Alternating Current Circuits Chapter 33 Note This topic

Alternating Current Circuits Chapter 33 Note: This topic requires you to understand manipulate sinusoidal quantities which include phase differences. It is a good idea to review these topics in Chapters 15 and 16.

Alternating Current Circuits An “AC” circuit is one in which the driving voltage and hence the current are sinusoidal in time. V(t) Vp fv p 2 p wt V = VP sin (wt - fv ) I = IP sin (wt - f. I ) -Vp w is the angular frequency (angular speed) [radians per second]. Sometimes instead of w we use the frequency f [cycles per second] Frequency f [cycles per second, or Hertz (Hz)] w = 2 p f

Alternating Current Circuits V = VP sin (wt - fv ) I = IP sin (wt - f. I ) I(t) Vp Vrms fv -Vp p 2 p wt Ip Irms f. I/w t -Ip Vp and Ip are the peak current and voltage. We also use the “root-mean-square” values: Vrms = Vp / and Irms=Ip / fv and f. I are called phase differences (these determine when V and I are zero). Usually we’re free to set fv=0 (but not f. I).

Example: household voltage In the U. S. , standard wiring supplies 120 V at 60 Hz. Write this in sinusoidal form, assuming V(t)=0 at t=0.

Example: household voltage In the U. S. , standard wiring supplies 120 V at 60 Hz. Write this in sinusoidal form, assuming V(t)=0 at t=0. This 120 V is the RMS amplitude: so Vp=Vrms = 170 V.

Example: household voltage In the U. S. , standard wiring supplies 120 V at 60 Hz. Write this in sinusoidal form, assuming V(t)=0 at t=0. This 120 V is the RMS amplitude: so Vp=Vrms = 170 V. This 60 Hz is the frequency f: so w=2 p f=377 s -1.

Example: household voltage In the U. S. , standard wiring supplies 120 V at 60 Hz. Write this in sinusoidal form, assuming V(t)=0 at t=0. This 120 V is the RMS amplitude: so Vp=Vrms = 170 V. This 60 Hz is the frequency f: so w=2 p f=377 s -1. So V(t) = 170 sin(377 t + fv). Choose fv=0 so that V(t)=0 at t=0: V(t) = 170 sin(377 t).

Resistors in AC Circuits R E ~ EMF (and also voltage across resistor): V = VP sin (wt) Hence by Ohm’s law, I=V/R: I = (VP /R) sin(wt) = IP sin(wt) (with IP=VP/R) V I p 2 p wt V and I “In-phase”

Capacitors in AC Circuits C Start from: q = C V [V=Vpsin(wt)] Take derivative: dq/dt = C d. V/dt So I = C d. V/dt = C VP w cos (wt) E ~ I = C w VP sin (wt + p/2) V I p 2 p wt This looks like IP=VP/R for a resistor (except for the phase change). So we call Xc = 1/(w. C) the Capacitive Reactance The reactance is sort of like resistance in that IP=VP/Xc. Also, the current leads the voltage by 90 o (phase difference). V and I “out of phase” by 90º. I leads V by 90º.

Capacitor Example A 100 n. F capacitor is connected to an AC supply of peak voltage 170 V and frequency 60 Hz. C E What is the peak current? What is the phase of the current? What is the dissipated power? ~

Inductors in AC Circuits ~ L V = VP sin (wt) Loop law: V +VL= 0 where VL = -L d. I/dt Hence: d. I/dt = (VP/L) sin(wt). Integrate: I = - (VP / Lw) cos (wt) or V Again this looks like IP=VP/R for a resistor (except for the phase change). I p I = [VP /(w. L)] sin (wt - p/2) 2 p wt So we call the XL = w L Inductive Reactance Here the current lags the voltage by 90 o. V and I “out of phase” by 90º. I lags V by 90º.

Inductor Example A 10 m. H inductor is connected to an AC supply of peak voltage 10 V and frequency 50 k. Hz. What is the peak current? What is the phase of the current? What is the dissipated power? E ~ L

Phasor Diagrams A phasor is an arrow whose length represents the amplitude of an AC voltage or current. The phasor rotates counterclockwise about the origin with the angular frequency of the AC quantity. Phasor diagrams are useful in solving complex AC circuits. The “y component” is the actual voltage or current. Resistor Ip Vp wt

Phasor Diagrams A phasor is an arrow whose length represents the amplitude of an AC voltage or current. The phasor rotates counterclockwise about the origin with the angular frequency of the AC quantity. Phasor diagrams are useful in solving complex AC circuits. The “y component” is the actual voltage or current. Resistor Ip Vp Capacitor Ip wt wt Vp

Phasor Diagrams A phasor is an arrow whose length represents the amplitude of an AC voltage or current. The phasor rotates counterclockwise about the origin with the angular frequency of the AC quantity. Phasor diagrams are useful in solving complex AC circuits. The “y component” is the actual voltage or current. Resistor Ip Vp Capacitor Inductor Vp Ip wt wt Vp

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Analyzing the L-C Circuit Total energy in the circuit: Differentiate : N o change in energy

Analyzing the L-C Circuit Total energy in the circuit: Differentiate : N o change in energy

Analyzing the L-C Circuit Total energy in the circuit: Differentiate : N o change in energy

Analyzing the L-C Circuit Total energy in the circuit: Differentiate : N o change in energy The charge sloshes back and forth with frequency w = (LC)-1/2