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Chapter 24

Chapter 24. Time-Varying Currents and Fields. AC Circuit. An AC circuit consists of a combination of circuit elements and an AC generator or source The output of an AC generator is sinusoidal and varies with time according to the following equation v = v 0 sin 2  ƒt

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Chapter 24

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  1. Chapter 24 Time-Varying Currents and Fields

  2. AC Circuit • An AC circuit consists of a combination of circuit elements and an AC generator or source • The output of an AC generator is sinusoidal and varies with time according to the following equation • v = v0 sin 2ƒt • v is the instantaneous voltage • V0 is the maximum voltage of the generator • ƒ is the frequency at which the voltage changes, in Hz

  3. Root Mean Square (rms) You have a variable x. Square, take average, and put square-root. Use rms values of v and i for AC to evaluate DC equivalent Power.

  4. Period = 1/60 = 16.7 ms 170 V -170 V From the wall outlet, you will see voltage signal like this with an amplitude of 170 V. However, when you estimate average Power, use vrms = 170/0.707 = 120 V.

  5. Case1: Seinfeld used a heater connected to a wall-outlet for 30 min. Case2: Kramer somehow found a road-kill DC voltage source which produces 120 V. Kramer sneaked in Jerry’s place and took his heater and run for 30 mins. Used the same amount of electric power and $$!!!

  6. R C L http://www.educatorscorner.com/

  7. Resistor in an AC Circuit • Consider a circuit consisting of an AC source and a resistor • The graph shows the current through and the voltage across the resistor • The current and the voltage reach their maximum values at the same time • The current and the voltage are said to be in phase

  8. More About Resistors in an AC Circuit • The direction of the current has no effect on the behavior of the resistor • The rate at which electrical energy is dissipated in the circuit is given by • P = i2 R • where i is the instantaneous current • the heating effect produced by an AC current with a maximum value of Imax is not the same as that of a DC current of the same value • The maximum current occurs for a small amount of time

  9. rms Current and Voltage • The rms current is the direct current that would dissipate the same amount of energy in a resistor as is actually dissipated by the AC current • Alternating voltages can also be discussed in terms of rms values

  10. Ohm’s Law in an AC Circuit • rms values will be used when discussing AC currents and voltages • AC ammeters and voltmeters are designed to read rms values • Many of the equations will be in the same form as in DC circuits • Ohm’s Law for a resistor, R, in an AC circuit • V = I R • Also applies to the maximum values of v and i

  11. Capacitors in an AC Circuit • Consider a circuit containing a capacitor and an AC source • The current starts out at a large value and charges the plates of the capacitor • There is initially no resistance to hinder the flow of the current while the plates are not charged • As the charge on the plates increases, the voltage across the plates increases and the current flowing in the circuit decreases

  12. More About Capacitors in an AC Circuit • The current reverses direction • The voltage across the plates decreases as the plates lose the charge they had accumulated • The voltage across the capacitor lags behind the current by 90°(or T/4)

  13. io V v A V lags 90 deg behind i. AC i v= vosin(wt) Q = C V i= C w vo cos(wt) io = Cwvo vo = (1/wC)io

  14. Capacitive Reactance and Ohm’s Law • The impeding effect of a capacitor on the current in an AC circuit is called the capacitive reactance and is given by • When ƒ is in Hz and C is in F, XC will be in ohms • Ohm’s Law for a capacitor in an AC circuit • V = I XC

  15. RC Circuits • A DC circuit may contain capacitors and resistors, the current will vary with time • When the circuit is completed, the capacitor starts to charge • The capacitor continues to charge until it reaches its maximum charge (Q = Cε) • Once the capacitor is fully charged, the current in the circuit is zero

  16. Charging Capacitor in an RC Circuit • The charge on the capacitor varies with time • q = Q(1 – e-t/RC) • i=i0 e-t/RC • The time constant, =RC • The time constant represents the time required for the charge to increase from zero to 63.2% of its maximum • i=i0 e-1 = 0.368 i0

  17. Discharging Capacitor in an RC Circuit • When a charged capacitor is placed in the circuit, it can be discharged • q = Qe-t/RC • The charge decreases exponentially • At t =  = RC, the charge decreases to 0.368 Qmax • In other words, in one time constant, the capacitor loses 63.2% of its initial charge

  18. - + - + dQ/dt = C (dV/dt) dV/dt = (1/C) (dQ/dt) - + - + - + - + I Q V slope RC Circuit Q = C V V Constant I-source

  19. Vc V Vc t V = 0 t = 0: Vc = 0 I0 = (V – Vc)/R = V/R t = t1: Vc = V1 (>0) I1 = (V –V1)/R (< I0) t = t2: Vc = V2 (> V1 >0) I2 = (V – V2)/R (< I1 < I2)

  20. Level Time

  21. Vc Vc V V 0.63V t t RC: time constant Vc = V e-t/RC 0.37V RC Vc = V (1 – e-t/RC) t = RC Vc = V (1 – e-1) = 0.63 V [RC] = [(V/I)(Q/V)] = [Q/I] = C/(C/s) = s

  22. Q. A 100 microF capacitor is fully charged with 5 V DC source. This capacitor is discharged through 10 K resistor for 1 s. How much of charge is left in the capacitor in C? Total amount of charge: Q = C V = (100 x 10-6 F)(5 V) = 5 x 10-4 C Time constant: t = RC = (100 x 10-6 F)(10 x 103 Ohm) = 1 s Since Q is proportional to V, after one time constant (100 – 63)% of initial charge is left: Q = 0.37 (5 x 10-4) C

  23. Self-inductance • Self-inductance occurs when the changing flux through a circuit arises from the circuit itself • As the current increases, the magnetic flux through a loop due to this current also increases • The increasing flux induces an emf that opposes the current • As the magnitude of the current increases, the rate of increase lessens and the induced emf decreases • This opposing emf results in a gradual increase of the current

  24. Self-inductance cont • The self-induced emf must be proportional to the time rate of change of the current • L is a proportionality constant called the inductance of the device • The negative sign indicates that a changing current induces an emf in opposition to that change

  25. Self-inductance, final • The inductance of a coil depends on geometric factors • The SI unit of self-inductance is the Henry • 1 H = 1 (V · s) / A

  26. http://www.bugman123.com/Physics/Physics.html For infinitely long solenoid B = monI n: number of turns/m

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