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**Unformatted text preview: **Physics 212
Lecture 20
AC Circuits Phasors Physics 212 Lecture 20, Slide 1 I ( t ) = I max sin ( t ) { I max t Actual current I(t) is the projection of the "phasor" onto the y-axis at time t. VR = I R
I ( t) VR = RI max sin ( t ) Vmax = R I max { Vmax t R I ( t ) = I max sin ( t )
VL = LdI/dt L dI VL = L = LI max cos ( t ) dt Inductor phasor ( VL ) max = LI max ( VL ) max = LI max = X L I max I max
t 90o Inductive "reactance" I ( t ) = I max sin ( t ) C Q VC = C , dQ =I dt I max VC = - cos ( t ) C 90o I max
t ( VC ) max I max = = X C I max C Summary
R
Vmax = R Imax VR in phase with I C Vmax = XC Imax XC = 1/C Current comes first since it charges capacitor Like a short circuit at high VC 90o behind I L Vmax = XL Imax XL = L VL 90o ahead of I
Like a short circuit at low Physics 212 Lecture 20, Slide 5 RL ACT
An RL circuit is driven by an AC generator as shown in the figure. XL = L L As 0, so does XL As 0, resistance of circuit R current gets bigger R
For what driving frequency of the generator will the current through the resistor be largest A) large B) Current through R doesn't depend on C) small Physics 212 Lecture 20, Slide 6 Add phasor voltages around the loop. These add like vectors. r VC r C R r r r r = VR + VC + VL
If phasors satisfy Khirchoff's Law, then so will their projections. L r VL r VL r r VR r VR r VC The projections are the actual voltages.
Physics 212 Lecture 20, Slide 7 Imax XC r r r r = VR + VC + VL max C R
Imax R L Imax XL Imax XL Imax XL r VL r VC r VR max
Imax R Imax R Imax XL Imax R Imax XC Imax XC Imax XC max
Physics 212 Lecture 20, Slide 8 Imax XC Add component by component. r r r r = VR + VC + VL max C R
Imax R L Imax XL Imax XL Imax R r Imax(XLXC) Imax R Imax XC Physics 212 Lecture 20, Slide 9 r = max = I max Z
Imax(XLXC) Imax R Define total circuit impedance Z() max = Imax Z Impedance Phasors (XLXC) R XL -XC tan ( ) = R
Physics 212 Lecture 20, Slide 10 Summary:
VCmax= Imax XC VLmax= Imax XL VRmax= Imax R Imax XC max C R
Imax R L Imax XL max = Imax Z
Imax = max / Z R ( XLXC) XL -XC tan ( ) = R
Physics 212 Lecture 20, Slide 11 Example: RL Circuit Xc=0 L R
Imax R Imax XL max
Imax XL max
Imax R Physics 212 Lecture 20, Slide 12 Checkpoint 1a
Draw Voltage Phasors Imax XL max
Imax R A B C Physics 212 Lecture 20, Slide 13 Checkpoint 1b Draw Voltage Phasors Imax XL max
Imax R A B C Physics 212 Lecture 20, Slide 14 Checkpoint 1c The CURRENT is THE CURRENT Imax XL max
Imax R A B C D is the phase between generator and current Physics 212 Lecture 20, Slide 15 Checkpoint 2a A B C IXL IR IXc
Physics 212 Lecture 20, Slide 16 Checkpoint 2b A B C IR IXL IXc Physics 212 Lecture 20, Slide 17 Checkpoint 2c A B C IR IXL IXc Physics 212 Lecture 20, Slide 18 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. Calculation
V ~ C L R Conceptual Analysis What is XL, the reactance of the inductor, at this frequency? 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
Physics 212 Lecture 20, Slide 19 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? Calculation
V ~ C L R 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 45
VR (phase of current) V leads IR VL = ImaxXL VC = ImaxXC
VC V Physics 212 Lecture 20, Slide 20 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? Calculation
V ~ C L R What is Z, the total impedance of the circuit? 50 k (A) (B) (C) (D) 35.4 k 70.7 k 21.1 k Vmax 100V Z= = = 50k I max 2mA Physics 212 Lecture 20, Slide 21 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. Calculation
V ~ C L R Z = 50k sin(45)=.707 cos(45)=.707 What is R? 70.7 k (A) (B) (C) (D) 35.4 k 50 k Determined from impedance triangle
45 What is XL, the reactance of the inductor, at this frequency? 21.1 k R Z=50k (XCXL) R cos(45) = Z R = Z cos(45o) = 50 k x 0.707 = 35.4 k
Physics 212 Lecture 20, Slide 22 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. Calculation
V ~ C L R Z = 50k R = 35.4k 70.7 k 35.4 k (A) (B) (C) (D) 50 k
We start with the impedance triangle: 45 What is XL, the reactance of the inductor, at this frequency? 21.1 k R XC - X L = tan 45 = 1 R XL = XC R
What is XC ? Z (XCXL) XL = 56.5 k 35.4 k VCmax = ImaxXC 113 XC = = 56.5k 2 Physics 212 Lecture 20, Slide 23 ...

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