31 lecture outline

Alternating Current

Learning Goals for Chapter 31

Looking forward at …

• How phasors make it easy to describe sinusoidally varying quantities.

• How to use reactance to describe the voltage across a circuit element that carries an alternating current.

• How to analyze an L-R-C series circuit with sinusoidal emfs of different frequencies.

Introduction

• Waves from a broadcasting station produce an alternating current in the circuits of a

radio (like the one in this classic car).

• How does a radio tune to a particular station?

• How are ac circuits different from dc circuits?

AC sources

• Most present-day household and industrial power

distribution systems operate with alternating current (ac). • Any appliance that you plug

into a wall outlet uses ac.

• An ac source is a device that supplies a sinusoidally

AC sources and currents

• A sinusoidal voltage might be described by a function such as:

• Here v is the instantaneous potential difference, V is the voltage amplitude, and ω = 2πf is the angular frequency.

• In the United States and Canada, commercial electric-power distribution systems use a frequency f = 60 Hz.

Phasor diagrams

• To represent sinusoidally varying voltages and currents, we define rotating vectors called phasors.

• Shown is a phasor diagram

Root-mean-square values

a sinusoidal current:

1. Graph current i versus time.

2. Square the instantaneous current i.

3. Take the average (mean) value of i2_.

Root-mean-square values

• For sinusoidal ac sources, the rms current and voltage values are:

• This wall socket has a voltage amplitude of V = 170 V, meaning that the voltage alternates

between +170 V and −170 V.

Resistor in an ac circuit: Slide 1 of 3

•

_{When a resistor is}

connected with an

ac source, the voltage

and current amplitudes

are related by

Ohm’s law:

Inductor in an ac circuit: Slide 1 of 3

connected with an

ac source, the voltage and current amplitudes are related by:

• The inductive reactance is X_L = ωL; the greater the inductance and the higher the frequency, the greater the inductive

Capacitor in an ac circuit: Slide 1 of 3

connected with an

ac source, the voltage and current amplitudes are related by:

• The capacitive reactance is X_C = 1/ωC; the greater the

Comparing ac circuit elements

• The graph shows how the resistance of a resistor and the reactances of an inductor and a capacitor vary with angular frequency ω.

• Resistance R is

independent of frequency.

• If ω = 0, corresponding to a dc circuit, there is no

current through a capacitor because X_C → ∞.

A useful application: The loudspeaker

• In order to route signals of different frequency to the

appropriate speaker shown, the woofer and tweeter are

connected in parallel across the amplifier output.

• _{The capacitor in the tweeter}

branch blocks the low-frequency components of sound but passes the higher frequencies.

high-The

L-R-C

series circuit: Slide 1 of 3

capacitor are connected in series with an ac source, the voltage and current amplitudes are related by:

Measuring body fat by bioelectric impedance

analysis

• The electrodes attached to this overweight patient’s chest are applying a small ac voltage of frequency 50 kHz.

• The attached instrumentation

measures the amplitude and phase angle of the resulting current

through the patient’s body. • These depend on the relative

Power in a resistor

is a pure resistor, the voltage and current are in phase.

Power in an inductor

element is a pure inductor, the

voltage leads the current by 90°. • The power is

negative when v

and i have

Power in a capacitor

element is a pure capacitor, the

voltage lags the current by 90°. • The power is

negative when v

and i have

Power in a general ac circuit

combination of resistors,

Power in a general ac circuit

combination of resistors, capacitors, and inductors, the voltage v across the entire circuit has some phase angle ϕ with respect to the current i. • The factor cos ϕ is called the

power factor of the circuit. • For a pure resistor, the power

Resonance in ac circuits

X_C, and Z as functions of log ω.

• As the frequency increases,

X_L increases and X_C

decreases; hence there is always one frequency at which X_L and X_C are equal and X_L − X_C is zero.

Resonance in ac circuits

• As we vary the angular frequency ω of the source, the maximum value of I occurs at the frequency at which the impedance Z is minimum.

• This peaking of the current amplitude at a certain frequency is called resonance.

• The angular frequency ω₀ at which the resonance peak occurs is called the resonance angular frequency.

• At ω = ω₀ the inductive reactance X_L and capacitive reactance

Resonance in ac circuits

• Shown is a graph of current amplitude I as a function of angular frequency ω for an L-R-C series circuit with

Transformers

power is supplied to a primary coil, and then the secondary coil

delivers power to a resistor.

Transformers

• In an ideal transformer, the ratio of the voltages across the primary and secondary coils is equal to the ratio of the

number of turns in the coils: