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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
β’ An AC circuit consists of circuit elements and a power source that provides an alternating voltage
βπ. This time-varying voltage is described by:
βπ = βππππ₯ sinππ‘
β’ Where βππππ₯ is the maximum output voltage of the AC source, or the voltage amplitude.
β’ The angular frequency of the AC voltage is:
π = 2ππ =2π
π
β’ Where π is the frequency of the source and π is the period.
β’ 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.
β’ 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 π.
24/11/2020 Phys 104 - Ch. 33/I - lecture 36 - Dr. Alismail 3
33.02sec. Resistors in an AC Circuit
β’ Consider a simple AC circuit consisting of a resistor and an AC source .
β’ At any instant, the algebraic sum of the voltages around a closed loop in a circuit must be zero
(Kirchhoffβs loop rule).
β’ Therefore,βπ + βππ = 0, so that the magnitude of the source voltage equals the magnitude of the voltage
across the resistor:
βπ = βππ = βππππ₯ sinππ‘
β’ Where βππ is the instantaneous voltage across the resistor.
β’ Therefore, the instantaneous current in the resistor is:
ππ =βππ π
=βππππ₯
π sinππ‘ = πΌπππ₯ sinππ‘
β’ Where πΌπππ₯ is the maximum current:
πΌπππ₯ =βππππ₯
π
β’ The instantaneous voltage across the resistor is:
βππ = πΌπππ₯π sinππ‘
A circuit consisting of a resistor of
resistance π 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
β’ A plot of voltage and current versus time for this circuit is shown in Fig. (a):
βͺ At point π, the current has a maximum value in one direction, arbitrarily
called the positive direction.
βͺ Between points π and π, the current is decreasing in magnitude but is still in
the positive direction.
βͺ At π, the current is momentarily zero; it then begins to increase in the
negative direction between points π and π.
βͺ At π, the current has reached its maximum value in the negative direction.
β’ The current and voltage are in step with each other because they vary identically
with time.
β’ Because ππ and βππ both vary as sinππ‘ and reach their maximum values at the
same time, as shown in Fig. (a), they are said to be in phase.
β’ Thus, for a sinusoidal applied voltage, the current in a resistor is always in
phase with the voltage across the resistor.
β’ 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 ππ and instantaneous voltage
βππ 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 π‘ = π, 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
β’ To simplify our analysis of circuits containing two or more elements, we use
graphical constructions called phasor diagrams:
βͺ A phasor is a vector whose length is proportional to the maximum value of
the variable it represents (βππππ₯ for voltage and πΌπππ₯ for current in the
present discussion) and which rotates counterclockwise at an angular speed
equal to the angular frequency associated with the variable.
βͺ The projection of the phasor onto the vertical axis represents the
instantaneous value of the quantity it represents.
β’ 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.
β’ For example, the projection of the current phasor in Fig. (b) is πΌπππ₯ sinππ‘.
β’ Thus, we can use the projections of phasors to represent current values that vary
sinusoidally in time.
β’ We can do the same with time-varying voltages.
β’ 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 ππ and instantaneous voltage
βππ 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 π‘ = π, 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
β’ For the simple resistive circuit, the average value of the current over one cycle is
zero.
β’ The collisions between electrons and the fixed atoms of the resistor result in an
increase in the resistorβs temperature. Although this temperature increase depends
on the magnitude of the current, it is independent of the direction of the current.
β’ We can understand this by recalling that the rate at which energy is delivered to a
resistor is the power π« = π2π , where π is the instantaneous current in the resistor.
β’ Because this rate is proportional to the square of the current, it makes no difference
whether the current is direct or alternatingβthat is, whether the sign associated with
the current is positive or negative.
β’ However, the temperature increase produced by an alternating current having a
maximum value πΌπππ₯ is not the same as that produced by a direct current equal to
πΌπππ₯.
β’ 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 Ξ€πΌπππ₯2 2 have the same area as the gray shaded
regions above the curve and below the dashed line for Ξ€πΌπππ₯2 2.
Thus, the average value of π2 is Ξ€πΌπππ₯2 2.
24/11/2020 Phys 104 - Ch. 33/I - lecture 36 - Dr. Alismail 8
33.02sec. Resistors in an AC Circuit
β’ In an AC circuit, the average value of current is referred to as the rms current.
β’ 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: πΌπππ = ΰ΄₯π2.
β’ Because π2 varies as sin2ππ‘ and because ΰ΄₯π2 is1
2πΌπππ₯2 (see Fig. (b)), the rms current
is:
πΌπππ =πΌπππ₯
2= 0.707πΌπππ₯
β’ Thus, the average power delivered to a resistor that carries an alternating current is:
π«ππ£ = πΌπππ 2 π = πΌπππ βππππ =
βππππ 2
π
β’ Alternating voltage is also best discussed in terms of rms voltage, and the
relationship is identical to that for current:
βππππ =βππππ₯
2= 0.707βππππ₯
(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 Ξ€πΌπππ₯2 2 have the same area as the gray shaded
regions above the curve and below the dashed line for Ξ€πΌπππ₯2 2.
Thus, the average value of π2 is Ξ€πΌπππ₯2 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 βπ = 200 V sinππ‘. Find the rms current in the circuit when this source is connected to a
100 β Ξ© resistor.
Example 33.01
Comparing this expression for voltage output with the general form βπ = βππππ₯ sinππ‘, we see that βππππ₯
= 200 V. Thus, the rms voltage is:
βππππ =βππππ₯
2=200
2= 141 V
Therefore,
πΌπππ =βππππ
π =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 β Hz power source having a maximum voltage of
170 V? (b) What If? What is the resistance of a 100 βW bulb?
Problem 33.02
Resistors in an AC Circuit
βππππ =βππππ₯
2=170
2= 120 V
π«ππ£ =βππππ
2
π
π =βππππ
2
π«ππ£=
120 2
75.0= 193 Ξ©
π =βππππ
2
π«ππ£=
120 2
100= 144 Ξ©
(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 βππππ₯= 100 V. This power supply is connected to a 24.0 β Ξ© 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.
βππππ =βππππ₯
2=100
2= 70.7 V
πΌπππ =βππππ
π =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 β V AC (rms) household supply voltage. Lamps 1 and 2 have 150 βW bulbs; lamp 3 has a 100βW bulb. Find the rms current and resistance of each bulb.
Problem 33.06
βππππ = βππππ 1= βππππ 2= βππππ 3
π«ππ£ = πΌπππ βππππ , πΌπππ =βππππ
π
πΌπππ 1 =π«1
βππππ =150
120= 1.25 A, π 1 =
βππππ
πΌπππ 1=120
1.25= 96.0 Ξ©
πΌπππ 2 =π«2
βππππ =150
120= 1.25 A, π 2 =
βππππ
πΌπππ 2=120
1.25= 96.0 Ξ©
πΌπππ 3 =π«3
βππππ =100
120= 0.833 A, π 3 =
βππππ
πΌπππ 3=
120
0.833= 144 Ξ©
Resistors in an AC Circuit