1. Chapter 22: AC Circuits
Figure (a) Direct current. (b) Alternating current.

2. DC means direct current
DC Circuit Summary
DC Circuits
DC means direct current
The source of electrical energy is usually a battery
If only resistors are in the circuit, the current is independent of time
Ohm’s Law holds: I = V/R
If the circuit contains capacitors & resistors, the current can vary with time but always approaches a constant value (I = V/R) a long enough time after closing the switch.

3. AC Circuit Introduction
AC Circuits
AC means alternating current
The power source is a device that produces an electric potential that varies with time.
There will be a frequency and peak voltage associated with the potential.
Household electrical energy is supplied by an AC source:
The standard frequency is f = 60 Hz
AC power has numerous advantages over DC power

4. DC vs. AC Sources

5. Direct Current, or DC. or AC. Alternating Current,
Current from a battery
flows steadily in one
direction. This is called
Direct Current,
or DC.
By contrast, current
from a power plant
varies sinusoidally
with time. This is called
Alternating Current,
or AC.
Figure (a) Direct current. (b) Alternating current.

6. Generating AC Voltages
Most sources of AC voltage use a generator based on magnetic induction
A shaft holds a coil with many loops of wire
The coil is positioned between the poles of a permanent magnet
The magnetic flux through the coil varies with time as the shaft turns
This changing flux induces a voltage in the coil
This induced voltage is the generator’s output

7. Generators
Generators of electrical energy convert the mechanical energy of the rotating shaft into electrical energy.
The principle of conservation of total energy still applies.
The source of electrical energy in a circuit enables the transfer of electrical energy from a generator to an attached circuit.

8. Resistors in AC Circuits
Assume a circuit with an AC generator & a resistor
The voltage across the output of the AC source varies with time:
V = Vmaxsin(2πƒt)
V is the instantaneous potential difference
Vmax is the amplitude of the AC voltage

9. Apply Ohm’s “Law”
Since the voltage varies sinusoidally, so does the current

10. RMS Voltage
To specify current & voltage values when they vary with time, rms values are adopted
RMS means Root Mean Squared
For the voltage:

11. RMS Current
The root-mean-square value can be defined for any quantity that varies with time. For the current:
The root-mean-square values of the voltage and current are typically used to specify the properties of an AC circuit

12. P = VmaxImaxsin2(2πƒt) P = I V Power
The instantaneous power is the product of the instantaneous voltage &
the instantaneous current
P = I V
Since both I & V vary with time, the power also varies with time:
P = VmaxImaxsin2(2πƒt)

13. Average Power = (½)Maximum Power Pavg = ½(VmaxImax) = Vrms Irms
The instantaneous power varies between
Vmax Imax & 0
Average Power = (½)Maximum Power
Pavg = ½(VmaxImax) = Vrms Irms
This has the same mathematical form as the power in a DC circuit.
Ohm’s Law can again be used to express the power in different ways:

14. AC Circuit Notation
It is important to distinguish between instantaneous & average values of voltage, current & power.
The amplitudes of the AC variations and the rms values are also important

15. Phasors The angle varies with time according to θ= 2πƒt
AC circuits can be analyzed graphically with Phasors.
An arrow has a length Vmax
The arrow’s tail is tied to the origin. Its tip moves along a circle. The arrow makes an angle of θ with the horizontal.
The angle varies with time according to θ= 2πƒt
The rotating arrow represents the voltage in an AC circuit
The arrow is called a phasor. A phasor is not a vector
A phasor diagram provides a convenient way to illustrate and think about the time dependence in an AC circuit

16. The current in an AC circuit can also be represented by a phasor
The 2 phasors always make the same angle with the horizontal axis as time passes
The current and voltage are in phase
For a circuit with only resistors

17. AC Circuits with Capacitors
Assume an AC circuit containing a single capacitor
The instantaneous charge on the plates is:
q = CV = CVmaxsin(2πƒt)
The capacitor’s voltage and charge are in phase with each other.

18. Current in Capacitors The current is the slope of the q-t plot.
The instantaneous current is the rate at which charge flows onto the capacitor plates in a short time interval
The current is the slope of the q-t plot.
A plot of the current as a function of time can be obtained from these slopes

19. I = Imax cos (2πƒt) I = Imax sin (2πƒt + ϕ)
Current in Capacitors
The current is a cosine function
I = Imax cos (2πƒt)
Equivalently, due to the relationship between sine & cosine functions
I = Imax sin (2πƒt + ϕ)
where ϕ = π/2

20. Capacitor Phasor Diagram
The current is out of phase with the voltage
The angle π/2 radians is called the phase angle, ϕ, between V & I
For this circuit, the current & voltage are out of phase by 90°= π/2 radians

21. Current Value for a Capacitor
The peak value of the current is ll
Xc is called the reactance of the capacitor
The SI unit of reactance is Ohms
Reactance and resistance are different because the reactance of a capacitor depends on the frequency
If the frequency is increased, the charge oscillates more rapidly & Δt is smaller, giving a larger current
At high frequencies, the peak current is larger and the reactance is smaller

22. Vmax Imaxsin (2πƒt)cos (2πƒt)
Power In A Capacitor
For an AC circuit with a capacitor,
P = VI =
Vmax Imaxsin (2πƒt)cos (2πƒt)
The average of the power over many oscillations is 0
Energy is transferred from the generator during part of the cycle and from the capacitor in other parts
Energy is stored in the capacitor as electric potential energy and not dissipated by the circuit

23. AC Circuits with Inductors
Assume an AC circuit containing an AC generator and a single inductor.
The voltage drop is
V = L(ΔI/Δt) =
Vmaxsin (2πƒt)
The inductor’s voltage is proportional to the slope of the current-time relationship

24. Current in Inductors
The instantaneous current oscillates in time according to a cosine function
I = -Imaxcos(2πƒt)
A plot of the current-time relationship is shown

25. I = Imaxsin (2πƒt – π/2) I = Imax sin (2πƒt + Φ)
Current in Inductors
The current equation can be rewritten as
I = Imaxsin (2πƒt – π/2)
Equivalently,
I = Imax sin (2πƒt + Φ)
where Φ = -π/2

26. Inductor Phasor Diagram
The current is out of phase with the voltage
For this circuit, the current and voltage are out of phase
by -90°
Remember, for a capacitor, the phase difference was +90°

27. Current Value for an Inductor
The peak value of the current is i l
XL is called the reactance of the inductor
SI unit of inductive reactance is Ohms
As with the capacitor, inductive reactance depends on the frequency
As the frequency is increased, the inductive reactance increases

28. -Vmax Imax sin (2πƒt) cos (2πƒt)
Power in an Inductor
For an AC circuit with an inductor
P = VI =
-Vmax Imax sin (2πƒt) cos (2πƒt)
The average value of the power over many oscillations is 0
Energy is transferred from the generator during part of the cycle and from the inductor in other parts of the cycle
Energy is stored in the inductor as magnetic potential energy

29. Properties of AC Circuits