Electricity & Magnetism
Lecture 20
Today’s Concept:
AC Circuits
Maximum currents & voltages
Phasors: A Simple Tool
Electricity & Magne?sm Lecture 20, Slide 1
Other videos:
Prof. W. Lewin, MIT Open Courseware
Mechanical Universe, Driven LRC circuits
Resistors
ε = Vmaxsin(ωt)
R
I = VR/R = Vmax/R sin(ωt)
Amplitude = Vmax/R
Electricity & Magne?sm Lecture 20, Slide 3
Capacitors
Q = CV = CVmaxsin(ωt)
I = dQ/dt
ε = Vmaxsin(ωt)
C
I = VmaxωC cos(ωt)
Amplitude = Vmax/XC
90o
where XC = 1/ωC
is like the “resistance”
of the capacitor XC depends on ω
Electricity & Magne?sm Lecture 20, Slide 4
Inductors
dI/dt = VL = Vmaxsin(ωt)
ε = Vmaxsin(ωt)
L
I = − (Vmax/ωL) cos(ωt)
Amplitude = Vmax/XL
90o
where XL = ωL
is like the “resistance”
of the inductor XL depends on ω
Electricity & Magne?sm Lecture 20, Slide 5
RL Clicker Question
An RL circuit is driven by an AC generator as shown in the figure.
XL = ωL
L
R
As ω → 0, so does XL
As ω → 0,
resistance of circuit → R
current gets bigger impedance
For what driving frequency ω of the generator will the current through the resistor be largest A) ω large
B) Current through R doesn’t depend on ω C) ω small Electricity & Magne?sm Lecture 20, Slide 6
Summary
R
Imax = Vmax/R
VR in phase with I
Because resistors are simple
C
Imax = Vmax/XC
XC = 1/ωC
L
Imax = Vmax/XL
XL = ωL
VC 90o behind I
Current comes first since it
charges capacitor
Like a wire at high ω
VL 90o ahead of I
Opposite of capacitor
Like a wire at low ω
Electricity & Magne?sm Lecture 20, Slide 7
Makes sense to write everything in terms of I since this is the same everywhere in a one-­‐loop circuit:
Phasors make this simple to see
Imax XL
Vmax = Imax XC
V 90o behind I
Imax R
Vmax = Imax XL
εmax
V 90o ahead of I
Imax XC
Vmax = Imax R
V in phase with I
“Do you have any fancy-­‐schmancy simula?ons for to show me?”
Always looks the same. Only the lengths will change
Electricity & Magne?sm Lecture 20, Slide 8
The Voltages still Add Up
But now we are adding vectors: Imax XL
Imax XL
Imax R
εmax
Imax R
Imax R
Imax XC
Imax XC
Imax XC
Imax XL
εmax
Electricity & Magne?sm Lecture 20, Slide 9
Make this Simpler
Imax XL
Imax XL
εmax
Imax R
Imax XC
Imax R
Imax XC
Electricity & Magne?sm Lecture 20, Slide 10
Make this Simpler
Imax XL
εmax = Imax Z
Imax R
Imax(XL − XC)
Imax R
Imax XC
Electricity & Magne?sm Lecture 20, Slide 11
Make this Simpler
εmax = Imax Z
Imax(XL − XC)
Imax R
Electricity & Magne?sm Lecture 20, Slide 12
Make this Simpler
εmax = Imax Z
Imax(XL − XC)
φ
Imax R
φ
R
(X
L
−X
C)
Impedance Triangle
Electricity & Magne?sm Lecture 20, Slide 13
Summary
VCmax = Imax XC
VLmax = Imax XL
VRmax = Imax R
εmax
= Imax Z
Imax = εmax / Z
φ
R
(X
L
−X
C)
Electricity & Magne?sm Lecture 20, Slide 14
Example: RL Circuit Xc = 0
Imax XL
εmax
Imax R
Electricity & Magne?sm Lecture 20, Slide 15
CheckPoint 2
Draw Voltage Phasors
Imax XL
εmax
Imax R
60
45
A
B
C
30
15
0
1
Electricity & Magne?sm Lecture 20, Slide 16
CheckPoint 4
Draw Voltage Phasors
Imax XL
εmax
Imax R
60
A
B
C
45
30
15
0
1
Electricity & Magne?sm Lecture 20, Slide 17
CheckPoint 6
The CURRENT is THE CURRENT
Imax XL
φ
εmax
Imax R
φ is the phase between generator and current
50
A
B
C
D
38
25
13
0
1
Electricity & Magne?sm Lecture 20, Slide 18
CheckPoint 8
50
38
25
13
0A
B
C
IXL
1
IR
ε
IR
IXL
What does the voltage phasor diagram look like when the current is a maximum?
ε
IXc
IXc
Electricity & Magne?sm Lecture 20, Slide 19
IXL
IXc
CheckPoint 10
50
ε
IR
IR
38
ε
IXc
IXL
25
13
A
B0
C
1
What does the voltage phasor diagram look like when the capacitor is fully charged?
Electricity & Magne?sm Lecture 20, Slide 20
IXL
IXc
CheckPoint 12
50
ε
IR
IR
38
ε
IXc
IXL
25
13
A
B
0
C
1
What does the voltage phasor diagram look like when the voltage across capacitor is at its posi?ve maximum?
Electricity & Magne?sm Lecture 20, Slide 21
Calculation
Consider the harmonically driven series LCR circuit shown. Vmax = 100 V
Imax = 2 mA
VCmax = 113 V
The current leads generator voltage by 45o
L and R are unknown.
C
V ~
L
R
What is XL, the reactance of the inductor, at this frequency?
Conceptual Analysis
The maximum voltage for each component is related to its reactance and to the maximum current.
The impedance triangle determines the rela?onship between the maximum voltages for the components
Strategic Analysis
Use Vmax and Imax to determine Z
Use impedance triangle to determine R Use VCmax and impedance triangle to determine XL
Electricity & Magne?sm Lecture 20, Slide 22
Calculation
Consider the harmonically driven series LCR circuit shown. Vmax = 100 V
Imax = 2 mA
VCmax = 113 V
The current leads generator voltage by 45o
L and R are unknown.
C
V ~
L
R
What is XL, the reactance of the inductor, at this frequency?
Compare XL and XC at this frequency:
A) XL < XC
B) XL = XC
VL
C) XL > XC
D) Not enough informa?on
This informa?on is determined from the phase
Current leads voltage
IR
45ο
VL = ImaxXL
VC = ImaxXC
VR (phase of current) VC
V
V
leads
Electricity & Magne?sm Lecture 20, Slide 23
Calculation
Consider the harmonically driven series LCR circuit shown. Vmax = 100 V
Imax = 2 mA
VCmax = 113 V
The current leads generator voltage by 45o
L and R are unknown.
C
V ~
L
R
What is XL, the reactance of the inductor, at this frequency?
What is Z, the total impedance of the circuit?
A) 70.7
kΩ
B) 50
kΩ
C) 35.4
kΩ
D) 21.1 kΩ
Electricity & Magne?sm Lecture 20, Slide 24
Calculation
Consider the harmonically driven series LCR circuit shown. Vmax = 100 V
Imax = 2 mA
VCmax = 113 V
The current leads generator voltage by 45o
L and R are unknown.
C
V ~
L
R
Z = 50 kΩ
What is XL, the reactance of the inductor, at this frequency?
What is R?
sin(45°) =0.707
A) 70.7
kΩ
B) 50
kΩ
C) 35.4
kΩ
D)
21.1 kΩ
cos(45°) =0.707
Determined from impedance triangle
R
45ο
Z
=
(XC − XL)
R = Z cos(45o)
kΩ
50
= 50 kΩ × 0.707
= 35.4 kΩ
Electricity & Magne?sm Lecture 20, Slide 25
Calculation
Consider the harmonically driven series LCR circuit shown. Vmax = 100 V
Imax = 2 mA
VCmax = 113 V
The current leads generator voltage by 45o
L and R are unknown.
C
V ~
R
Z = 50 kΩ
What is XL, the reactance of the inductor, at this frequency?
A) 70.7
kΩ
B) 50
kΩ
C) 35.4
kΩ
D) 21.1 kΩ
We start with the impedance triangle:
Z
R = 35.4kΩ
XL = XC − R
R
45ο
L
What is XC?
VCmax = ImaxXC
(XC − XL)
XL = 56.5 kΩ − 35.4 kΩ
Electricity & Magne?sm Lecture 20, Slide 26
Practical Test Hints
You will have






4 banana plug wires, ~4 alligator clips
1 Scope Probe (x1/x10)
1 BNC wire
adaptors: Tee, BNC-­‐Male Banana, BNC-­‐Female Banana
proto-­‐board (Use it!)
scope, func?on generator, DMM
We will have two 60 min sessions:


12:30 — 13:30
13:40 — 14:30