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 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 The output of an AC generator is sinusoidal and varies with time according to the following equation v = v 0 sin 2ƒtv = v 0 sin 2ƒt v is the instantaneous voltage v is the instantaneous voltage V 0 is the maximum voltage of the generator V 0 is the maximum voltage of the generator ƒ is the frequency at which the voltage changes, in Hz ƒ is the frequency at which the voltage changes, in Hz

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.

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 v rms = 170x0.707 = 120 V. Period = 1/60 = 16.7 ms

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 $$!!!

R C L

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

More About Resistors in an AC Circuit The direction of the current has no effect on the behavior of the resistor 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 The rate at which electrical energy is dissipated in the circuit is given by P = i 2 RP = i 2 R where i is the instantaneous current where i is the instantaneous current the heating effect produced by an AC current with a maximum value of I max is not the same as that of a DC current of the same value the heating effect produced by an AC current with a maximum value of I max is not the same as that of a DC current of the same value The maximum current occurs for a small amount of time The maximum current occurs for a small amount of time

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 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 Alternating voltages can also be discussed in terms of rms values

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

Capacitors in an AC Circuit Consider a circuit containing a capacitor and an AC source 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 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 chargedThere 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 As the charge on the plates increases, the voltage across the plates increases and the current flowing in the circuit decreases

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

AC A V v= v o sin(  t) Q = C V i= C  v o cos(  t) ioio i o = C  v o v o = (1/  C)i o v i V lags 90 deg behind i.

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 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, X C will be in ohmsWhen ƒ is in Hz and C is in F, X C will be in ohms Ohm’s Law for a capacitor in an AC circuit Ohm’s Law for a capacitor in an AC circuit V = I X CV = I X C

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

Charging Capacitor in an RC Circuit The charge on the capacitor varies with time The charge on the capacitor varies with time q = Q(1 – e -t/RC )q = Q(1 – e -t/RC ) i=i 0 e -t/RCi=i 0 e -t/RC The time constant, =RCThe time constant, =RC The time constant represents the time required for the charge to increase from zero to 63.2% of its maximum The time constant represents the time required for the charge to increase from zero to 63.2% of its maximum i=i 0 e -1 = i 0 i=i 0 e -1 = i 0

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

RC Circuit Q = C V V Constant I-source I Q V  Q/  t = C (  V/  t)  V/  t = (1/C) (  Q/  t) slope

V = 0 VcVc t = 0: V c = 0I 0 = (V – V c )/R = V/R t = t 1 : V c = V 1 (>0)I 1 = (V –V 1 )/R (< I 0 ) VcVc t t = t 2 : V c = V 2 (> V 1 >0) I 2 = (V – V 2 )/R (< I 1 < I 2 ) V

Level Time

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

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 F)(5 V) = 5 x C Time constant: t = RC = (100 x F)(10 x 10 3 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 ) C

Self-inductance Self-inductance occurs when the changing flux through a circuit arises from the circuit itself 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 increasesAs the current increases, the magnetic flux through a loop due to this current also increases The increasing flux induces an emf that opposes the currentThe 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 decreasesAs 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 currentThis opposing emf results in a gradual increase of the current

Self-inductance cont The self-induced emf must be proportional to the time rate of change of the current 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 deviceL 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 changeThe negative sign indicates that a changing current induces an emf in opposition to that change

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

For infinitely long solenoid B =  o nI n: number of turns/m