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

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

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V p and I p are the peak current and voltage. We also use the “root-mean-square” values: V rms = V p / and I rms =I p / v and I are called phase differences (these determine when V and I are zero). Usually we’re free to set v =0 (but not I ). Alternating Current Circuits V = V P sin ( t - v ) I = I P sin ( t - I ) vv V(t) tt VpVp -V p V rms I/I/ I(t) t IpIp -Ip-Ip I rms

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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.

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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 V p =V rms = 170 V.

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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 V p =V rms = 170 V. This 60 Hz is the frequency f: so =2 f=377 s -1.

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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 V p =V rms = 170 V. This 60 Hz is the frequency f: so =2 f=377 s -1. So V(t) = 170 sin(377t + v ). Choose v =0 so that V(t)=0 at t=0: V(t) = 170 sin(377t).

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Resistors in AC Circuits E R ~ EMF (and also voltage across resistor): V = V P sin ( t) Hence by Ohm’s law, I=V/R: I = (V P /R) sin( t) = I P sin( t) (with I P =V P /R) V and I “In-phase” V tt I

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

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

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

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

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Phasor Diagrams VpVp IpIp t Resistor 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. 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.

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Phasor Diagrams VpVp IpIp t VpVp IpIp ResistorCapacitor 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. 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.

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Phasor Diagrams VpVp IpIp t VpVp IpIp VpVp IpIp ResistorCapacitor Inductor 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. 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.

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+ + i + + + + i i i i i LC Circuit time

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

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

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

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

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