Chapter 24 Alternating-Current Circuits. Units of Chapter 24 Alternating Voltages and Currents...

Chapter 24

Alternating-Current Circuits

Units of Chapter 24

• Alternating Voltages and Currents

• Capacitors in AC Circuits

• RC Circuits

• Inductors in AC Circuits

• RLC Circuits

• Resonance in Electrical Circuits

24-1 Alternating Voltages and Currents

Wall sockets provide current and voltage that vary sinusoidally with time.

Here is a simple ac circuit:

24-1 Alternating Voltages and Currents

The voltage as a function of time is:

24-1 Alternating Voltages and Currents

Since this circuit has only a resistor, the current is given by:

Here, the current and voltage have peaks at the same time – they are in phase.

24-1 Alternating Voltages and Currents

In order to visualize the phase relationships between the current and voltage in ac circuits, we define phasors – vectors whose length is the maximum voltage or current, and which rotate around an origin with the angular speed of the oscillating current.

The instantaneous value of the voltage or current represented by the phasor is its projection on the y axis.

24-1 Alternating Voltages and Currents

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); we square the value, find the mean value, and then take the square root:

120 volts is the rms value of household ac.

24-1 Alternating Voltages and Currents

By calculating the power and finding the average, we see that:

24-1 Alternating Voltages and Currents

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.

24-1 Alternating Voltages and Currents

A ground fault circuit interrupter can cut off the current in a short circuit within a millisecond.

24-2 Capacitors in AC Circuits

How is the rms current in the capacitor related to its capacitance and to the frequency? The answer, which requires calculus to derive:

24-2 Capacitors in AC Circuits

In analogy with resistance, we write:

24-2 Capacitors in AC Circuits

The voltage and current in a capacitor are not in phase. The voltage lags by 90°.

24-3 RC Circuits

In an RC circuit, the current across the resistor and the current across the capacitor are not in phase. This means that the maximum current is not the sum of the maximum resistor current and the maximum capacitor current; they do not peak at the same time.

24-3 RC CircuitsThis phasor diagram illustrates the phase relationships. The voltages across the capacitor and across the resistor are at 90° in the diagram; if they are added as vectors, we find the maximum.

24-3 RC Circuits

This has the exact same form as V = IR if we define the impedance, Z:

24-3 RC Circuits

There is a phase angle between the voltage and the current, as seen in the diagram.

24-3 RC Circuits

The power in the circuit is given by:

Because of this, the factor cos φ is called the power factor.

24-4 Inductors in AC Circuits

Just as with capacitance, we can define inductive reactance:

24-4 Inductors in AC Circuits

The voltage across an inductor leads the current by 90°.

24-4 Inductors in AC Circuits

The power factor for an RL circuit is:

Currents in resistors, capacitors, and inductors as a function of frequency:

24-5 RLC CircuitsA phasor diagram is a useful way to analyze an RLC circuit.

24-5 RLC Circuits

The phase angle for an RLC circuit is:

If XL = XC, the phase angle is zero, and the voltage and current are in phase.

The power factor:

24-5 RLC Circuits

At high frequencies, the capacitive reactance is very small, while the inductive reactance is very large. The opposite is true at low frequencies.

24-6 Resonance in Electrical CircuitsIf a charged capacitor is connected across an inductor, the system will oscillate indefinitely in the absence of resistance.

24-6 Resonance in Electrical Circuits

The rms voltages across the capacitor and inductor must be the same; therefore, we can calculate the resonant frequency.

24-6 Resonance in Electrical Circuits

In an RLC circuit with an ac power source, the impedance is a minimum at the resonant frequency:

24-6 Resonance in Electrical Circuits

The smaller the resistance, the larger the resonant current:

Summary of Chapter 24

• The voltage from an ac generator varies sinusoidally:

• Phasor represents voltage or current in ac circuit; as it rotates, its y component gives the instantaneous value.

• Root mean square (rms) of a sinusoidally varying quantity:

Summary of Chapter 24

• rms current in a capacitor:

• Capacitive reactance:

• Voltage across capacitor lags current by 90°

• Impedance in an RC circuit:

• Average power:

Summary of Chapter 24

• Inductive reactance:

• Impedance of an RL circuit:

• Impedance of an RLC circuit:

• Resonant frequency of an LC circuit: