A simple introduction of the AC circuit.

1. ALTERNATINALTERNATIN
G CURRENTG CURRENT
(AC)(AC)
CIRCUITCIRCUIT
By Jenerwin M. ColumnaBy Jenerwin M. Columna

2. Fig 1.Fig 1. The instantaneousThe instantaneous
value of the potentialvalue of the potential
difference across thedifference across the
terminals of the sourceterminals of the source
varies sinusoidally withvaries sinusoidally with
time according totime according to
Where:Where:
v=instantaneous value of the potential difference across the sourcev=instantaneous value of the potential difference across the source
(also known as the source voltage)(also known as the source voltage)
VVmm = maximum value of the source voltage (also referred to as the= maximum value of the source voltage (also referred to as the
peak voltage)peak voltage)
f= frequency of the AC sourcef= frequency of the AC source
t= timet= time
Resistor as Circuit elementResistor as Circuit element

3. Fig 2.Fig 2. The resistance in a purely resistive circuitThe resistance in a purely resistive circuit
has the same value at all frequencies. Thehas the same value at all frequencies. The
maximum potential difference is Vmaximum potential difference is Vm.m.

4. The instantaneous value of the currentThe instantaneous value of the current ii passing through the resistor maypassing through the resistor may
be found by applying Ohm’s law to the resistor.be found by applying Ohm’s law to the resistor.
Since the AC source and the resistor are connected in parallelSince the AC source and the resistor are connected in parallel
And we may write therefore,And we may write therefore,

5. • But is the maximum value of currentBut is the maximum value of current
passing through the resistor.passing through the resistor.
• That is,That is,
• Thus we can writeThus we can write

6. ExampleExample
The instantaneous voltage output of an AC source is given byThe instantaneous voltage output of an AC source is given by
the equationthe equation
The source is connected to a 100The source is connected to a 100ΩΩ resistor. What is (a)resistor. What is (a)
frequency of the source and (b) maximum instantaneousfrequency of the source and (b) maximum instantaneous
current flowing through the resistor?current flowing through the resistor?
(A)(A)From the given equation, the argument of the sineFrom the given equation, the argument of the sine
expression is 120expression is 120ΠΠt, which is equal to 2t, which is equal to 2ΠΠft. Therefore,ft. Therefore,
120120ΠΠt= 2t= 2ΠΠftft
f=60 Hzf=60 Hz
(B) The max current occurs at the instant of the maximum(B) The max current occurs at the instant of the maximum
voltage is Ivoltage is Imm =V=Vmm /R= 30 A/R= 30 A

7. Relationship of Voltage, Current, andRelationship of Voltage, Current, and
Power in a Resistive circuitPower in a Resistive circuit
• Fig. 3.Fig. 3. shows graphs of voltageshows graphs of voltage
and current versus time in aand current versus time in a
resistive circuit. These graphsresistive circuit. These graphs
indicate that when only resistanceindicate that when only resistance
is present, the voltage and currentis present, the voltage and current
are proportional to each other atare proportional to each other at
every moment. For example, whenevery moment. For example, when
the voltage increases fromthe voltage increases from A to BA to B
on the graph, the current followson the graph, the current follows
along in step, increasing from A’along in step, increasing from A’
to B’ duringto B’ during the same timethe same time
interval. Likewise, when theinterval. Likewise, when the
voltage decreases fromvoltage decreases from B to C, theB to C, the
currentcurrent decreases fromdecreases from B’ to C’.B’ to C’.
For this reason, the current in aFor this reason, the current in a
resistance R is said to beresistance R is said to be in phasein phase
with the voltage across thewith the voltage across the
resistance.

8. Instantaneous PowerInstantaneous Power
• The instantaneousThe instantaneous
power is neverpower is never
negative but variesnegative but varies
from a low of zerofrom a low of zero
(when I is zero) to a(when I is zero) to a
high of Ihigh of Imm
22
R(when IR(when I
has a peak value).has a peak value).

9. • The average of sinThe average of sin22
(2(2ΠΠft) over one cycle can beft) over one cycle can be
shown to ½. So the average power dissipated in theshown to ½. So the average power dissipated in the
resistor isresistor is
• Defining effective current I by the equationDefining effective current I by the equation
• We may write the average power dissipated by aWe may write the average power dissipated by a
resistor isresistor is

10. Capacitors as AC Circuit elementsCapacitors as AC Circuit elements
The potential differenceThe potential difference
across the a capacitoracross the a capacitor
depends on the amountdepends on the amount
of charge on its of itsof charge on its of its
plates (recall that Vplates (recall that Vcc
=Q/C. Hence V=Q/C. Hence Vcc will bewill be
zero when the charge Qzero when the charge Q
on its of the plates ison its of the plates is
zero, and Vzero, and Vcc will be atwill be at
maximum when C is atmaximum when C is at
max.max.

11. • The charge flowing into and out of theThe charge flowing into and out of the
capacitor varies sinusoidally with time. Thecapacitor varies sinusoidally with time. The
charge does not flow through the capacitorcharge does not flow through the capacitor
(that is, directly from one plate to the other(that is, directly from one plate to the other
through the space separating the plates).through the space separating the plates).
However, the plates are alternately chargedHowever, the plates are alternately charged
and discharged so that a flow +Q onto oneand discharged so that a flow +Q onto one
plate means that a charge +Q flows from theplate means that a charge +Q flows from the
other plate, leaving that plate with a netother plate, leaving that plate with a net
charge -Qcharge -Q

12. • The variation of charge causes aThe variation of charge causes a
variation in the voltage drop across avariation in the voltage drop across a
capacitor’s plate. The figure shows thatcapacitor’s plate. The figure shows that
the voltage drop across a capacitor lagsthe voltage drop across a capacitor lags
behind the current by ¼ of a cycle (90behind the current by ¼ of a cycle (9000
))

13. • No power is dissipated by the capacitor duringNo power is dissipated by the capacitor during
the process. During the part of the cycle whenthe process. During the part of the cycle when
the capacitor is being charged, the electricthe capacitor is being charged, the electric
field between its plates increases, and thefield between its plates increases, and the
capacitor absorbs energy from the AC source.capacitor absorbs energy from the AC source.
• During the remainder of the cycle when theDuring the remainder of the cycle when the
capacitor discharges, the electric fieldcapacitor discharges, the electric field
between its plates collapses, and thebetween its plates collapses, and the
capacitor completely returns this energy to thecapacitor completely returns this energy to the
source.source.

14. • A capacitor impedes the flow of alternatingA capacitor impedes the flow of alternating
current because of the reverse potentialcurrent because of the reverse potential
difference that appears across it as chargedifference that appears across it as charge
builds up on its plates. This potentialbuilds up on its plates. This potential
difference affects the current, just as thedifference affects the current, just as the
potential difference across a resistor in DCpotential difference across a resistor in DC
circuit affects the current.circuit affects the current.

15. • The extent which a capacitor impedes the flow ofThe extent which a capacitor impedes the flow of
alternating current depends on the quantity calledalternating current depends on the quantity called
Capacitive reactance, which is found experimentallyCapacitive reactance, which is found experimentally
to be inversely proportional to both frequency andto be inversely proportional to both frequency and
the capacitance.the capacitance.

16. • The capacitive reactance plays a roleThe capacitive reactance plays a role
similar to that of resistance in the flow ofsimilar to that of resistance in the flow of
current, so we may rewrite Ohm’s law forcurrent, so we may rewrite Ohm’s law for
a purely capacitive circuit asa purely capacitive circuit as

17. ExampleExample
The capacitance of the capacitor is 1.5The capacitance of the capacitor is 1.5µµFF
and the rms voltage of the generator isand the rms voltage of the generator is
25.0 V. What is the rms current in the25.0 V. What is the rms current in the
circuit when the frequency of thecircuit when the frequency of the
generator isgenerator is (a) 1.00 X 10(a) 1.00 X 1022
Hz andHz and
(b) 5.00 x 10(b) 5.00 x 1033
Hz?Hz?

18. Inductors as AC circuitInductors as AC circuit
elementselements
• Recall that changing current inRecall that changing current in
the circuit induces a potentialthe circuit induces a potential
difference whose instantaneousdifference whose instantaneous
magnitude is given bymagnitude is given by
In this case vIn this case vLL and i cannot be in phase with eachand i cannot be in phase with each
other because the instantaneous current changesother because the instantaneous current changes
most rapidly when it is equal to zeromost rapidly when it is equal to zero

19. • The current reaches its maximumThe current reaches its maximum after the voltageafter the voltage
does, and it is said thatdoes, and it is said that the current in anthe current in an
inductor lags behind the voltage by a phaseinductor lags behind the voltage by a phase
angle of 90° (/2 radians). In a purelyangle of 90° (/2 radians). In a purely
capacitive circuit, in contrast, the currentcapacitive circuit, in contrast, the current
leads theleads the voltage by 90°.voltage by 90°.

20. • An inductor impedes the flow of AC becauseAn inductor impedes the flow of AC because
of the potential difference that appears acrossof the potential difference that appears across
it from changing current. This potentialit from changing current. This potential
difference across an inductor in an AC circuitdifference across an inductor in an AC circuit
affects the current just as the potentialaffects the current just as the potential
difference across a resistor in DC cicuit.difference across a resistor in DC cicuit.
• The extent to which the inductor impedes theThe extent to which the inductor impedes the
flow of AC is called theflow of AC is called the inductiveinductive
reactance (Xreactance (XLL )) and is directly proportionaland is directly proportional
to frequency (f) and inductance (L)to frequency (f) and inductance (L)

22. • The inductive reactance plays a roleThe inductive reactance plays a role
similar to that of resistance in the flow ofsimilar to that of resistance in the flow of
current, so we can rewrite Ohm’s Lawcurrent, so we can rewrite Ohm’s Law
intointo

23. Question.Question.
• The drawing shows three ac circuits: oneThe drawing shows three ac circuits: one
contains a resistor, one a capacitor, andcontains a resistor, one a capacitor, and
one an inductor. The frequency of eachone an inductor. The frequency of each
ac generator is reduced to one-half itsac generator is reduced to one-half its
initial value. Which circuit experiencesinitial value. Which circuit experiences
(a) the greatest increase in current(a) the greatest increase in current
and (b) the least change inand (b) the least change in
current?current?