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AC Circuits – page 1 of 12 RESISTORS, CAPACITORS AND INDUCTORS
IN AC CIRCUITS
INTRODUCTION AND THEORY
Many electric circuits use batteries and involve direct current (dc). However, there are
considerably more circuits that operate with alternating current (ac), when the charge flow
reverses direction periodically. In an “ac” circuit, the most common generators serve the same
purpose as a battery serves in a dc circuit: they give energy to the moving electric charges
but they change the direction of magnetic forces periodically.
Since electrical outlets in a house provide alternating current, we all use ac circuits routinely.
If the voltage and the current alternate sinusoidally with time we can write:
v = v(t) = Vm ⋅ sin(ϖt + ϕ )
i = i(t) = Im ⋅ sin(ϖt)
€
Where
• v and€i represent the instantaneous voltage and current when we are considering
their variation with time explicitly.
• Vm and Im are the amplitude or peak value of the voltage and current
• V = Vm/21/2 and I = Im/21/2 without subscripts refer to the RMS values.
• f is the ordinary frequency and represents the number of complete oscillations per
second
• ω = 2πf is the angular frequency.
• φ is the phase difference between the voltage and current.
Resistors And Ohm's Law In AC Circuits
The schematic below is that of an ac circuit formed by plugging a toaster into a wall socket.
The heating element of the toaster is essentially a thin wire of resistance R and becomes red
hot when the electrical energy is dissipated in it According to Ohm’s Law, the instantaneous voltage v across a resistor is proportional to the
instantaneous current i flowing through it.
R V i Figure 1: Resistor in an AC circuit
AC Circuits – page 2 of 12 The next two graphs show the voltage across the resistor and the current flowing through the
resistor as a function of time.
Instantaneous voltage V – (blue), and current i – (red) V(t), i(t) t Figure 2: V and I: Sinusoidal variation in phase
Im Vm Figure 3: Phasor Diagram
From the graph in Figure 2 the peak value of voltage across a pure resistor is reached at the
same time with the peak value of the current. Therefore, the current and voltage are said to
be in phase, and mathematically this is expressed as
In the Figure 3, the radial vectors also called phasors rotate with angular velocity ω
representing the current and the voltage across the resistance. The lengths of these phasors
represent the peak current Im and voltage Vm. The y components of these phasors are
and they are equal to the y – values from the graph in figure 2 at any time. The phasor
diagram shows that the voltage and the current are in phase.
Resistance, Reactance And Impedance
The ratio of the voltage to the current in a resistor is called its resistance. As well, in a circuit
where the current is proportional to the voltage, the circuit is called a linear circuit. This is
happening when the circuit contains only resistors, capacitors and/or inductors. Resistance
does not depend on frequency, and in a resistor the voltage and the current are in phase.
However, circuits with only resistors, capacitors, and solenoids are not very useful in some
AC Circuits – page 3 of 12 cases. If the circuit contains also, diodes or transistors, the circuit is no longer linear.
In most practical cases, the ratio of the voltage to the current depends on the frequency and
in general there is a phase difference between the voltage and the current. In this general
case, the ratio of the voltage to the current is called Impedance and it is denoted with the
symbol Z. Resistance is a special case of impedance. A very important case is that in which
the voltage and the current are out of phase by 90°: this is an important case because when
this happens, no power is lost in the circuit and the ratio of the voltage to the current is called
the reactance, denoted with the symbol X. Reactance can be caused by capacitors or by
inductors.
Capacitors In AC Circuits
Capacitors store electric charge. They are used with resistors in timing circuits because it
takes time for a capacitor to fill with charge. They are also used to smooth varying DC
supplies by acting as a reservoir of charge. They are also used in filter circuits because
capacitors easily pass AC signals but they block DC signals. The voltage on a capacitor
depends on the amount of charge stored on its plates. If we denote the instantaneous value
of the current with i(t):
The current flowing off the positive plate is equal to the current flowing into the negative plate
and by definition is the rate at which the charge Q is being stored. From the definition of the
capacitance as a function of the charge Q and the potential across the capacitor,
it follows that
But the charge Q on the capacitor equals the integral of the current with respect to time.
In the indefinite integral above, the constant of integration was set to zero so that the average
charge on the capacitor would be zero (we are starting with an uncharged capacitor).
AC Circuits – page 4 of 12 Therefore, the voltage across the capacitor:
The last equation shows that the current and the voltage are out of phase by 900.
The capacitive reactance XC is equal to:
and it can be defined as the ratio of the magnitude of the voltage to magnitude of the current
in a capacitor (and that is Ohm’s Law for the capacitor!)
Looking at the difference of phase, the voltage across the capacitor is 90°, or one quarter of
period, behind the current. The same phase difference φ = 90° is reflected in the phasor
diagrams. Since the vertical component of any phasor arrow represents the instantaneous
value of its quantity and the phasors are rotating counter clockwise the phasor representing
VC is 90° behind the current.
Instantaneous voltage V – (blue), and current i – (red) Figure 4: V and I: Sinusoidal variation for Capacitor
Figure 5: Capacitor Phasor Diagram
As we have seen before, when the voltage and the current differ in phase by 900, the
resistance is called reactance. Another important difference between reactance and resistance
is that the reactance is frequency dependent and for a capacitor, it decreases with frequency.
AC Circuits – page 5 of 12 Inductors In AC Circuits
An inductor is usually a coil of wire. The resistance of an ideal inductor is negligible, as is its
capacitance. However, the voltage across an inductor is influenced by changes of its own
magnetic field. Faraday's law of electromagnetic induction states that the current i(t) in the coil
sets up a magnetic field, whose magnetic flux ΦB is proportional to the field strength B, which
in turn is proportional to the current.
Therefore, the self inductance of the coil, denoted L is defined as:
However, Faraday's law gives the emf induced in a coil due to a change in the magnetic flux
According to Kirchhoff’s First Law the emf is a voltage rise; therefore, the voltage drop vL
across the inductor should be:
vL (t) = −eL =
dΦ B
dt
d
[ L ⋅ i(t)]
dt
d
= L ⋅ [ I m ⋅ sin(ϖ t)]
dt
= ϖ L ⋅ I m ⋅ cos(ϖ t)
=
π
= Vm ⋅ sin(ϖ t + )
2
As in the case of the capacitor, we define the inductive reactance XL as the ratio of the
magnitudes of the voltage and the current, and from the equation above we see that
XL = ωL.
It is worth noting the analogy to Ohm's law: the voltage is proportional to the current, and the
peak voltage and currents are related by
Vm = XL.Im
AC Circuits – page 6 of 12 V(t), i(t) t Figure 6: V and I: Sinusoidal variation for Solenoid
Figure 7: Solenoid Phasor Diagram
From figures 6 and 7 it follows that the voltage and the current through the solenoid are in
phase: the voltage across the inductor has its maximum when the current is changing most
rapidly, which is when the current is passing through zero. Therefore, the voltage across the
ideal inductor is 90° (or
) ahead of the current, (i.e. it reaches its maximum one quarter of
the cycle before the current does). The same conclusion is drawn from the phasor diagram.
We should also note that for a coil the reactance is frequency dependent in the sense that it
increases with frequency.
Summary:
Resistance, Reactance and Impedance
The following is a summary of the relationship between voltage and current in linear circuits:
• The impedance is the general term for the ratio of the voltage to the current.
• Resistance is the special case of impedance when φ = 0,
• Reactance the special case when φ = ± 90°.
Component
Difference of
Phase between
Voltage and
Current
Ohm’s Law
Resistor
Voltage and
Current are in
phase
R=
VR
I
Capacitor
Inductor
Voltage lags
Current lags
behind Current by behind Voltage
π/2
by π/2
Xc =
VC
1
=
I ϖC
XL =
VL
=ϖL
I
AC Circuits – page 7 of 12 Resistor and Capacitor connected in Series
When we connect components together, Kirchhoff's laws apply at any instant. So the voltage
vs(t) across a resistor and capacitor in series is just
vs(t) = vR(t) + vC(t)
However, the addition is more complicated because the two are not in phase: they add to give
a new sinusoidal voltage, but the amplitude VS is less than VR + VC as it can be seen from
figure 11.
VC
C
VR R VS VR
I ϕ i vseri
es =
s
vR
er
+
ie
vCSeries
Figure 10: Resistor and Capacitor in
v
VC s
VRCS Figure 11: R-C Series Phasor Diagram
=
vseri vtheorem in figure 11 we have:
However, using Pythagoras'
but
es =
R
2
2
seri2
vR V+ = V V+V
S
R
C
es >
+
v
VR
vC
C
Using Ohm’s Law and expressing
the
+ three voltages and substituting in the equation above,
we obtain the impedance ZRCS andVthe
phase difference ϕ between voltage VRCS and the
C.
current I:
but
Vseri
2
2
2
es > (b
I ⋅ Z RCS ) = ( I ⋅ R) + ( I ⋅ XC )
VR u
+
t
2
VC. V
! 1 $
2
Z RCS = R + #
&
"ϖC %
s
er
ie
and
The
V
X
1
s ϕ =− C =− C =−
tan
am
V
R
ϖ RC
>
plitR
V
ude
R
s
+
and
V
the
The
C
RM
am .
S
plit
volt
AC Circuits – page 8 of 12 Resistor and Inductor connected in Series
Similarly, when we connect a solenoid in series with a resistor, the instantaneous voltage vs(t)
across the resistor and inductor in series will be:
vs(t) = vR(t) + vL(t)
And again, the amplitude VRLS will be always less than VR + VL as it can be seen from figure
13.
L
VL VL
VRLS R VR
i vs
eri
es
I ϕ VS VR vseri
es =
vR
+
vC
=
vR
Figure 12: Resistor and Inductor in Series
Figure 13: R-L Series Phasor Diagram
+
vseri
vC
es =
but
vR Pythagoras' theorem, from figure 13 we have:
Applying again
Vseri
+
es >
2
2
2
vC
V
b
V=R VR +VL
RLS
ut
+
Vsexpress V
Using Ohm’s Law to
the
C. three voltages in the equation above, we obtain the
but
impedanceVZRLS and
eri the phase difference ϕ between voltage VRLS and the current I:
seri
es
>
VR
+
VC.
es
>
V
2
2
= ( I ⋅ R) + ( I ⋅ X L )
R
+
V
C.
and
The
am
plit
ude
s
and
( I ⋅ Z RCS )
T
h
e
a
Z RCS = R 2 + (ϖ L )
2
V
X
ϖL
tanThe
ϕ= L = L =
R
am VR R
plit
ude
s
and
the
RM
S
volt
age
sV
2
AC Circuits – page 9 of 12 Resistor, Capacitor and Inductor connected in Series
Now we connect a capacitor, and solenoid in series with a resistor, the instantaneous voltage
vs(t) across the resistor, capacitor and inductor in series will be:
VRCLS(t) = vR(t) + vC(t) + vL(t)
But, the amplitude VRCLS will be always less than VR + VC + VL because of the same reason
explained before.
VC
VL VRLS R VS ϕ i I VR vseri
VL
es =
vR
+
vC
Figure 14: Resistor, Capacitor and Inductor in Series
VC
Figure 15: R-C-L Series Phasor Diagram
but
Vseri
seri
Applying again Pythagoras'> theorem, from figure 15 wevhave:
es
es =
VRvseri
2
vR
2
= VR2 + (VL −VC )
+ esV=RCLS
+
VCv.R
vC
+
Using Ohm’s Law to express the four voltages in the equation above, we obtain the
vC
impedance ZRCLS and the phase difference ϕ between voltage VRCLS and the current I:
but
2
2
2
but
⋅seri
XC )
( I ⋅ Z RCLS ) = ( I ⋅ R) + ( I ⋅ X L − IV
es >
Vseri
VR
es >
+
2
VR
"
1 %
2
VC.
'
+ Z RCS = R + $ϖ L −
#
ϖC &
VC.
The
am
1
plit
ϖL−
V
−V
X
−
X
C
ϖC
udetan ϕ = L C = L
and
=
VR
R
R
s
and
the
Since the inductive and capacitive phasors are 180° out of phase, their reactances tend to
RM
cancel each other. This happens
at resonance when XL = XC. At resonance ϕ = 0, the
S
Thethe circuit can reach very large
impedance Z = R has a minimum and the current through
volt
am
The
age
plit
am
AC Circuits – page 10 of 12 values that could be damaging to the circuit.
APPARATUS
•
RLC circuit board with resistors, capacitors and inductor
o Resistors: 100 Ω, 1 W; 33 Ω, 5 W; 10 Ω, 10 W
o Capacitors: 100 µF, 16 V and 330 µF, 16 V (capacitance
values may vary by ±20 %)
o Inductor: 8.2 mH @ 1 kHz, 6.5 Ω maximum DC resistance,
0.8 A current rating RMS, 3/4” I.D. x 1-3/4” O.D.
• Dual channel oscilloscope with sinusoidal voltage signal generator incorporated.
• BNC cables.
• Banana plug patch cords.
PROCEDURE
In this experiment you will be applying a sinusoidal signal to different circuits and will analyze
the effect of the resistors, capacitors and inductors on the current and the relative phase
between the voltage applied and the current.
Familiarize yourself with the apparatus to be used. The function generator and the
oscilloscope are a single unit. There should be a “TEE” connected to the output of the
function generator portion of the unit, with one end of the “TEE” connected directly to CH1 of
the oscilloscope (to give you the input signal) and the other end going to the points of the
circuit where the source is going to be connected. The voltage collected across different
components of the circuit will be going to CH2 of the oscilloscope.
1. Select the mode of the function generator to “sinusoidal” and then select a signal in the
range of 100KHz.
AC Circuits – page 11 of 12 2. Using the “T” splitter to apply this signal to “Chanel 1” of the oscilloscope and to the
“source” in the circuit in figure 10.
3. Adjust the amplitude of the sinusoidal signal, and make sure that the ‘OFFSET’ knob is
pushed in. Setup the time/div so that only one cycle appears on the oscilloscope.
4. Collect the voltage across the resistor and the capacitor and apply it on “Chanel 2” of
the oscilloscope. Compare with what you see on CH1. Are CH1 and CH2 in phase for
both the resistor and the capacitor? Explain.
5. Read the relative phase for each component from the oscilloscope by comparing the
position of the signal on CH1 relative to CH2.
6. Measure the resistance of the resistor R with an ohm-meter and calculate the capacitive
reactance XC and the capacitance of the capacitor C.
Table # 1: Data for Resistor and Capacitor in Series
Component
ϕ
(µs)
ϕ
(rad)
tan
ϕ
R
(Ω)
f
(Khz)
ϖ=
2πf
(S-1)
C
(F)
Capacitive
Reactance
XC
(Ω)
Resistor
Capacitor
7. Repeat steps 2 through 6 for the circuits in figure 12 and determine the relative phase,
the inductive reactance and the induction of the solenoid.
Table # 2: Data for Resistor and Inductor in Series
Component
ϕ
(µs)
Resistor
Inductor
ϕ
(rad)
tan
ϕ
R
(Ω)
Inductive
Reactance
XL
(Ω)
f
(Khz)
ϖ=
2πf
(S-1)
L
(H)
AC Circuits – page 12 of 12 8. Repeat steps 2 through 6 for the circuit in figure 14 and determine the relative phase,
the overall reactance XL - XC and the impedance Z of the circuit.
Table # 3: Data for Resistor, Capacitor and Inductor in Series
Component
ϕ
(µs)
Resistor
Capacitor &
Inductor
ϕ
(rad)
tan
ϕ
R
(Ω)
Overall
Reactance
XL - XC
(Ω)
f
(Khz)
ϖ=
2πf
(S-1)
Z
(Ω)
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