PHYS219 Fall semester 2014 Lecture 16: AC Circuits with Inductors and Capacitors Dimitrios Giannios Purdue University

Reminder: Resistor in AC circuit θ = 2 π ƒ t V(t) = V max sin (2 π ƒ t) Applying Ohm ’ s Law: I = V/ R

Assume an AC circuit with an AC generator and a single inductor The supply voltage is V(t)=V max sin(2πft) The instantaneous voltage across the inductor is V L = V max sin (2πft) = L (ΔI / Δt) V L is related to the change in current with time The slope of the current-time relationship determines V L Inductive AC Circuit V=V max sin(2πft)

The current is lagging the voltage For an inductive circuit, the current and voltage are out of phase by -90 ° The Current and Voltage are Out of Phase

The maximum value (peak value) of the current is given by The factor X L is called the reactance of the inductor SI unit of inductive reactance is Ohms Reactance and resistance are different because the reactance of an inductor depends on frequency f (in Hz) X L, R (in Ω) R XLXL The Reactance of inductor X L

For an AC circuit with an inductor, P = VI = -V max I max sin (2πƒt) cos (2πƒt) The average value of the power over many oscillations is 0 – Energy is transferred from the generator to the inductor during part of the cycle and from the inductor to the generator in the other part of the cycle Energy is stored in the inductor as magnetic potential energy Power Lost in an IDEAL Inductor Energy flows into the inductor Energy flows from the inductor back to the generator

VRVR I VLVL R Impedance Triangle Z X L The impedance Z in a series LR circuit L=0.1 mH R=1000 Ω f=1 MHz V max =10 V V max sin(2πft) Electrical components in series: Impedances “add as phasors” to provide the equivalent impedance Z “Vector” Impedance

The capacitor’s voltage and charge are out of phase with each other Assume an AC circuit containing a single capacitor The supply voltage is V(t)=V max sin(2πft) The instantaneous charge is q(t) = C V(t) = C V max sin (2πft) The current is given by Capacitive AC Circuit When the slope of q vs. t is zero, I=0 I is large when the slope of q is large Slope=maximum Slope=0 V= V max sin(2πft) C D

The current is ahead of the voltage For a capacitive circuit, the current and voltage are out of phase by +90 ° The Current and Voltage are Out of Phase

The maximum value (peak value) of the current is The factor X c is called the reactance of the capacitor SI unit of capacitive reactance is Ohms Reactance and resistance are different because the reactance of a capacitor depends on frequency f (in Hz) X c, R (in Ω) R XCXC The Reactance of capacitor X C

For an AC circuit with a capacitor, P = VI P = VI = V max I max sin (2πƒt) cos (2πƒt) The average value of the power over many oscillations is 0 – Energy is transferred from the generator to the capacitor during part of the cycle and from the capacitor to the generator in the other part of the cycle Energy is stored in the capacitor as electric potential energy and is not lost. Power Lost in a Capacitor is zero

V max sin(2πft ) C=100 pF R=1000Ω f=1 MHz V max =10 V R I VCVC XCXC Z V Impedance Triangle “Vector” Impedance The impedance Z in a series RC circuit Electrical components in series: Impedances “add as phasors” to provide the equivalent impedance Z: R

Application: Power Dissipation in a Series RC Circuit V max = 170V f= 60Hz V max = 170V f= 60Hz R 1 =5600 Ω R 2 =1000 Ω C =0.48 μ F ????? a) P ave =I 2 rms R 2 b) P ave =I 2 rms X C c) P ave =I 2 rms Z d) P ave =0

Summary