Slide 1 Fig 33-CO, p.1033

Slide 2 Fig 33-1, p.1034

.. the basic principle of the ac generator is a direct consequence of Faraday’s law of induction. When a conducting loop is rotated in a magnetic field at constant angular frequency ω , a sinusoidal voltage (emf) is induced in the loop. This instantaneous voltage Δv is

where ΔV max is the maximum output voltage of the ac generator, or the voltageamplitude, the angular frequency is

The voltage supplied by an AC source is sinusoidal with a period T.

where f is the frequency of the generator (the voltage source) and T is the period.

Commercial electric power plants in the United States use a frequency of 60 Hz, which corresponds to an angular frequency of 377 rad/s.

Slide 3

To simplify our analysis of circuits containing two or more elements, we use

graphical constructions called phasor diagrams.

In these constructions, alternating (sinusoidal) quantities, such as current and

voltage, are represented by rotating vectors called phasors.

The length of the phasor represents the amplitude (maximum value) of the

quantity, and the projection of the phasor onto the vertical axis represents the

instantaneous value of the quantity.

As we shall see, a phasor diagram greatly simplifies matters when we must

combine several sinusoidally varying currents or voltages that have different

phases.

Slide 4 Fig 33-2, p.1035

At any instant, the algebraic sum of the voltages around a closed loop in a circuit must be zero (Kirchhoff’s loop rule).

where ΔvR is the instantaneous voltage across the resistor. Therefore, the instantaneous current in the resistor is

the maximum current:

Slide 5 Fig 33-3, p.1035

Slide 6 Fig 33-3a, p.1035

Plots of the instantaneous current iR

and instantaneous voltage vR across a

resistor as functions of time.

The current is in phase with the

voltage, which means that the current

is zero when the voltage is zero,

maximum when the voltage is

maximum, and minimum when the

voltage is minimum.

At time t = T, one cycle of the time-

varying voltage and current has been

completed.

Slide 7 Fig 33-3b, p.1035

Phasor diagram for the resistive circuit

showing that the current is in phase with

the voltage.

What is of importance in an ac circuit is an average value of current, referred to as the rms current

Slide 8 Fig 33-5, p.1037

(a) Graph of the current in a resistor as a function of time

(b) Graph of the current squared in a resistor as a function of time.

Notice that the gray shaded regions under the curve and above the dashed line

for I 2max/2 have the same area as the gray shaded regions above the curve and

below the dashed line for I 2 max/2. Thus, the average value of i 2 is I 2max/2.

Slide 9

The voltage output of a generator is given by Δv = (200 V)sin ωt. Find the rms current in the circuit when this generator is connected to a 100 Ω- resistor.

Slide 10 Fig 33-6, p.1038

is the self-induced instantaneous voltage across the inductor.

Slide 11

the inductive reactance

Slide 12 Fig 33-7a, p.1039

max

max

sin

sin( )2

L

L

dIV L V t

dtV

I tL

Slide 13 Fig 33-7b, p.1039

Slide 14

In a purely inductive ac circuit, L = 25.0 mH and the rms voltage is 150 V.

Calculate the inductive reactance and rms current in the circuit if the

frequency is 60.0 Hz.

Slide 15

Slide 16

Slide 17 Fig 33-9, p.1041

Slide 18 Fig 33-10, p.1041

Slide 19 Fig 33-10a, p.1041

Slide 20 Fig 33-10b, p.1041

Slide 21

capacitive reactance:

Slide 22

Slide 23 Fig 33-13a, p.1044

Φ the phase angle between the current and the applied voltage

the current at all points in a series ac circuit has the same amplitude and phase

Slide 24

Slide 25 Fig 33-13b, p.1044

Slide 26 Fig 33-14, p.1044

Slide 27 Fig 33-14a, p.1044

Slide 28 Fig 33-14b, p.1044

Slide 29 Fig 33-14c, p.1044

Slide 30 Fig 33-15, p.1045

(a) Phasor diagram for the series RLC circuit The phasor VR is in phase with the current phasor Imax, the phasor VL leads Imax by 90°, and the phasor VC lags Imax by 90°. The total voltage Vmax makes an Angle with Imax. (b) Simplified version of the phasor diagram shown in part (a)

Slide 31 Fig 33-16, p.1045

An impedance triangle for a series RLC circuit gives the relationship Z R2 + (XL - XC)2

Slide 32 Table 33-1, p.1046

Slide 33

Slide 34

the phase angle

Slide 35

Slide 36

Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

No power losses are associated with pure capacitors and pure inductors inan ac circuit

When the current begins to increase in one direction in an ac circuit,

charge begins to accumulate on the capacitor, and a voltage drop

appears across it. When this voltage drop reaches its maximum value,

the energy stored in the capacitor is

However, this energy storage is only momentary. The capacitor is

charged and discharged twice during each cycle: Charge is delivered to

the capacitor during two quarters of the cycle and is returned to the

voltage source during the remaining two quarters. Therefore, the

average power supplied by the source is zero. In other words, no

power losses occur in a capacitor in an ac circuit.

Slide 42

For the RLC circuit , we can express the instantaneous power P

The average power

the quantity cos φ is called the power factor

the maximum voltage drop across the resistor is given by

Slide 43

In words, the average power delivered by the generator is converted to

internal energy in the resistor, just as in the case of a dc circuit. No

power loss occurs in an ideal inductor or capacitor.

When the load is purely resistive, then φ= 0, cos φ= 1, and

Slide 44

Slide 45

A series RLC circuit is said to be in resonance when the current has its maximum value. In general, the rms current can be written

Because the impedance depends on the frequency of the source, the

current in the RLC circuit also depends on the frequency. The frequency ω0

at which XL-XC=0 is called the resonance frequency of the circuit. To find ω0 ,

we use the condition XL = XC ,from which we obtain , ω0 L =1/ ω0 C or

Slide 46 Fig 33-19, p.1050

(a) The rms current versus frequency for a series RLC circuit, for three values of R. The current reaches its maximum value at the resonance frequency . (b) Average power delivered to the circuit versus frequency for the series RLC circuit, for two values of R.

Slide 47 Fig 33-19a, p.1050

Slide 48 Fig 33-19b, p.1050

Slide 49 Fig 33-20, p.1051

Slide 50 Fig 33-21, p.1052

Slide 51 Fig 33-22, p.1052

Slide 52 p.1053

Slide 53 Fig 33-23, p.1053

Slide 54 p.1053

Slide 55 Fig 33-24, p.1055

Slide 56 Fig 33-24a, p.1055

Slide 57 Fig 33-24b, p.1055

Slide 58 Fig 33-25, p.1055

Slide 59 Fig 33-25a, p.1055

Slide 60 Fig 33-25b, p.1055

Slide 61 Fig 33-26, p.1056

Slide 62 Fig 33-26a, p.1056

Slide 63 Fig 33-26b, p.1056

Slide 64 Fig Q33-2, p.1058

Slide 65 Fig Q33-22, p.1058

Slide 66 Fig P33-3, p.1059

Slide 67 Fig P33-6, p.1059

Slide 68 Fig P33-7, p.1059

Slide 69 Fig P33-25, p.1060

Slide 70 Fig P33-26, p.1060

Slide 71 Fig P33-30, p.1061

Slide 72 Fig P33-36, p.1061

Slide 73 Fig P33-47, p.1062

Slide 74 Fig P33-55, p.1062

Slide 75 Fig P33-56, p.1062

Slide 76 Fig P33-58, p.1063

Slide 77 Fig P33-61, p.1063

Slide 78 Fig P33-62, p.1063

Slide 79 Fig P33-64, p.1063

Slide 80 Fig P33-69, p.1064

Slide 81 Fig P33-69a, p.1035

Slide 82 Fig P33-69b, p.1035

Slide 83