Chapter 30 Inductance, Electromagnetic Oscillations and...

Chapter 30Inductance, Electromagnetic Oscillations and AC CircuitsOscillations, and AC Circuits

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30-7 AC Circuits with AC SourceResistors, capacitors, and inductors have diff t hdifferent phase relationships between current and voltagecurrent and voltage when placed in an ac circuit.

The current through i t i i ha resistor is in phase

with the voltage.


30-7 AC Circuits with AC Source

The voltage across the inductor is given by

or

.

Therefore, the current through an inductor


t oug a ductolags the voltage by 90°.


The voltage across the inductor is related to the current through it:g

.

The quantity XL is called the inductive reactance, and has units of ohms:reactance, and has units of ohms:



Example 30-9: Reactance of a coil.

A coil has a resistance R = 1 00 Ω andA coil has a resistance R = 1.00 Ω and an inductance of 0.300 H. Determine the current in the coil if (a) 120-V dc is ( )applied to it, and (b) 120-V ac (rms) at 60.0 Hz is applied.

Solution p797


Solution p797


The voltage across the capacitor is given bycapacitor is given by

.

Therefore, in a capacitor, the current leads the voltage by 90°.



The voltage across the capacitor is related to the current through it:g

.

The quantity XC is called the capacitive reactance, and (just like the inductivereactance, and (just like the inductive reactance) has units of ohms:


30-7 AC Circuits with AC SourceExample 30-10: Capacitor reactance.

What is the rms current in the circuit shown if C = 1.0 μF and Vrms = 120 V? Calculate (a) for f = 60 Hz and then (b) for f = 6.0 x 105 Hz.


Solution p799


This figure shows a high-pass filter (allows an ac signal to pass but blocks a dc voltage) g g )and a low-pass filter (allows a dc voltage to be maintained but blocks higher-frequency fl t ti )fluctuations).


30-8 LRC Series AC CircuitAnalyzing the LRC series AC circuit is complicated as the voltages are not in phasecomplicated, as the voltages are not in phase – this means we cannot simply add them. Furthermore, the reactances depend on the pfrequency.


30-8 LRC Series AC CircuitWe calculate the voltage (and current) using what are called phasors – these are vectors representing the individual voltages.

Here at t = 0 theHere, at t = 0, the current and voltage are both at ga maximum. As time goes on, the phasors will rotate counterclockwise.


30-8 LRC Series AC Circuit

Some time t laterSome time t later, the phasors have rotated.



The voltages across each device are given b th t fby the x-component of each, and the current by its x-componentby its x-component. The current is the same throughout the gcircuit.



We find from the ratio of voltage to current that the effective resistance, called the impedance, of the circuit is given byis given by


30-8 LRC Series AC CircuitTh h l b h l dThe phase angle between the voltage and the current is given by

or

The factor cos φ is called the power factor of the circuit.


power factor of the circuit.


Example 30-11: LRC circuit.

Suppose R = 25.0 Ω, L = 30.0 mH, and C = 12.0 μF, and they are connected in series to a 90.0-V ac (rms) 500-Hz source. Calculate (a) the current in the circuit (b) the voltmeter readingscircuit, (b) the voltmeter readings (rms) across each element, (c) the phase angle φ and (d) the powerphase angle φ, and (d) the power dissipated in the circuit.


Solution p801

30-9 Resonance in AC Circuits

The rms current in an ac circuit is

Cl l I d d th fClearly, Irms depends on the frequency.


W th t ill b i h

30-9 Resonance in AC CircuitsWe see that Irms will be a maximum when XC= XL; the frequency at which this occurs is

f0 = ω0/2π is called the resonant frequency.q y


Chapter 30 Inductance, Electromagnetic Oscillations and...

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