LECTURE 19 RL CIRCUITS and AC CIRCUITS I Lecture 19

LECTURE 19 RL CIRCUITS AND AC CIRCUITS I Lecture 19

! Reading chapter 28-8, 29-1, and 29-4.

! Concepts: ! RL circuits (DC) ! LC circuits ! RLC circuits ! AC circuit with resistor ! RMS RL circuit

! A circuit containing a resistor and an inductor is called an RL circuit. ! Current in a RL circuit cannot change discontinuously. ! In a dc circuit, an inductor acts as a short circuit long time after the switch is closed/opened. Energizing an RL circuit

! The switch is closed at t = 0. dI E0 − IR − L = 0 From Kirchhoff’s loop rule dt ! The current in the circuit grows and

approaches the final value If. E I t 0 1 e−t /( L/ R) I 1 e−t /τ ( ) = ( − ) = f ( − ) R ! The time constant of the circuit is given by L τ = R Current after short time

! The switch was open, then closed.

! Initially the inductor can be treated as an open circuit. Current after long time

! A long time after the switch is closed, the inductor can be treated as a short circuit. Quiz: 19-1 & 19-2 De-energizing an RL circuit

! The switch has been at position e for a long time, then switched to position f at t = 0.

! The current in the inductor starts out

at I0 and exponentially decays. dI −IR − L = 0 From Kirchhoff’s loop rule dt E I t = I e−t /τ = 0 e−t /τ ( ) 0 R + R 1 Demo: 1

! RL circuit ! An inductor in series with a resistor, a battery and a switch. ! When the switch is closed the current rises slowly from zero to a steady-state value. E I t 0 1 e−t /( L/ R) I 1 e−t /τ ( ) = ( − ) = f ( − ) R Example: 1

! The switch shown has been in position a for a long time. It changes to position b at t = 0. a) What is the current in the circuit at t = 5 µs? b) At what time has the current decayed to 1% of its original value? Make-before-break switch

! A make-before-break switch allows us to remove the battery from the circuit without sparks.

! If the contact e was broken before making a contact with f, the current in the inductor has no where to go. Demo: 2

! Lamp in parallel with solenoid ! A large DC current introduced suddenly to a large inductor cannot pass through the coil at first, so a lamp in parallel with the coil lights up. ! After the current becomes steady, the coil draws more current and bulb dims. ! When the current is switched off suddenly, the induced voltage in the coil (back EMF) again lights the lamp. ! A separate neon lamp in parallel with the coil shows that the direction of the second voltage surge is the opposite of the first. Example: 2

! The switch is open initially.

a) Find the current in the circuit a long time after the switch is closed.

b) Describe the behavior of the light bulb from the time the switch is closed until the current reaches the value found in part a).

c) Explain the behavior of the light bulb if the switch is opened after having been closed for a long time.

d) Find the voltage across the light bulb just before and just after the switch is opened. LC circuits

! Consider an LC circuit with initial charge Q0 stored in the upper capacitor plate and with the switch open. At t = 0, the switch is closed. 1 Q t Q cos t, where ( ) = 0 ω ω = LC dQ I t Q sin t ( ) = = −ω 0 ω dt

! Q lags I by 90°.

! Total energy remains constant.

2 2 1 Q 1 2 1 Q0 Ue + Um = + LI = 2 C 2 2 C Demo: 2

! LC circuit ! Demonstration of oscillatory behavior of LC circuit Example: 3

! You have a 10 mH inductor. What capacitor should you use with it in series to make an oscillator with a frequency of 920 kHz? (This frequency is near the center of the AM radio band.) RLC circuits

! Consider an RLC circuit with an initially charged

charge Q0 capacitor and with the switch open. After the switch is closed, if R is small, Q and I oscillate

with the natural frequency, ω0, but the oscillation is damped. 1 ω0 = LC ! The initial energy stored in the capacitor dissipates through the resistor. The higher the resistance of the resistor, the more damping the circuit has. Demo: 3

! Damped RLC oscillation Alternating current

! When you plug a lamp into a wall socket, the voltage and current supplied to the light bulb vary sinusoidally.

! The voltage oscillate with a fixed frequency of 60 cycles per second in the US.

! The current therefore also periodically reverses direction, so the wall socket provides an alternating current (ac). AC generator

! An alternating current (ac) generator produces an emf given by E = E cosωt peak

where Epeak is the largest value attained by the voltage during a cycle, ω is the angular frequency, and t is time.

! In circuit diagrams, an ac generator is represented by the symbol: Angular frequency, ω

! Angular frequency ω is used for how fast sinusoidal oscillation is happening because the y-component of a rotating vector is a sine function.

! Angular frequency and frequency f are related by ω = 2π f where frequency is measured in Hz (cycles per second). Resistors in AC circuits

! The potential drop across the

resistor, VR, and current through the circuit, I, are given by

VR = VR peak cosωt V VR R peak I = = cosωt ≡ Ipeak cosωt R R

! VR and I are in phase. Power delivered to the resistor

! The power, P, and average

power, Pav, delivered to the resistor are given by

2 2 2 P = I R = Ipeak Rcos ωt

1 P = I 2 R = I 2 R av ( )av peak 2 rms values

! The voltage and current in an ac circuit both average to zero, making the average useless in describing their behavior.

! We use instead the root mean square (rms).

! In general, the rms value of x is given by

x = x2 rms ( )av rms current and voltage

! The rms values of sinusoidally varying current and voltage are given by:

Ipeak Vpeak Irms = Vrms = 2 2 ! 120 volts is the rms value of household ac.

! The rms or peak current is related to the rms or peak voltage by

V Vrms peak Irms = Ipeak = R R ! The average power delivered to a resistor by an AC generator is given by

P = I 2 R = E I av rms rms rms Electrical fuse and circuit breaker

! Electrical fires can be started by improper or damaged wiring because of the heat caused by a too-large current or resistance.

! A fuse is designed to be the hottest point in the circuit – if the current is too high, the fuse melts.

! A circuit breaker is similar, except that it is a bimetallic strip that bends enough to break the connection when it becomes too hot. When it cools, it can be reset. Polarized and grounded plugs

! ~0.1 A of current through a body could be fatal.

! With a polarized plug, the case of an appliance is always connected to the low- potential side, essentially at ground. The high- potential extends only from the wall outlet to Polarized plug the switch when the appliance is turned off.

! With a three-prong grounded plug, the round prong is connected to the ground and the case of an appliance. If a high-potential wire accidentally touches the case, the current flows through the third prong. Grounded plug