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Understanding Time-Varying Currents and Fields in AC Circuits

This chapter explores Alternating Current (AC) circuits, focusing on their components, such as resistors and capacitors, and their relationship with AC generators. It covers key concepts including the behavior of AC voltages and currents, root mean square (RMS) calculations for power evaluation, and the significance of phase relationships in AC circuits. Additionally, the chapter explains how capacitors charge and discharge in RC circuits, detailing the time constant and its impact on current and voltage behavior. Ideal for students and enthusiasts of electrical engineering.

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Understanding Time-Varying Currents and Fields in AC Circuits

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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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