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Alternating Current Circuits
Ch. 23
Phys 104 - Ch. 33/I - lecture 36
1442 - 1st semester
Dr. Ayman Alismail
33Ch.
24/11/2020 Phys 104 - Ch. 33/I - lecture 36 - Dr. Alismail 2
33.01sec. AC Sources
โข An AC circuit consists of circuit elements and a power source that provides an alternating voltage
โ๐. This time-varying voltage is described by:
โ๐ = โ๐๐๐๐ฅ sin๐๐ก
โข Where โ๐๐๐๐ฅ is the maximum output voltage of the AC source, or the voltage amplitude.
โข The angular frequency of the AC voltage is:
๐ = 2๐๐ =2๐
๐
โข Where ๐ is the frequency of the source and ๐ is the period.
โข Because the output voltage of an AC source varies sinusoidally with time, the voltage is positive
during one half of the cycle and negative during the other half.
โข Likewise, the current in any circuit driven by an AC source is an alternating current that also
varies sinusoidally with time.The voltage supplied by an AC source is sinusoidal
with a period ๐.
24/11/2020 Phys 104 - Ch. 33/I - lecture 36 - Dr. Alismail 3
33.02sec. Resistors in an AC Circuit
โข Consider a simple AC circuit consisting of a resistor and an AC source .
โข At any instant, the algebraic sum of the voltages around a closed loop in a circuit must be zero
(Kirchhoffโs loop rule).
โข Therefore,โ๐ + โ๐๐ = 0, so that the magnitude of the source voltage equals the magnitude of the voltage
across the resistor:
โ๐ = โ๐๐ = โ๐๐๐๐ฅ sin๐๐ก
โข Where โ๐๐ is the instantaneous voltage across the resistor.
โข Therefore, the instantaneous current in the resistor is:
๐๐ =โ๐๐ ๐
=โ๐๐๐๐ฅ
๐ sin๐๐ก = ๐ผ๐๐๐ฅ sin๐๐ก
โข Where ๐ผ๐๐๐ฅ is the maximum current:
๐ผ๐๐๐ฅ =โ๐๐๐๐ฅ
๐
โข The instantaneous voltage across the resistor is:
โ๐๐ = ๐ผ๐๐๐ฅ๐ sin๐๐ก
A circuit consisting of a resistor of
resistance ๐ connected to an AC source,
designated by the symbol .
24/11/2020 Phys 104 - Ch. 33/I - lecture 36 - Dr. Alismail 4
33.02sec. Resistors in an AC Circuit
โข A plot of voltage and current versus time for this circuit is shown in Fig. (a):
โช At point ๐, the current has a maximum value in one direction, arbitrarily
called the positive direction.
โช Between points ๐ and ๐, the current is decreasing in magnitude but is still in
the positive direction.
โช At ๐, the current is momentarily zero; it then begins to increase in the
negative direction between points ๐ and ๐.
โช At ๐, the current has reached its maximum value in the negative direction.
โข The current and voltage are in step with each other because they vary identically
with time.
โข Because ๐๐ and โ๐๐ both vary as sin๐๐ก and reach their maximum values at the
same time, as shown in Fig. (a), they are said to be in phase.
โข Thus, for a sinusoidal applied voltage, the current in a resistor is always in
phase with the voltage across the resistor.
โข For resistors in AC circuits, there are no new concepts to learn. Resistors behave
essentially the same way in both DC and AC circuits.
(a) Plots of the instantaneous current ๐๐ and instantaneous voltage
โ๐๐ 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 ๐ก = ๐, one cycle
of the time-varying voltage and current has been completed. (b)
Phasor diagram for the resistive circuit showing that the current is
in phase with the voltage.
24/11/2020 Phys 104 - Ch. 33/I - lecture 36 - Dr. Alismail 5
33.02sec. Resistors in an AC Circuit
โข To simplify our analysis of circuits containing two or more elements, we use
graphical constructions called phasor diagrams:
โช A phasor is a vector whose length is proportional to the maximum value of
the variable it represents (โ๐๐๐๐ฅ for voltage and ๐ผ๐๐๐ฅ for current in the
present discussion) and which rotates counterclockwise at an angular speed
equal to the angular frequency associated with the variable.
โช The projection of the phasor onto the vertical axis represents the
instantaneous value of the quantity it represents.
โข The projections of the phasor arrows onto the vertical axis are determined by a sine
function of the angle of the phasor with respect to the horizontal axis.
โข For example, the projection of the current phasor in Fig. (b) is ๐ผ๐๐๐ฅ sin๐๐ก.
โข Thus, we can use the projections of phasors to represent current values that vary
sinusoidally in time.
โข We can do the same with time-varying voltages.
โข The advantage of this approach is that the phase relationships among currents and
voltages can be represented as vector additions of phasors.
(a) Plots of the instantaneous current ๐๐ and instantaneous voltage
โ๐๐ 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 ๐ก = ๐, one cycle
of the time-varying voltage and current has been completed. (b)
Phasor diagram for the resistive circuit showing that the current is
in phase with the voltage.
24/11/2020 Phys 104 - Ch. 33/I - lecture 36 - Dr. Alismail 6
33.02sec. Resistors in an AC Circuit
Quick Quiz 33.1 Consider the voltage phasor in the following figure, shown at three
instants of time. Choose the part of the figure that represents the instant of time at which
the instantaneous value of the voltage has the largest magnitude.
Answer (a). The phasor in part (a) has the largest projection onto the vertical axis.
Quick Quiz 33.2 For the voltage phasor in the following figure, choose the part of the
figure that represents the instant of time at which the instantaneous value of the voltage
has the smallest magnitude.
Answer (b). The phasor in part (b) has the smallest-magnitude projection onto the
vertical axis.
A voltage phasor is shown at three instants
24/11/2020 Phys 104 - Ch. 33/I - lecture 36 - Dr. Alismail 7
33.02sec. Resistors in an AC Circuit
โข For the simple resistive circuit, the average value of the current over one cycle is
zero.
โข The collisions between electrons and the fixed atoms of the resistor result in an
increase in the resistorโs temperature. Although this temperature increase depends
on the magnitude of the current, it is independent of the direction of the current.
โข We can understand this by recalling that the rate at which energy is delivered to a
resistor is the power ๐ซ = ๐2๐ , where ๐ is the instantaneous current in the resistor.
โข Because this rate is proportional to the square of the current, it makes no difference
whether the current is direct or alternatingโthat is, whether the sign associated with
the current is positive or negative.
โข However, the temperature increase produced by an alternating current having a
maximum value ๐ผ๐๐๐ฅ is not the same as that produced by a direct current equal to
๐ผ๐๐๐ฅ.
โข This is because the alternating current is at this maximum value for only an instant
during each cycle (Fig. (a)).
(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 ฮค๐ผ๐๐๐ฅ2 2 have the same area as the gray shaded
regions above the curve and below the dashed line for ฮค๐ผ๐๐๐ฅ2 2.
Thus, the average value of ๐2 is ฮค๐ผ๐๐๐ฅ2 2.
24/11/2020 Phys 104 - Ch. 33/I - lecture 36 - Dr. Alismail 8
33.02sec. Resistors in an AC Circuit
โข In an AC circuit, the average value of current is referred to as the rms current.
โข The notation rms stands for root-mean square, which in this case means the square
root of the mean (average) value of the square of the current: ๐ผ๐๐๐ = เดฅ๐2.
โข Because ๐2 varies as sin2๐๐ก and because เดฅ๐2 is1
2๐ผ๐๐๐ฅ2 (see Fig. (b)), the rms current
is:
๐ผ๐๐๐ =๐ผ๐๐๐ฅ
2= 0.707๐ผ๐๐๐ฅ
โข Thus, the average power delivered to a resistor that carries an alternating current is:
๐ซ๐๐ฃ = ๐ผ๐๐๐ 2 ๐ = ๐ผ๐๐๐ โ๐๐๐๐ =
โ๐๐๐๐ 2
๐
โข Alternating voltage is also best discussed in terms of rms voltage, and the
relationship is identical to that for current:
โ๐๐๐๐ =โ๐๐๐๐ฅ
2= 0.707โ๐๐๐๐ฅ
(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 ฮค๐ผ๐๐๐ฅ2 2 have the same area as the gray shaded
regions above the curve and below the dashed line for ฮค๐ผ๐๐๐ฅ2 2.
Thus, the average value of ๐2 is ฮค๐ผ๐๐๐ฅ2 2.
24/11/2020 Phys 104 - Ch. 33/I - lecture 36 - Dr. Alismail 9
33.02sec.
Example 25.01
The voltage output of an AC source is given by the expression โ๐ = 200 V sin๐๐ก. Find the rms current in the circuit when this source is connected to a
100 โ ฮฉ resistor.
Example 33.01
Comparing this expression for voltage output with the general form โ๐ = โ๐๐๐๐ฅ sin๐๐ก, we see that โ๐๐๐๐ฅ
= 200 V. Thus, the rms voltage is:
โ๐๐๐๐ =โ๐๐๐๐ฅ
2=200
2= 141 V
Therefore,
๐ผ๐๐๐ =โ๐๐๐๐
๐ =141
100= 1.41 A
Resistors in an AC Circuit
24/11/2020 Phys 104 - Ch. 33/I - lecture 36 - Dr. Alismail 10
33.02sec.
Example 25.01
(a) What is the resistance of a lightbulb that uses an average power of 75.0 W when connected to a 60.0 โ Hz power source having a maximum voltage of
170 V? (b) What If? What is the resistance of a 100 โW bulb?
Problem 33.02
Resistors in an AC Circuit
โ๐๐๐๐ =โ๐๐๐๐ฅ
2=170
2= 120 V
๐ซ๐๐ฃ =โ๐๐๐๐
2
๐
๐ =โ๐๐๐๐
2
๐ซ๐๐ฃ=
120 2
75.0= 193 ฮฉ
๐ =โ๐๐๐๐
2
๐ซ๐๐ฃ=
120 2
100= 144 ฮฉ
(a)
(b)
24/11/2020 Phys 104 - Ch. 33/I - lecture 36 - Dr. Alismail 11
33.02sec.
Example 25.01
An AC power supply produces a maximum voltage โ๐๐๐๐ฅ= 100 V. This power supply is connected to a 24.0 โ ฮฉ resistor, and the current and resistor
voltage are measured with an ideal AC ammeter and voltmeter, as shown in the following figure. What does each meter read? Note that an ideal ammeter
has zero resistance and that an ideal voltmeter has infinite resistance.
Problem 33.03
Each meter reads the rms value.
โ๐๐๐๐ =โ๐๐๐๐ฅ
2=100
2= 70.7 V
๐ผ๐๐๐ =โ๐๐๐๐
๐ =70.7
24.0= 2.95 A
Resistors in an AC Circuit
24/11/2020 Phys 104 - Ch. 33/I - lecture 36 - Dr. Alismail 12
33.02sec.
Example 25.01
The following figure shows three lamps connected to a 120 โ V AC (rms) household supply voltage. Lamps 1 and 2 have 150 โW bulbs; lamp 3 has a 100โW bulb. Find the rms current and resistance of each bulb.
Problem 33.06
โ๐๐๐๐ = โ๐๐๐๐ 1= โ๐๐๐๐ 2= โ๐๐๐๐ 3
๐ซ๐๐ฃ = ๐ผ๐๐๐ โ๐๐๐๐ , ๐ผ๐๐๐ =โ๐๐๐๐
๐
๐ผ๐๐๐ 1 =๐ซ1
โ๐๐๐๐ =150
120= 1.25 A, ๐ 1 =
โ๐๐๐๐
๐ผ๐๐๐ 1=120
1.25= 96.0 ฮฉ
๐ผ๐๐๐ 2 =๐ซ2
โ๐๐๐๐ =150
120= 1.25 A, ๐ 2 =
โ๐๐๐๐
๐ผ๐๐๐ 2=120
1.25= 96.0 ฮฉ
๐ผ๐๐๐ 3 =๐ซ3
โ๐๐๐๐ =100
120= 0.833 A, ๐ 3 =
โ๐๐๐๐
๐ผ๐๐๐ 3=
120
0.833= 144 ฮฉ
Resistors in an AC Circuit