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### Physics 122B Electricity and Magnetism

Lecture 24 (Knight: 33.9, 34.1-5)

LC and AC Circuits

Martin Savage

Lecture 24 Announcements

- Lecture HW is due tonight at 10 PM.
- Midterm Exam 3 is this coming Friday. Covers explicitly everything not covered in the previous exam…and assumes understanding of all previous material.
- Lecture question and lab question are multiple-choice, tutorial is long answer.

Physics 122C - Lecture 23

LC Circuits

A charged capacitor bears a certain resemblance to a stretched spring (remember the rubber diaphragm), storing energy even when the charge is not moving.

An inductor similarly resembles a moving mass (remember the flywheel), storing energy only when charge is in motion.

We know that a mass and spring can make an oscillator. What about a capacitor and inductor. Consider the circuit shown in the diagram. What happens when the switch is closed?

The capacitor discharges by creating a current in the inductor. But where does the energy go that had been stored in the inductor? There are no dissipative elements in the system. Therefore, when the charge of the capacitor goes to zero, all of its previous energy must reside in the inductor. The current in the inductor falls while charging the capacitor in the opposite direction. And so on …

Physics 122C - Lecture 23

The Oscillation Cycle

Physics 122C - Lecture 23

Example:An AM Radio Oscillator

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

Physics 122C - Lecture 23

Plumber’s LC Analogy

Valve

V1

V2

P1

P2

V3

RubberDiaphragm

Flywheel

The “plumber’s analogy” of an LC circuit is a rubber diaphragm that has been stretched by pressure on the top (P1) side. When the valve starts the flow, the diaphragm forces water past the flywheel, which begins to spin. After the diaphragm has become flat, the momentum of the flywheel forces the diaphragm to be stretched in the other direction, and the cycle repeats.

P3

Valve = SwitchRubber Diaphragm = Capacitor

Flywheel = InductorPressure = PotentialWater Flow = Current

Physics 122C - Lecture 23

Chapter 33 - Summary (1)

Physics 122C - Lecture 23

Chapter 33 - Summary (2)

Physics 122C - Lecture 23

Chapter 33 - Summary (3)

Physics 122C - Lecture 23

AC Sources and Phasors

You can think of an AC generator as a battery-like object with an emf that varies sinusoidally as E (t) = E0cos wt, where E0 is the maximum emf and w is the angular frequency, with w=2pf, where f is the frequency in Hz.

Alternatively, the emf and other oscillatory quantities can be represented by a phasor diagram. The phasor is a vector of length E0 that rotates counterclockwise around the origin with angular frequency w, so that the angle it makes with the horizontal axis at any time is wt. The projection of the phasor on the horizontal axis at any time gives the emf.

Physics 122C - Lecture 23

Resistor AC Circuits

Consider an AC current iR through a resistor. Ohm’s Law gives the potential drop across the resistor, which we will call the resistor voltagevR.

If the resistor is connected in an AC circuit as shown, then Kirschoff’s loop law tells us that:

source

In the phasor diagram, the phasors for vR and iR are parallel.

Physics 122C - Lecture 23

Example:Finding Resistor Voltages

In the circuit shown, find (a) the peak voltage across each resistor, and(b) the instantaneous resistor voltages at t=20 ms.

Physics 122C - Lecture 23

Capacitor AC Circuits (1)

Consider an AC current iC through a capacitor as shown. The capacitor voltage vC = E = E0cos wt = VCcos wt. The charge on the capacitor will be q = CvC = CVCcos wt.

The AC current through a capacitor leads the capacitor voltage by p/2 rad or 900.

Physics 122C - Lecture 23

Capacitor AC Circuits (2)

The AC current through a capacitor leads the capacitor voltage by p/2 rad or 900. The phasors for vC and iCare perpendicular, with the iC phasor ahead of the vCphasor.

This is analogous to the behavior of the position and velocity of a mass-and-spring harmonic oscillator.

Physics 122C - Lecture 23

Capacitive Reactance

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

We can then 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:

Physics 122C - Lecture 23

Question

- The instantaneous value of the emf E represented by this phasor is:
- Increasing;
- (b) Decreasing;
- (c) Constant;
- (d) It is not possible to tell without knowing t.

Physics 122C - Lecture 23

Example: Capacitive Reactance

What is the capacitive reactance of a 0.10 mF capacitor at a 100 Hz audio frequency and at a 100 MHz FM radio frequency?

Physics 122C - Lecture 23

Example: Capacitive Current

A 10 mF capacitor is connected to a 1000 Hz oscillator with a peak emf of 5.0 V.

What is the peak current in the capacitor?

Physics 122C - Lecture 23

Voltage Dividers

r

Vout

The circuit indicates a potentiometer, a resistor with a sliding contact. The overall resistance of the unit is R, while the resistance from the sliding tap to the bottom is r.

What is the voltage Vout delivered between the output terminals?

Thus, the potentiometer divides the input voltage and delivers some fraction of it proportional to r/R. This is a voltage divider.

Physics 122C - Lecture 23

Analyzing an RC Circuit

Draw the current vector I at some arbitrary angle. All elements of the circuit will have this current.

Draw the resistor voltage VR in phase with the current. Draw the capacitor voltage VC 900 behind the current. Make sure all phasor lengths scale properly.

The phasors VR and VC form the sides of a right triangle, with E0 as the hypotenuse. Therefore,E02 = VR2+VC2.

Draw the emf E0 as the vector sum of VR and VC. The angle of this phasor is wt, where the time-dependent emf is E0 cos wt.

Physics 122C - Lecture 23

RC Filter Circuits

Now consider a circuit that includes both a resistor and a capacitor. Because the capacitor voltage VC and the resistor voltage VR are 900 apart in the phasor diagram, they must be added like the sides of a right triangle:

Physics 122C - Lecture 23

Filters and Transmission

An RC filter is a circuit that passes a signal with attenuation of some frequencies. Define the transmission of an RC filter asT = vout/vin with wC = 1/(RC):

Low Pass

High Pass

Cross-over Point

Note log scale

Physics 122C - Lecture 23

Example: Designing a Filter

For a science project you have built a radio to listen to AM radio broadcasts at frequencies near 1 MHz. The basic circuit is an antenna, which produces a very small oscillating voltage when it absorbs energy from an electromagnetic wave, and an amplifier. Unfortunately, your neighbor’s short wave broadcast at 10 MHz interferes with your reception. You decide to place a filter between the antenna and the amplifier. You have a 500 pF capacitor.

What frequency should you select for the filter’s cross over frequency?

What value of resistance should be used in the filter?

Tlow (w1) = T high (w2)

Physics 122C - Lecture 23

Question

Which of these RC filter circuits has the largest cross-over frequency wC?

Physics 122C - Lecture 23

AC Inductor Circuits

Consider an AC current iR through an inductor. The changing current produces an instantaneous inductor voltagevL.

If the inductor is connected in an AC circuit as shown, then Kirschoff’s loop law tells us that:

In the phasor diagram, the inductor current iLlags the voltage vL by 900, so that iL peaks T/4 later than vL.

Physics 122C - Lecture 23

Inductive Reactance

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

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:

Physics 122C - Lecture 23

Example: Current and Voltageof an Inductor

A 25 mH inductor is used in a circuit thatis driven at 100 kHZ. The current throughthe inductor reaches a peak value of 20 mAat t=5.0 ms.

What is the peak inductor voltage, and when, closest to t=5.0 ms, does it occur?

The voltage peaks ¼ cycle before the current, which peaks at 5 ms. For f = 100 kHz, T = 10 ms, so T/4 = 2.5 ms. Therefore, the voltage peaks at t =(5.0-2.5) ms = 2.5 ms.

Physics 122C - Lecture 23

Analyzing an LRC Circuit

Draw the current vector I at some arbitrary angle. All elements of the circuit will have this current.

Draw the resistor voltage VR in phase with the current. Draw the inductor and capacitor voltages VL and VC 900 before and behind the current, respectively.

The phasors VR and VL-VC form the sides of a right triangle, with E0 as the hypotenuse. Therefore, E02= VR2+(VL-VC)2.

Draw the emf E0 as the vector sum of VR and VL-VC. The angle of this phasor is wt, where the time-dependent emf is E0 cos wt.

Physics 122C - Lecture 23

The Series RLC Circuit

The figure shows a resistor, inductor, and capacitor connected in series. The same current i passes through all of the elements in the loop. From Kirchhoff’s loop law, E= vR + vL + vC.

Because of the capacitive and inductive elements in the circuit, the current i will not in general be in phase with E, so we will have i = I cos(wt-f) where f is the phase angle between current and voltage. If VL>VC then the current i will lag E and f>0.

Physics 122C - Lecture 23

Impedance and Phase Angle

We can define the impedanceZ of the circuit as:

From the phasor diagram ,we see that the phase anglef of the current is given by:

Physics 122C - Lecture 23

Resonance

The current I will be a maximum when wL=1/wC. This defines the resonant frequency of the system w0:

Physics 122C - Lecture 23

Example:Designing a Radio Receiver

- An AM radio antenna picks up a 1000 kHz signal with a peak voltage of 5.0 mV. The tuning circuit consists of a 60 mH inductor in series with a variable capacitor. The inductor coil has a resistance of 0.25 W, and the resistance of the rest of the circuit is negligible.
- To what capacitance should the capacitor be tuned to listen to this radio station.
- What is the peak current through the circuit at resonance?
- A stronger station at 1050 kHz produces a 10 mV antenna signal. What is the current in the radio at this frequency when the station is tuned to 1000 kHz.

Physics 122C - Lecture 23

Lecture 24 Announcements

- Lecture HW is due tonight at 10 PM.
- Midterm Exam 3 is this coming Friday. Covers explicitly everything not covered in the previous exam…and assumes understanding of all previous material.
- Lecture question and lab question are multiple-choice, tutorial is long answer.

Physics 122C - Lecture 23

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