Alternating current circuit

A simple introduction of the AC circuit.

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

2. Fig 1. Fig 1. The instantaneous The instantaneous
value of the potential value of the potential
difference across the difference across the
terminals of the source terminals of the source
varies sinusoidally with varies sinusoidally with
time according totime according to
Where:Where:
v=instantaneous value of the potential difference across the source v=instantaneous value of the potential difference across the source
(also known as the source voltage)(also known as the source voltage)
VV
mm = 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 circuit The resistance in a purely resistive circuit
has the same value at all frequencies. The has the same value at all frequencies. The
maximum potential difference is Vmaximum potential difference is V
m.m.

4. The instantaneous value of the current The instantaneous value of the current i i passing through the resistor may passing 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 current But is the maximum value of current
passing through the resistor. passing through the resistor.
•That is,That is,
•Thus we can write Thus we can write

6. ExampleExample
The instantaneous voltage output of an AC source is given by The 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 instantaneous frequency 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 sine From 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 I
mm =V =V
mm /R= 30 A /R= 30 A

7. Relationship of Voltage, Current, and Relationship of Voltage, Current, and
Power in a Resistive circuitPower in a Resistive circuit
•Fig. 3.Fig. 3. shows graphs of voltage shows graphs of voltage
and current versus time in a and current versus time in a
resistive circuit. These graphs resistive circuit. These graphs
indicate that when only resistance indicate that when only resistance
is present, the voltage and current is present, the voltage and current
are proportional to each other at are proportional to each other at
every moment. For example, when every moment. For example, when
the voltage increases from the voltage increases from A to B A to B
on the graph, the current follows on the graph, the current follows
along in step, increasing from A’ along in step, increasing from A’
to B’ during to B’ during the same time the same time
interval. Likewise, when the interval. Likewise, when the
voltage decreases from voltage decreases from B to C, the B to C, the
current current decreases from decreases from B’ to C’. B’ to C’.
For this reason, the current in a For this reason, the current in a
resistance R is said to be resistance R is said to be in phase in phase
with the voltage across the with the voltage across the
resistance.resistance.

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

9. •The average of sinThe average of sin
22
(2 (2ΠΠft) over one cycle can be ft) over one cycle can be
shown to ½. So the average power dissipated in the shown 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 a We may write the average power dissipated by a
resistor isresistor is

10. Capacitors as AC Circuit elementsCapacitors as AC Circuit elements
The potential difference The potential difference
across the a capacitor across the a capacitor
depends on the amount depends on the amount
of charge on its of its of charge on its of its
plates (recall that Vplates (recall that V
cc
=Q/C. Hence V=Q/C. Hence V
cc will be will be
zero when the charge Q zero when the charge Q
on its of the plates is on its of the plates is
zero, and Vzero, and V
cc will be at will be at
maximum when C is at maximum when C is at
max.max.

11. •The charge flowing into and out of the The charge flowing into and out of the
capacitor varies sinusoidally with time. The capacitor varies sinusoidally with time. The
charge does not flow through the capacitor charge 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 charged However, the plates are alternately charged
and discharged so that a flow +Q onto one and discharged so that a flow +Q onto one
plate means that a charge +Q flows from the plate means that a charge +Q flows from the
other plate, leaving that plate with a net other plate, leaving that plate with a net
charge -Qcharge -Q

12. •The variation of charge causes a The variation of charge causes a
variation in the voltage drop across a variation in the voltage drop across a
capacitor’s plate. The figure shows that capacitor’s plate. The figure shows that
the voltage drop across a capacitor lags the voltage drop across a capacitor lags
behind the current by ¼ of a cycle (90behind the current by ¼ of a cycle (90
00
))

13. •No power is dissipated by the capacitor during No power is dissipated by the capacitor during
the process. During the part of the cycle when the process. During the part of the cycle when
the capacitor is being charged, the electric the capacitor is being charged, the electric
field between its plates increases, and the field 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 the During the remainder of the cycle when the
capacitor discharges, the electric field capacitor discharges, the electric field
between its plates collapses, and the between its plates collapses, and the
capacitor completely returns this energy to the capacitor completely returns this energy to the
source.source.

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

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

16. •The capacitive reactance plays a role The capacitive reactance plays a role
similar to that of resistance in the flow of similar to that of resistance in the flow of
current, so we may rewrite Ohm’s law for current, 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 1.5mmF F
and the rms voltage of the generator is and the rms voltage of the generator is
25.0 V. What is the rms current in the 25.0 V. What is the rms current in the
circuit when the frequency of the circuit when the frequency of the
generator is generator is (a) 1.00 X 10(a) 1.00 X 10
22
Hz and Hz and
(b) 5.00 x 10(b) 5.00 x 10
33
Hz? Hz?

18. Inductors as AC circuit Inductors as AC circuit
elementselements
•Recall that changing current in Recall that changing current in
the circuit induces a potential the circuit induces a potential
difference whose instantaneous difference whose instantaneous
magnitude is given bymagnitude is given by
In this case vIn this case v
LL and i cannot be in phase with each and i cannot be in phase with each
other because the instantaneous current changes other 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 maximum The current reaches its maximum after the voltage after the voltage
does, and it is said that does, and it is said that the current in an the current in an
inductor lags behind the voltage by a phase inductor lags behind the voltage by a phase
angle of 90° (/2 radians). In a purely angle of 90° (/2 radians). In a purely
capacitive circuit, in contrast, the current capacitive circuit, in contrast, the current
leads the leads the voltage by 90°.voltage by 90°.

20. •An inductor impedes the flow of AC because An inductor impedes the flow of AC because
of the potential difference that appears across of the potential difference that appears across
it from changing current. This potential it from changing current. This potential
difference across an inductor in an AC circuit difference across an inductor in an AC circuit
affects the current just as the potential affects 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 the The extent to which the inductor impedes the
flow of AC is called the flow of AC is called the inductive inductive
reactance (Xreactance (X
LL) ) and is directly proportional and is directly proportional
to frequency (f) and inductance (L)to frequency (f) and inductance (L)

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

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