Physics 115 General Physics II Session 33 AC: RL vs RC circuits - - PowerPoint PPT Presentation

Physics 115

General Physics II Session 33

AC: RL vs RC circuits Phase relationships RLC circuits

6/2/14 1

• R. J. Wilkes
• Email: phy115a@u.washington.edu
• Home page: http://courses.washington.edu/phy115a/

Today

Lecture Schedule

6/2/14 2 Physics 115

Announcements

• Final exam is one week from today! 2:30 pm,

Monday 6/9, here

• 2 hrs allowed, will probably take you about 1 hr
• Comprehensive, but with extra items on material

covered after exam 3

• Final exam will contain ONLY Ch. 24 topics covered

in class

• Usual arrangements, procedures
• Homework set 9 is NOT due Weds night, but Friday

6/6, 11:59pm

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EXAM 3: Scores back from scan shop, should be posted soon AVG: 66

• STD. DEV: 17

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5)

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9)

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Root-Mean-Square V and I

2 1 2 2 RMS 2

( ) 2

R R

I P I R R I R ⎛ ⎞ = = = ⎜ ⎟ ⎝ ⎠

RMS

2

R

I I ≡

RMS

2

R

V V ≡

RMS

2

R

≡ E E

2 2 RMS RMS RMS RMS

( ) ( )

R

V P I R I V R = = =

source RMS RMS

P I = E

“RMS” stands for “root-mean-square”

• 1. square the quantity (gets rid of polarity)
• 2. average the squared values over time,
• 3. take the square root of the result.

P is proportional to I2 , so RMS gives us the equivalent DC voltage in terms of power: DC voltage that would dissipate the same power in the resistor (same joules/sec heating effect)

IR and VR = peak values

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Inductors in AC Circuits

For AC current iR through an inductor: The changing current produces an induced EMF = voltage vL. vL = L ΔiL Δt For the AC circuit as shown, Kirchhoff’s loop law tells us: ΔVsource + ΔVL = E − vL = 0

E (t) = E0 cosωt = vL

Notice: the inductor current iL lags the voltage vL by π/2 radians =900, so that iL peaks T/4 later than vL. ΔiL = 1 L " # $ % & 'vLΔt → iL = VL ωL sinωt = VL ωL cos ωt − π 2 " # $ % & ' = IL cos ωt − π 2 " # $ % & '

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“It can be shown” (calculus) that: For inductors, ωL acts like R in Ohm’s Law: I = V/ R à IL=V/(ωL)

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We can then use a form of Ohm’s Law to relate the peak voltage VL, the peak current IL, and the inductive reactance XL in an AC circuit:

Inductive Reactance

For AC circuits we define a resistance-like quantity, measured in ohms, for inductance. It is called the inductive reactance XL:

2

L

X L f L ω π ≡ =

and

L L L L L L

V I V I X X = =

Reactance is not the same as resistance: it depends on f (time variation), and current is not in phase with voltage!

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Capacitors in AC Circuits

For AC current iC through a capacitor as shown, the capacitor voltage vC = E = E0 cos ωt = VC cos ωt. The charge on the capacitor will be q = C vC = C VC cos ωt.

( )

cos sin

C C C

dq d i CV t CV t dt dt ω ω ω = = = −

cos( / 2)

C C

i CV t ω ω π = +

So: AC current through a capacitor leads the capacitor voltage by π/2 rad or 900.

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Again, we use a form of Ohm’s Law to relate the peak voltage VC, the peak current IC, and the capacitive reactance XC in an AC circuit:

Capacitive Reactance

For AC circuits we also define a resistance-like quantity, measured in ohms, for capacitance. It is called the capacitive reactance XC :

1 1 2

C

X C f C ω π ≡ =

and

C C C C C C

V I V I X X = =

Capacitive vs inductive reactance: XL is proportional to f and L XC is proportional to 1/f and 1/C Current leads in capacitors, lags in inductors

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Example: Capacitive Current

A 10 µF capacitor is connected to a 1000 Hz oscillator with a peak emf of 5.0 V. What is the peak current in the capacitor?

• 1
• 5

1 (1000 Hz) 15.9 2 (1000 s )(1.0 10 F)

C

X π = = Ω ×

(5.0 V) 0.314 A (15.9 )

C C C

V I X = = = Ω

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Capacitors and springs

AC current through a capacitor leads the capacitor voltage by π/ 2 rad or 900. This is analogous to the behavior of the position and velocity of a mass-and- spring harmonic oscillator.

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LC Circuits

A charged capacitor is analogous to a stretched spring : stores energy even when the charge is not moving. An inductor resembles a moving mass (remember the flywheel), stores energy only when charge is in motion. Mass + spring = an oscillator. What about a capacitor + inductor? When the switch is closed in the circuit shown in the diagram:

• 1. The capacitor discharges, creating a current in the inductor.
• 2. There is no dissipative element (resistor = friction) in this system, so

its energy is conserved.

• 3. So, when the capacitor charge reaches 0, all of its stored (E field)

energy must now be stored in the inductor’s B field.

• 4. Then the current in the inductor falls as it charges the capacitor in the
• pposite direction.

And so on …

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The Oscillation Cycle

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Clicker Question

In the circuit above, the frequency is initially f Hz If the generator’s frequency is doubled, to 2f, what happens to the inductor’s reactance XL ?

• A. It doubles XL is proportional to f
• B. It quadruples
• C. It is unchanged
• D. It is 50% smaller
• E. It is ¼ the original value