1. chapter 33 Alternating Current Circuits
33.1 AC Sources
33.2 Resistors in an AC Circuit
33.3 Inductors in an AC Circuit
33.4 Capacitors in an AC Circuit
33.5 The RLC Series Circuit
33.6 Power in an AC Circuit
33.7 Resonance in a Series RLC
Circuit

2. What is direct current?
Direct current is travels in one direction only
Cells and batteries supply direct current
Direct current is also known as d.c. current or d.c. supply

3. Direct current With d.c. current the voltage is constant
The cathode ray oscilloscope (CRO) trace will show d.c. supply as a horizontal line
voltage

4. What is alternating current?
Alternating current is also known as a.c. current of a.c supply
This current is always changing direction, so the voltage goes up and down all the time too

5. Alternating current
a.c. supply will produce a wave on the cathode ray oscilloscope
The voltage with a.c. supply is constantly going up and down

6. AC Circuit
An AC circuit consists of a combination of circuit elements and an AC generator or source
The output of an AC generator is sinusoidal and varies with time according to the following equation
Δv = ΔVmax sin 2ƒt
Δv is the instantaneous voltage
ΔVmax is the maximum voltage of the generator
ƒ is the frequency at which the voltage changes, in Hz

7. AC Voltage The angular frequency is
ƒ is the frequency of the source
T is the period of the source
The voltage is positive during one half of the cycle and negative during the other half
Commercial electric power plants in Canada/US use a frequency of 60 Hz
This corresponds with an angular frequency of 377 rad/s

8. Resistors in an AC Circuit
For a sinusoidal applied voltage, the current in a resistor is always in phase with the voltage across the resistor
The direction of the current has no effect on the behavior of the resistor
Resistors behave essentially the same way in both DC and AC circuits

9. RMS Current and Voltage
The average current in one cycle is zero
The rms current is the average of importance in an AC circuit
rms stands for root mean square
Alternating voltages can also be discussed in terms of rms values

10. Resistor in an AC Circuit
Consider a circuit consisting of an AC source and a resistor
The graph shows the current through and the voltage across the resistor
The current and the voltage reach their maximum values at the same time
The current and the voltage are said to be in phase

11. Example: In the U.S., standard wiring supplies 120 V at 60 Hz.
Write this in sinusoidal form, assuming V(t)=0 at t=0.
This 120 V is the RMS amplitude: so Vp=Vrms = 170 V.
This 60 Hz is the frequency f: so w=2p f=377 s -1.
So V(t) = 170 sin(377t + fv).
Choose fv=0 so that V(t)=0 at t=0: V(t) = 170 sin(377t).

12. More About Resistors in an AC Circuit
The direction of the current has no effect on the behavior of the resistor
The rate at which electrical energy is dissipated in the circuit is given by
P = i2 R
where i is the instantaneous current
the heating effect produced by an AC current with a maximum value of Imax is not the same as that of a DC current of the same value
The maximum current occurs for a small amount of time

13. rms Values
It is useful to compare ac circuits and dc circuits. We cannot use, however, the average ac current or voltage which are both zero. Instead we must use the root mean square or rms values.
The rms value of current or voltage in an ac circuit can be compared to the equivalent quantities in a simple dc circuit.
Vrms = IrmsR
Pav = Irms2R = Vrms2/R

14. Root Mean Square Voltage and Current
Vrms = Square root of the mean (average) of V-squared.
Average AC Power consumed by a resistor
independent of form of V vs t curve

15. rms Current and Voltage
The rms current is the direct current that would dissipate the same amount of energy in a resistor as is actually dissipated by the AC current
Alternating voltages can also be discussed in terms of rms values

16. A 3.33-kW resistor is connected to a generator with a maximum voltage of 101 V. Find (a) the average and (b) the maximum power delivered to this circuit.

17. Ohm’s Law in an AC Circuit
rms values will be used when discussing AC currents and voltages
AC ammeters and voltmeters are designed to read rms values
Many of the equations will be in the same form as in DC circuits
Ohm’s Law for a resistor, R, in an AC circuit
ΔVrms = Irms R
Also applies to the maximum values of v and i

18. Capacitors in an AC Circuit
Consider a circuit containing a capacitor and an AC source
The current starts out at a large value and charges the plates of the capacitor
There is initially no resistance to hinder the flow of the current while the plates are not charged
As the charge on the plates increases, the voltage across the plates increases and the current flowing in the circuit decreases

19. More About Capacitors in an AC Circuit
The current reverses direction
The voltage across the plates decreases as the plates lose the charge they had accumulated
The voltage across the capacitor lags behind the current by 90°

20. Capacitors in an AC Circuit
The circuit contains a capacitor and an AC source
Kirchhoff’s loop rule gives:
v + vc = 0 and so
v = vC = Vmax sin t
vc is the instantaneous voltage across the capacitor

21. Phasor Diagram for Capacitor
The phasor diagram shows that for a sinusoidally applied voltage, the current always leads the voltage across a capacitor by 90o
This is equivalent to saying the voltage lags the current

22. Capacitive Reactance
The maximum current in the circuit occurs at cos t = 1 which gives
The impeding effect of a capacitor on the current in an AC circuit is called the capacitive reactance and is given by

23. Voltage Across a Capacitor
The instantaneous voltage across the capacitor can be written as
vC = Vmax sin t = Imax XC sin t
As the frequency of the voltage source increases, the capacitive reactance decreases and the maximum current increases
As the frequency approaches zero, XC approaches infinity and the current approaches zero
This would act like a DC voltage and the capacitor would act as an open circuit

24. Capacitive Reactance and Ohm’s Law
The impeding effect of a capacitor on the current in an AC circuit is called the capacitive reactance and is given by
When ƒ is in Hz and C is in F, XC will be in ohms
Ohm’s Law for a capacitor in an AC circuit
ΔVrms = Irms XC

25. Capacitive Reactance
The larger the capacitance, the smaller the capacitive reactance
As frequency increases, reactance decreases
DC: capacitor is an “open circuit” and
high frequency: capacitor is a “short circuit”
and

26. Capacitive Reactance
A particular example:

27. If frequency is doubled, XC drops by factor of 2,
An rms voltage of 10.0 V with a frequency of 1.00 kHz is applied to a mF capacitor. (a) What is the rms current in this circuit? (b) By what factor does the current change if the frequency of the voltage is doubled? (c) Calculate the current for a frequency of 2.00 kHz.
If frequency is doubled, XC drops by factor of 2,
Current is doubled: Irms = 49.6 kHz

28. Inductors in an AC Circuit
Consider an AC circuit with a source and an inductor
The current in the circuit is impeded by the back emf of the inductor
The voltage across the inductor always leads the current by 90°

29. Inductors in an AC Circuit
Kirchhoff’s loop rule can be applied and gives:

30. Current in an Inductor
The equation obtained from Kirchhoff's loop rule can be solved for the current
This shows that the instantaneous current iL in the inductor and the instantaneous voltage ∆vL across the inductor are out of phase by (p/2) rad = 90o

31. Phase Relationship of Inductors in an AC Circuit
The current in the circuit is impeded by the back emf of the inductor
For a sinusoidal applied voltage, the current in an inductor always lags behind the voltage across the inductor by 90° (π/2)

32. Phasor Diagram for an Inductor
The phasors are at 90o with respect to each other
This represents the phase difference between the current and voltage
Specifically, the current lags behind the voltage by 90o

33. Inductive Reactance
The factor wL has the same units as resistance and is related to current and voltage in the same way as resistance
Because wL depends on the frequency, it reacts differently, in terms of offering resistance to current, for different frequencies
The factor is the inductive reactance and is given by:
XL = wL

34. Inductive Reactance, cont.
Current can be expressed in terms of the inductive reactance
As the frequency increases, the inductive reactance increases
This is consistent with Faraday’s Law:
The larger the rate of change of the current in the inductor, the larger the back emf, giving an increase in the inductance and a decrease in the current

35. Voltage Across the Inductor
The instantaneous voltage across the inductor is

36. Inductive Reactance and Ohm’s Law
The effective resistance of a coil in an AC circuit is called its inductive reactance and is given by
XL = 2ƒL
When ƒ is in Hz and L is in H, XL will be in ohms
Ohm’s Law for the inductor
ΔVrms = Irms XL

37. Inductive Reactance
Larger inductance: larger reactance (more induced EMF to oppose the applied AC voltage)
Higher frequency: larger impedance (higher frequency means higher time rate of change of current, which means more induced EMF to oppose the applied AC voltage)

38. The RLC Series Circuit
The resistor, inductor, and capacitor can be combined in a circuit
The current in the circuit is the same at any time and varies sinusoidally with time

39. Current and Voltage Relationships in an RLC Circuit
The instantaneous voltage across the resistor is in phase with the current
The instantaneous voltage across the inductor leads the current by 90°
The instantaneous voltage across the capacitor lags the current by 90°

40. More About Voltage in RLC Circuits
VR is the maximum voltage across the resistor and VR = ImaxR
VL is the maximum voltage across the inductor and VL = ImaxXL
VC is the maximum voltage across the capacitor and VC = ImaxXC
In series, voltages add and the instantaneous voltage across all three elements would be
v = vR + vL+ vC
Easier to use the phasor diagrams

41. Resulting Phasor Diagram
The individual phasor diagrams can be combined
Here a single phasor Imax is used to represent the current in each element
In series, the current is the same in each element

42. Vector Addition of the Phasor Diagram
Vector addition is used to combine the voltage phasors
∆VL and ∆VC are in opposite directions, so they can be combined
Their resultant is perpendicular to ∆VR

43. Total Voltage in RLC Circuits
From the vector diagram, ∆Vmax can be calculated

44. Impedance The current in an RLC circuit is
Z is called the impedance of the circuit and it plays the role of resistance in the circuit, where
Impedance has units of ohms
Also, ∆Vmax = ImaxZ

45. Impedance Triangle
Since Imax is the same for each element, it can be removed from each term in the phasor diagram
The result is an impedance triangle

46. Impedance Triangle, cont.
The impedance triangle confirms that
The impedance triangle can also be used to find the phase angle, 
The phase angle can be positive or negative and determines the nature of the circuit
Also, cos  =

47. Phasor Diagrams
To account for the different phases of the voltage drops, vector techniques are used
Represent the voltage across each element as a rotating vector, called a phasor
The diagram is called a phasor diagram

48. Phasor Diagram for RLC Series Circuit
The voltage across the resistor is on the +x axis since it is in phase with the current
The voltage across the inductor is on the +y since it leads the current by 90°
The voltage across the capacitor is on the –y axis since it lags behind the current by 90°

49. Phasor Diagram, cont
The phasors are added as vectors to account for the phase differences in the voltages
ΔVL and ΔVC are on the same line and so the net y component is ΔVL - ΔVC

50. ΔVmax From the Phasor Diagram
The voltages are not in phase, so they cannot simply be added to get the voltage across the combination of the elements or the voltage source
 is the phase angle between the current and the maximum voltage

51. Impedance of a Circuit
The impedance, Z, can also be represented in a phasor diagram

52. Impedance and Ohm’s Law
Ohm’s Law can be applied to the impedance
ΔVmax = Imax Z

53. Problem
An AC circuit with R= 2 W, C = 15 mF, and L = 30 mH is driven by a generator with voltage V(t)=2.5 sin(8pt) Volts. Calculate the maximum current in the circuit, and the phase angle. L R C
Imax = Vgen,max /Z
Imax = 2.5/2.76 = .91 Amps

54. Problem
An AC circuit with R= 2 W, C = 15 mF, and L = 30 mH is driven by a generator with voltage V(t)=2.5 sin(8pt) Volts. Calculate the maximum current in the circuit, and the phase angle.
Imax = Vgen,max /Z L R C
Imax = 2.5/2.76 = .91 Amps

55. A 65.0-Hz generator with an rms voltage of 115 V is connected in series to a 3.35-kW resistor and a 1.50-mF capacitor. Find (a) the rms current in the circuit and (b) the phase angle, f, between the current and the voltage. I
VR= IR
VC= I XC
= I/(wC)
115 V

56. Power in AC Circuits: Example
Series RLC Circuit:
R = 15W, C = 4.7 mF, L = 25mH. The circuit is driven by a source of
Vrms = 75 V and f = 550 Hz.
At what average rate is energy dissipated by the resistor?

57. (a) Irms=(0.70748 V)/12 W Irms=2.83 A (b) P=(0.70748 V)(2.83 A)
Example: An AC power supply with Vmax=48 V is connected to a resistor with 12 W. Calculate (a) the rms current, (b) P and (c) Pmax.
(a) Irms=(0.70748 V)/12 W
Irms=2.83 A
(b) P=(0.70748 V)(2.83 A)
P=96 W
(c) Pmax=2P=192 W

58. Example
A coil has a resistance R = 1.00 Ω and an inductance of 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.
a. For dc current, there is no magnetic induction, so the current is determined by the resistance: I = V/R = 120 A.
b. The inductive reactance is XL = 2πfL = 113 Ω. The current is Irms = Vrms/XL = 1.06 A.

59. Example
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.
: a. XC = 1/2πfC = 2.7 kΩ; I = V/XC = 44 mA.
b. XC = 0.27 Ω and I = 440 A.

60. Example
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 readings (rms) across each element, (c) the phase angle , and (d) the power dissipated in the circuit.
a. XL = 2πfL = 94.2 Ω; XC = 1/(2πfC) = 26.5 Ω; so Z = 72.2 Ω and I = 1.25 A.
b. The voltages are the currents multiplied by the reactances (or resistance). VL = 118 V; VC = 33.1 V; VR = 31.2 V.
c. cos φ = 0.346, so φ = 69.7°.
d. P = IV cos φ = 39.0 W.

61. Summary of Circuit Elements, Impedance and Phase Angles

62. Problem Solving for AC Circuits
Calculate as many unknown quantities as possible
For example, find XL and XC
Be careful of units -- use F, H, Ω
Apply Ohm’s Law to the portion of the circuit that is of interest
Determine all the unknowns asked for in the problem

63. Power in an AC Circuit
No power losses are associated with capacitors and pure inductors in an AC circuit
In a capacitor, during one-half of a cycle energy is stored and during the other half the energy is returned to the circuit
In an inductor, the source does work against the back emf of the inductor and energy is stored in the inductor, but when the current begins to decrease in the circuit, the energy is returned to the circuit

64. Power in an AC Circuit, cont
The average power delivered by the generator is converted to internal energy in the resistor
Pav = IrmsΔVR = IrmsΔVrms cos 
cos  is called the power factor of the circuit
Phase shifts can be used to maximize power outputs

65. Resonance in an AC Circuit
Resonance occurs at the frequency, ƒo, where the current has its maximum value
To achieve maximum current, the impedance must have a minimum value
This occurs when XL = XC

66. Resonance in RLC circuit in series
PR = IDV = Irms2R
Where Irms = Vrms/Z
For a fixed AC voltage source, maximum power dissipation occurs when XL = XC, ie. When Z = R
i.e.

67. Resonance, cont
Theoretically, if R = 0 the current would be infinite at resonance
Real circuits always have some resistance
Tuning a radio
A varying capacitor changes the resonance frequency of the tuning circuit in your radio to match the station to be received
Metal Detector
The portal is an inductor, and the frequency is set to a condition with no metal present
When metal is present, it changes the effective inductance, which changes the current which is detected and an alarm sounds

68. Summary of Resonance At resonance At lower frequencies
Z is minimum (=R)
Imax is maximum (=Vgen,max/R)
Vgen is in phase with I
XL = XC VL(t) = -VC(t)
At lower frequencies
XC > XL Vgen lags I
At higher frequencies
XC < XL Vgen lead I
Imax(XL-XC)
ImaxXL
ImaxXC
ImaxR
Vgen,max f

69. Power in AC circuits The voltage generator supplies power.
Only resistor dissipates power.
Capacitor and Inductor store and release energy.
P(t) = I(t)VR(t) oscillates so sometimes power loss is large, sometimes small.
Average power dissipated by resistor:
P = ½ Imax VR,max
= ½ Imax Vgen,max cos(f)
= Irms Vgen,rms cos(f)

70. AC Summary Resistors: VR,max=Imax R In phase with I
Capacitors: VC,max =Imax XC Xc = 1/(2pf C)
Lags I
Inductors: VL,max=Imax XL XL = 2pf L
Leads I
Generator: Vgen,max=Imax Z Z = √R2 +(XL -XC)2
Can lead or lag I tan(f) = (XL-XC)/R
Power is only dissipated in resistor:
P = ½ImaxVgen,max cos(f)