Physics 102: Lecture 13, Slide 1
AC Circuit Phasors
•Today’s lecture will cover Textbook
Section 21.5
Physics 102: Lecture 13
L
R
C
•I = I
maxsin(2πft)
•V
R= ImaxRsin(2πft)
•V
C= ImaxXCsin(2πft-π/2)
•V
L= ImaxXLsin(2πft+ π/2)
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AC Circuit Phasors - Lecture Slides | PHYS 102, Study notes of Physics
Material Type: Notes; Class: College Physics: E&M & Modern; Subject: Physics; University: University of Illinois - Urbana-Champaign; Term: Unknown 1989;
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Physics 102: Lecture 13^ AC Circuit Phasors• Today’s lecture will cover TextbookSection 21.
L R C
- I = Imax
sin(2πft)
- V^ = IR^
R sin(2max πft)
- V^ = IC^
X^ sin(2max C^
πft-π/2)
- V^ = IL^
X^ sin(2max L^
πft+^ π/
A reminder about sines andcosines Recall: Physics 102: Lecture 13, Slide 2
y^ coordinatesof endpoints are• a sin(θ^ +^ π/2) • a sin(θ) • a sin(θ^ -^ π/2)
a^
a a
Peak valuesin AC Circuits
,max^ max C
C V^ I ,max^ max R^ X = V^ I
R = ,max max L L V^ I
X = ,max max gen V^
I^ Z =
(^1) X = CC ω X L ω= L^2
2 (^
) L C Z^ R^
X^ X =^ +^
−
voltage^
reactance orimpedance
(capacitive reactance)(inductive reactance)
L R C (impedance)
Peak + RMS valuesin AC Circuits (REVIEW)
L R C
,max^ max C
C V^ I ,max^ max R^ X = V^ I
R = ,max max L L V^ I
X = ,max max gen V^
I^ Z = When asking about RMS or Maximum valuesrelatively simple expresions
(^1) X = CC ω X L ω= L^2
2 (^
) L C Z^ R^
X^ X =^ +^
−
π/
Phasor Diagrams
V^ sin( π/3 R,max^ f =1/12t = 2 2 π ft =^ π /3 ) max V
- I = Imax
sin(π/3)
- V^ = VR^
sin(R,max^
π/3) Length of vector = V
across that componentmax^ Vertical component = instantaneous value of V
Phasor Diagrams
V^ sin( π/2 )=VR,max^^ π/ t = 3 2 π ft =^ π /2^0 max V
- I = Imax
sin(π/2)
- V^ = VR^
sin(R,max^
π/2) Length of vector = V
across that componentmax^ Vertical component = instantaneous value of V
Phasor Diagrams
- I = Imax
sin(2πft)
- V^ = IR^
R sin(2max πft)
R^ ImaxI^ R sin(2max^
π ft)
• V^ = IC^
X^ sin(2max C^
πft-π/2)
- V^ = IL^
X^ sin(2max L^
πft+π/ I^ X^ maxL sin(2 π ft+ π/2 ) )
I^ X^ maxC sin(2 π ft- π /2) ImaxX^ L^ I^ maxX^ C
Instantaneous Values: Voltage across resistor is always _______ with current!Voltage across capacitor always _______ current!Voltage across inductor always ________ current!
Phasor Diagram PracticeLabel the vectors that corresponds tothe resistor, inductor and capacitor.Which element has the largest voltageacross it at the instant shown?1) R^ 2) C^
- L Is the voltage across the inductorincreasing or decreasing?Which element has the largestmaximum voltage across it?
Drawing Phasor Diagrams
VL
(2) Capacitor vector: downwards•^ Length given by V
C
VC
(3) Inductor vector: upwards•^ Length given by V
L
VR
(1)^ Resistor vector: to the right•^ Length given by V
R (4) Generator vector: add first 3 vectors•^ Length given by V
gen
Vgen VR VC VL
(5) Rotate entire thing counter-clockwise•^ Vertical components give instantaneousvoltage across R, C, L, gen
Vgen
time 1^ Physics 102: Lecture 13, Slide 14
time 2
time 3^
time 4
ACTS 13.1, 13.2, 13.
When does VIs the phase angle positive or negative?
= Vgen R^ ? When does V
= 0 ?gen φ
AC Summary
Resistors:
V=I RR In phase with ICapacitors:^
V=ICmax^
Xmax C^ X= 1/(2c^
π f^ C)
Lags IInductors:^
V=ILmax^ max
XX L^
= 2π f^ LL
Leads IGenerator:^
V=Igenmax
Z^ max^
Z =^ √R
2 +(X-XL^
(^2) ) (^) C
Can lead or lag I
tan(φ) = (X
-X)/RL C
Power is only dissipated in resistor:
P = I^ Vrms
cos(φrms )
Problem Time!
An AC circuit with R= 2
Ω, C = 15 mF, and L = 30 mH is driven by a generator with voltage V(t)=2.5 sin(
πt)
Volts. Calculate the maximum current in the circuit, andthe phase angle.I= V^ max^ gen,max
/Z^
L R C
Z =
ACT: Voltage Phasor Diagram LX^ max^ I
R^ maxICX^ maxI φ gen,maxV
Rotates Counter ClockwiseAt this instant, the voltageacross the generator ismaximum.
What is the voltage across the resistor at this instant?1) V^ = IR^ max
R^ 2) V
= IR sin(R max
φ )^ 3) V
= IR cos(R max
φ )
See you next time!
- Read Sections 21.6, 22.1, 4-5, 9