Physics 2112 Unit 20 Outline: Driven AC Circuits Phase of V and I Conceputally Mathematically With phasors

e = Vmaxsin(wdt) Driving frequency = natural frequency (wo) AC Generator e = Vmaxsin(wdt) Driving frequency = natural frequency (wo)

IR = VR/R “Phase” between I and V Simple Case - Resistors Voltage goes up  current goes up “In phase”  Phase angle = 0o I= Vmax/R sin(wdt) Amplitude = Vmax/R

Q = CV = CVmaxsin(wt)  I = VmaxwC cos(wt) Capacitors C Amplitude = Vmax/XC 90o where XC = 1/wC is like the “resistance” of the capacitor XC depends on w

Inductors L Amplitude = Vmax/XL where XL = wL is like the “resistance” of the inductor XL depends on w

V and I “in phase” I “leads” V I “lags” V Phase Summary R C L “ELI the ICE man”

What does this look like together? Notice phase relationships

What does this look like together? 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? 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 Doesn’t depend on w w goes up, Cc goes down L w goes up, CL goes up

Example 20.1 (Inductor Reactance) A 60Hz signal with a Vmax = 5V is sent through a 50mH inductor. What is the maximum current, Imax, through the inductor?

Think of same material graphically using “phasors” Phasor just thinks of sine wave as rotating vector

Remember VR and I in phase Circuit using Phasors Represent voltage drops across elements as rotating vectors (phasors) Imax XL Imax XC Imax R VL and VC 180o out of phase VL and VR 90o out of phase Remember VR and I in phase

emax = Imax Z emax Make this Simpler f C L R Imax XC Imax XL Imax R L R C emax Imax R emax = Imax Z Imax(XL - XC) f R (XL - XC) Impedance Triangle

emax emax = Imax Z Summary f VCmax = Imax XC C VLmax = Imax XL L Imax R L R C emax VCmax = Imax XC VLmax = Imax XL VRmax = Imax R emax = Imax Z Imax = emax / Z f R (XL - XC)

CheckPoint 1(A) A RL circuit is driven by an AC generator as shown in the figure. The voltages across the resistor and generator are. always out of phase always in phase sometimes in phase and sometimes out of phase

CheckPoint 1(B) A RL circuit is driven by an AC generator as shown in the figure. The voltages across the resistor and inductor are. always out of phase always in phase sometimes in phase and sometimes out of phase

CheckPoint 1(C) 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 is always zero is always 90o depends on the value of L and R depends on L, R and the generator voltage

In the circuit to the right Example 20.2 (LCR) In the circuit to the right L=500mH Vmax = 6V C=47uF R=100W V C L R What is the maximum current and phase angle if w = 60rad/sec? What is the maximum current and phase angle if w = 400 rad/sec? The purpose of this Check is to jog the students minds back to when they studied work and potential energy in their intro mechanics class. What is the maximum current and phase angle if w = 206 rad/sec?

What does this look like graphically? The purpose of this Check is to jog the students minds back to when they studied work and potential energy in their intro mechanics class.

VL-max + VC-max + VR-max + e = 0 Point of confusion?? VL + VC + VR + e = 0 VL-max + VC-max + VR-max + e = 0 (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? VL = 0 VL = VL-max/2 VL = VL=max

CheckPoint 2(B) 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? VL = 0 VL = VL-max/2 VL = VL=max

CheckPoint 2(C) 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? VL = 0 VL = VL-max/2 VL = VL=max

Get your calculators out 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? ~ C R L V 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 The purpose of this Check is to jog the students minds back to when they studied work and potential energy in their intro mechanics class. Strategic Analysis Use Vmax and Imax to determine Z Use impedance triangle to determine R Use VCmax and impedance triangle to determine XL Get your calculators out