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Driven AC Circuits Phase of V and I
Conceputally Mathematically With phasors
Physics 2112Unit 20
Outline:
Electricity & Magnetism Lecture 20, Slide 1
AC Generator
e = Vmaxsin(wdt)
Electricity & Magnetism Lecture 20, Slide 2
Driving frequency = natural frequency (wo)
“Phase” between I and V
R
IR = VR/R
I= Vmax/R sin(wdt)
Amplitude = Vmax/R
Electricity & Magnetism Lecture 20, Slide 3
Voltage goes up current goes up
Simple Case - Resistors
“In phase” Phase angle = 0o
Capacitors
C
I = VmaxwC cos(wt)
where XC = 1/wCis like the “resistance”of the capacitor XC depends on w
Amplitude = Vmax/XC
Q = CV = CVmaxsin(wt)
90o
Unit 20, Slide 4
Inductors
L
where XL = wLis like the “resistance”of the inductor XL depends on w
90o
Electricity & Magnetism Lecture 20, Slide 5
)sin(max tVVdt
dIL L
)cos(max tL
VI
Amplitude = Vmax/XL
Phase Summary
L
Electricity & Magnetism Lecture 20, Slide 6
)sin(max tR
VI dR
)cos(max tCVI d
)90sin(max od
C
tV
C
)cos(max tL
VI
)90sin(max o
L
tV
I “leads” V
V and I “in phase”
I “lags” V
“ELI the ICE man”
What does this look like together?
Electricity & Magnetism Lecture 20, Slide 7
Notice phase relationships
What does this look like together?
Electricity & Magnetism Lecture 20, Slide 8
Capacitor and Inductor always 180o out of phase
Capacitor/Inductor and Resistor always 90o out of phase
Resistor is some unknown phase angle out of phase is signal generator
What about current?
Electricity & Magnetism Lecture 20, Slide 9
Current is always the same through all elements (in series)
Current and Voltage in phase across Resistor
Current and voltage out of phase by unknown phase angle across signal generator
(We’ll find this “phase angle” later.)
Reactance Summary
L
Electricity & Magnetism Lecture 20, Slide 10
RR
CX C
1C
LX L
w goes up, Cc goes down
Doesn’t depend on w
w goes up, CL goes up
Example 20.1 (Inductor Reactance)
L
A 60Hz signal with a Vmax = 5V is sent through a 50mH inductor. What is the maximum current, Imax, through the inductor?
Electricity & Magnetism Lecture 20, Slide 11
Think of same material graphically using “phasors”
Electricity & Magnetism Lecture 20, Slide 12
Phasors
Phasor just thinks of sine wave as rotating vector
Imax XL
Imax XC
Imax R
VL and VC 180o out of phase
Electricity & Magnetism Lecture 20, Slide 13
Circuit using Phasors
Represent voltage drops across elements as rotating vectors (phasors)
VL and VR 90o out of phase
Remember VR and I in phase
Imax R
emax = Imax Z
Imax(XL - XC)
R(X
L - XC )
f
f
Impedance Triangle
Make this Simpler
22 )( CL XXRZ
Electricity & Magnetism Lecture 20, Slide 14
R
XX CL )(tan
Imax XL
Imax XC
Imax R
L
R
C
emax
f
R (XL - X
C )
22 )( CL XXRZ
VCmax = Imax XC
VLmax = Imax XL
VRmax = Imax R
Summary
Imax = emax / Z
emax = Imax Z
Electricity & Magnetism Lecture 20, Slide 15
R
XX CL )(tan
22CL XXRZ
Imax XL
Imax XC
Imax R
L
R
C
emax
CheckPoint 1(A)
Electricity & Magnetism Lecture 20, Slide 16
A RL circuit is driven by an AC generator as shown in the figure.
The voltages across the resistor and generator are.
A. always out of phase B. always in phase C. sometimes in phase and sometimes
out of phase
CheckPoint 1(B)
Electricity & Magnetism Lecture 20, Slide 17
A RL circuit is driven by an AC generator as shown in the figure.
The voltages across the resistor and inductor are.
A. always out of phase B. always in phase C. sometimes in phase and sometimes
out of phase
CheckPoint 1(C)
Electricity & Magnetism Lecture 20, Slide 18
A RL circuit is driven by an AC generator as shown in the figure.
The phase difference between the CURRENT through the resistor and inductor
A. is always zero B. is always 90o C. depends on the value of L and R D. depends on L, R and the
generator voltage
Example 20.2 (LCR)
Electricity & Magnetism Lecture 20, Slide 19
C
R
V
In the circuit to the right• L=500mH• Vmax = 6V• C=47uF• R=100W
L
What is the maximum current and phase angle if w = 60rad/sec?
What is the maximum current and phase angle if w = 206 rad/sec?
What is the maximum current and phase angle if w = 400 rad/sec?
What does this look like graphically?
Electricity & Magnetism Lecture 20, Slide 20
Point of confusion??
Electricity & Magnetism Lecture 20, Slide 21
VL + VC + VR + e = 0
VL-max + VC-max + VR-max + e = 0
Z
VI max
max Z
VI
(Add like vectors)
(Imax and Vmax happen at different times.)
CheckPoint 2(A)
A driven RLC circuit is represented by the phasor diagram to the right.
The vertical axis of the phasor diagram represents voltage. When the current through the circuit is maximum, what is the potential difference across the inductor?
A. VL = 0 B. VL = VL-max/2 C. VL = VL=max
CheckPoint 2(B)
Electricity & Magnetism Lecture 20, Slide 23
A driven RLC circuit is represented by the above phasor diagram.
When the capacitor is fully charged, what is the magnitude of the voltage across the inductor?
A. VL = 0 B. VL = VL-max/2 C. VL = VL=max
CheckPoint 2(C)
Electricity & Magnetism Lecture 20, Slide 24
A driven RLC circuit is represented by the above phasor diagram.
When the voltage across the capacitor is at its positive maximum, VC = +VC-max, what is the magnitude of the voltage across the inductor?
A. VL = 0 B. VL = VL-max/2 C. VL = VL=max
Conceptual AnalysisThe maximum voltage for each component is related to its reactance and to the
maximum current.The impedance triangle determines the relationship between the maximum
voltages for the components
Strategic AnalysisUse Vmax and Imax to determine ZUse impedance triangle to determine R Use VCmax and impedance triangle to determine XL
~
C
R
LV
Example 20.3
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.
What is XL, the reactance of the inductor, at this frequency?
Electricity & Magnetism Lecture 20, Slide 25
Get your
calcu
lators out
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