Your Comments
This material is a little abstract. Can you explain these phase diagrams and how to draw the right triangles to find phi in more detail? Is there ever going to be more than one loop/ more than one of each piece of the circuit? If so, can we please go through an example?
So we just get done with a decently difficult exam and this is what you throw at us??!?!?!?! Between this and the exam, I kinda sorta really, really hate E&M now. And did we really set up a differential equation, examine three separate complex instances, introduce phasers (which confuse the int(B * dA) out of me), and graph phasers all to arrive at ... a problem solved by simple geometry and algebra? Not only do I hate E&M, E&M constantly confuses me with how it can be so complex and yet so simple.
I really like the pretty diagrams... I just wish I understood them haha.
Sorry, i didn't really do this prelecture/checkpoint. but i'll come back to it in a few days! studying for a CS exam :0
This pre-lecture went way over my head. Going over everything in class will definitely help because I need some general explanation of what things are in order to understand the details of the example.
Electricity & Magnetism Lecture 20, Slide 1 Physics 212 Lecture 20
Today’s Concept: AC Circuits Maximum currents & voltages Phasors: A Simple Tool
Electricity & Magnetism Lecture 20, Slide 2 Big Idea
Maximum Values (easy V=IR)
VRmax = Imax R
Imax = emax / Z VLmax = Imax XL
VCmax = Imax XC
Value at specific time (phasors)
y component gives voltage
V-Inductor Leads current V-Capacitor Lags current
Electricity & Magnetism Lecture 20, Slide 3 Resistors
R I = VR/R = Vmax/R sin(wt) e = Vmaxsin(wt)
Amplitude = Vmax/R
Electricity & Magnetism Lecture 20, Slide 4 Capacitors
Q = CV = CVmaxsin(wt) I = dQ/dt
I = VmaxwC cos(wt) e = Vmaxsin(wt) C
Amplitude = Vmax/XC
90o where XC = 1/wC is like the “resistance” of the capacitor
XC depends on w
Electricity & Magnetism Lecture 20, Slide 5 Inductors
L dI/dt = VL = Vmaxsin(wt)
I = - Vmax/wL cos(wt) e = Vmaxsin(wt) L
Amplitude = Vmax/XL
where XL = wL is like the “resistance” o 90 of the inductor
XL depends on w
Electricity & Magnetism Lecture 20, Slide 6 RL Clicker Question
An RL circuit is driven by an AC generator as shown in the figure.
XL = wL
L As w -> 0, so does XL
As w -> 0, resistance of circuit R R current gets bigger
For what driving frequency w of the generator will the current through the resistor be largest A) w large B) Current through R doesn’t depend on w C) w small
Electricity & Magnetism Lecture 20, Slide 7 Summary
R I = V /R max max VR in phase with I Because resistors are simple
o C Imax = Vmax/XC VC 90 behind I
XC = 1/wC Current comes first since it charges capacitor Like a wire at high w
I = V /X o L max max L VL 90 ahead of I
XL = wL Opposite of capacitor Like a wire at low w
Electricity & Magnetism Lecture 20, Slide 8 CheckPoint 1c
f
The phase difference between the CURRENT through the resistor and the CURRENT throughA the inductor is B C D The CURRENT is THE CURRENT There is only 1 current in this circuit Same everywhere in circuit
Electricity & Magnetism Lecture 20, Slide 9 Driven RLC Circuit
Makes sense to write everything in Phasors make this terms of I since this is the same simple to see everywhere in a one-loop circuit: Imax XL
Vmax = Imax XC V 90o behind I
C Imax R V = I X e L max max L max V 90o ahead of I R
Imax XC
Vmax = Imax R V in phase with I Always looks the same. Only the lengths will change
Electricity & Magnetism Lecture 20, Slide 10 The Voltages still Add Up
Imax XC
But now we are adding vectors: C
L Imax XL emax R I X max L Imax XL
Imax R
Imax R emax
Imax R Imax R Imax XC Imax XL
emax Imax XC Imax XC
Electricity & Magnetism Lecture 20, Slide 11 Make this Simpler
Imax XC
C
L Imax XL emax R Imax XL Imax XL
Imax R
emax
Imax R Imax R
Imax XC Imax XC
Electricity & Magnetism Lecture 20, Slide 12 Make this Simpler
Imax XC
C
L Imax XL emax R Imax XL
Imax R
emax = Imax Z
Imax(XL - XC) I R max Imax R
Imax XC
Electricity & Magnetism Lecture 20, Slide 13 Make this Simpler
Imax XC
C
L Imax XL emax R
Imax R
emax = Imax Z
Imax(XL - XC)
Imax R
Electricity & Magnetism Lecture 20, Slide 14 Make this Simpler
Imax XC
C L I X emax = Imax Z max L emax
f Imax(XL - XC) R
Imax R Imax R
(XL - XC)
R
Impedance Triangle X - X tan (f) = L C R
Electricity & Magnetism Lecture 20, Slide 15 Summary
Imax XC
VCmax = Imax XC C V = I X Lmax max L L I X e max L VRmax = Imax R max R emax = Imax Z
Imax R
Imax = emax / Z 2 2 Z = R (X L - X C )
2 2 Z = R X - X (XL - XC) L C f X - X tan (f) = L C R R
Electricity & Magnetism Lecture 20, Slide 16 Example: RL Circuit Xc = 0
L Imax XL emax R
Imax R Imax XL
emax
Imax R
Electricity & Magnetism Lecture 20, Slide 17 CheckPoint 1a
Draw Voltage Phasors
Imax XL
emax
Imax R
A B C
Electricity & Magnetism Lecture 20, Slide 18 CheckPoint 1b
Draw Voltage Phasors
Imax XL
emax
Imax R
A B C
Electricity & Magnetism Lecture 20, Slide 19 CheckPoint 2a
IXL e e IR
IR What does the voltage phasor diagram look like IX when the current is a maximum? IXL c
IX Electricity & Magnetism Lecture 20,c Slide 20 CheckPoint 2b
A driven RLC circuit is represented by the phasor diagram below.
c
IXL e IX
IR
IR
IXc
e
L IX
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
What does the voltage phasor diagram look like when the capacitor is fully charged?
Electricity & Magnetism Lecture 20, Slide 21 CheckPoint 2C
A driven RLC circuit is represented by the phasor diagram below.
c e
IXL IX
IR
IR
IXc
e
L IX
When the voltage across the capacitor is at its positive maximum, VC = +VC,max, what is the voltage across the inductor ?
A) VL = 0 B) VL = VL,max C) VL = -VL,max
What does the voltage phasor diagram look like when the voltage across capacitor is at its positive maximum?
Electricity & Magnetism Lecture 20, Slide 22 Calculation
Consider the harmonically driven series LCR circuit shown. C Vmax = 100 V Imax = 2 mA VCmax = 113 V The current leads generator voltage by 45o V ~ L L and R are unknown. 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 relationship 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 & Magnetism Lecture 20, Slide 23 Calculation
Consider the harmonically driven series LCR circuit shown. C Vmax = 100 V Imax = 2 mA VCmax = 113 V The current leads generator voltage by 45o V ~ L L and R are unknown. 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 C) XL > XC D) Not enough information
This information is determined from the phase Current leads voltage VL IR
45o VL = ImaxXL VR (phase of current) VC = ImaxXC V
VC V leads
Electricity & Magnetism Lecture 20, Slide 24 Calculation
Consider the harmonically driven series LCR circuit shown. C Vmax = 100 V Imax = 2 mA VCmax = 113 V The current leads generator voltage by 45o V ~ L L and R are unknown. R
What is XL, the reactance of the inductor, at this frequency?
What is Z, the total impedance of the circuit? A) 70.7 k W B) 50 k W C) 35.4 k W D) 21.1 kW
V 100V Z = max = = 50kW Imax 2mA
Electricity & Magnetism Lecture 20, Slide 25 Calculation
Consider the harmonically driven series LCR circuit shown. C Vmax = 100 V Imax = 2 mA VCmax = 113 V The current leads generator voltage by 45o V ~ L L and R are unknown. R
X What is L, the reactance of the inductor, at this frequency? Z = 50kW What is R? sin(45) =.707 A) 70.7 k W B) 50 k W C) 35.4 k W D) 21.1 kW cos(45) =.707
Determined from impedance triangle R R 45o cos(45) = R = Z cos(45o) (XC - XL) Z = 50 kW x 0.707 = 35.4 kW
Electricity & Magnetism Lecture 20, Slide 26 Calculation
Consider the harmonically driven series LCR circuit shown. C Vmax = 100 V Imax = 2 mA VCmax = 113 V The current leads generator voltage by 45o V ~ L L and R are unknown. R
What is XL, the reactance of the inductor, at this frequency? Z = 50kW R = 35.4kW A) 70.7 k W B) 50 k W C) 35.4 k W D) 21.1 kW
We start with the X - X impedance triangle: C L = tan45 =1 X = X - R R L C R What is XC? 45o V = I X (XC - XL) Cmax max C Z 113 X C = = 56.5kW XL = 56.5 kW - 35.4 kW 2
Electricity & Magnetism Lecture 20, Slide 27