature response is same frequency but it lags
Hurricane response is same frequency but it leads
f (=ω/2π) is once per year
Transient response has long since died out
Magnitude and phase of response are determined by the source (amount of sunlight) and the system (the earth and its atmosphere)
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Phasors

4. Steady State Assumption
A linear network, excited by a sinusoidal source of frequency ω …
…will respond with voltages and currents all of which will be at the same frequency ω …
…with magnitudes and phases determined by the source characteristics Vm, ω & φ…
…as well as the network characteristics
t=0 R L vs i
Now how do we find I and θ?
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Phasors

5. Phasors P P-1 It’s actually easier
This is how we’re going to model our source
We’re going to use a technique that focuses on amplitude and phase of currents and voltages
Seems crazy, no?
We start with
Euler’s formula
Given this, we may write So
We know ω will stay the same throughout. We want to focus on the magnitude and phase
We’re going to define So
It is known as the Phasor transform. That is P
Phasor Transform
P-1
Inverse Phasor Transform
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Phasors

6. Phasor or Frequency Domain
Phasor Transform P
Time domain
Phasor or Frequency domain
Inverse Phasor Transform
P-1
Phasor or Frequency domain
Time domain
Given that we know we’re using sinusoidal signals
The same information is present in both domains
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Phasors

7. Sinusoid
Phasor
lower ω
larger Vm
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Phasors

8. Back to our circuit We had this circuit… …where t=0 R iL vs i
By writing KVL we got
We said, in steady state, i must be sinusoidal
And asked, “How can we find I and θ?”
Note that we can write v & i in terms of the exponential
Apply the complex exponential to the DE
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Phasors

9. For linear circuits with continuous sinusoidal excitations
Solution Approach
t=0 R L vs i
…where
For linear circuits with continuous sinusoidal excitations
1. Transform all voltages and currents to phasors
2. Solve the circuit
3. Transform the phasors back to the time domain
Before showing you how to do that, we do a little work with phasors to see how to manipulate them
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Phasors

10. Consider a simple problem
Phasor Manipulations
Consider a simple problem
What’s their sum?
44.7
Traditional way
24.6
33.4
37.3
Wasn’t that fun?
Now what?
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Phasors

11. Notice mag and angle info is preserved
Phasor Approach
Notice mag and angle info is preserved P
Seems a little easier!
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Phasors

12. Find the phasor transformation of each trigonometric function
Assessment 9.1
Find the phasor transformation of each trigonometric function
(a)
(b)
(c)
(d)
Mag
angle
rads Re Im 5
36.9
0.64 4 3 10
-53.1
-0.93 6 -8
11.18
-26.57
-0.464 -5
300 45
0.79
212
-100
-60
-1.05
-50
86.6
339.7
61.52
1.0737
162
298.6
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Phasors

13. Find the time domain expression for each phasor
Assessment 9.1
Find the time domain expression for each phasor
(a)
(b)
Mag
angle
rads Re Im 20 45
0.79
14.1
-50
-30
-0.52
-43.3 25
48.8
126.7
2.21
-29.2
39.1
(c)
Mag
angle
rads Re Im 20 80
-30 15
0.26
-29
-7.76
72.8
97.08
1.69
-8.98
72.2
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Phasors

14. Recall Our Solution Approachvs i
…where
For linear circuits with continuous sinusoidal excitations
1. Transform all voltages and currents to phasors
2. Solve the circuit
3. Transform the phasors back to the time domain
Now we’ll look at how circuits react to phasors
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Phasors

15. Resistors in the Frequency Domainv i + - R
Ohm’s Law
These are so similar we normally use the terms interchangeably
Let’s apply a sinusoidal voltage source to the resistor so
Then clearly
In the phasor domain
In the frequency domain
In either case Or
That is, Ohm’s Law holds for phasors!
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Phasors

16. Ohm’s Law and Phasors i + Now use a strictly phasor approach I v R- R
Now use a strictly phasor approach V I + - R
Ohm’s Law
For phasors
Which we can actually tell from the phasor
The current has the same phase as the voltage if
Vm/R Vm
and so
An ac current through a resistor is always in phase with the voltage across it
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Phasors

17. Inductors in the Frequency Domainv i + - L
Similar approach
This is known as the impedance of the inductance
Let
Then clearly
In the phasor domain
but V I + - L so or
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Phasors

18. Inductors in the Frequency Domainv i + - L
This time apply
90° phase shift V I + - L Vm
Vm/ωL
90° t
voltage
current
The ac current through an inductor always lags the voltage across it by 90°
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Phasors

19. Capacitors in the Frequency Domain
Current leads the voltage v i + - C
Similar approach
Let V I + - C
phasor domain so Vm
ωC Vm
90°
voltage
current
and t
The ac current through a capacitor always leads the voltage across it by 90°
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Phasors