Lecture Outlines
Chapter 24
Physics, 3rd Edition
James S. Walker
© 2007 Pearson Prentice Hall
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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 Circuits
This 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 Circuits
A 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 Circuits
If 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: