RESISTORS, CAPACITORS AND INDUCTORS IN AC CIRCUITS

piquantdistractedElectronics - Devices

Oct 5, 2013 (3 years and 10 months ago)

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AC  Circuits  

 
page  
1
 
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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 s
erves 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
)

V
m

sin(
ϖ
t

ϕ
)












i

i
(
t
)

I
m

sin(
ϖ
t
)



Where




v and i
represent the
instantaneous

voltage and current when we are considering
their variation with time explicitly.



V
m

and I
m

are the
amplitude

or
peak value

of the voltage and current



V =

V
m
/2
1/2

and I =

I
m
/2
1/2

without subscripts refer to the RMS values.



f
is the
ordinary frequency

and represents the number of complete oscillations per
second



ω = 2
π
f 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
a
n

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, t
he
instantaneous
volta
ge v across a resistor is proportional to the
instantaneous
current

i flowing through it.











Figure 1: Resistor in an AC circuit

R
 
V
 
i
 
AC  Circuits  

 
page  
2
 
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The next
two
graph
s show

the voltage
across the resistor
and
the
current
flowing through the

resistor as a func
tion of time.









Figure 2: V and I: Sinusoidal variation in phase



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, t
he 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 I
m

and voltage V
m
. 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
r
esistor the voltage and the current

are in phase.

However, circuits with only resistors
,

capacitors, and solenoids

are not very
useful in some
Instantaneous  voltage  
V  

 
(blue
),  
 
and  current  
i
 

   
(
red
)
 
V(t),  
 
i
(t)
 
t
 
I
m
 
V
m
 
AC  Circuits  

 
page  
3
 
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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 depend
s

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
I
mpedance

and i
t
is denoted with the
symbol Z
. Resistance is a spe
c
ial 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 t
he circuit and
the ratio of
the
voltage to
the
current is called
th
e
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

circui
ts

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
capac
itors easily pass AC
sign
als but they block DC

signals.

The voltage on a capacitor
depends o
n 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 fol
lows 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  

 
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Therefore, the voltage across the capacitor:






The last equation shows that

the current and
the
voltage are out of phase

by 90
0
.

The
capacitive reactance

X
C

is equal to:










and it can be defined
as the r
atio of the magnitude of the voltage to magnitude of the curr
ent
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 t
he vertical component of
any

phasor arrow represents the instantaneous
value of its quantity

and the
phasors are rotating counter clockwise
t
he phasor representing
V
C

is

90°
behind

the

current
.






















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 90
0
, 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.

Instantaneous  voltage  
V  

 
(blue
),  and  current  
i
 

   
(
red
)
 
AC  Circuits  

 
page  
5
 
of  
12
 

Inductors In AC Circuits

An inductor is usually a coil of wire.
The
resistance
of an ideal indu
ctor

is negligible, as is its
capacitance.

However, the voltage across an inductor is influenced

by changes of
its own
magnetic field
.
Faraday's la
w of electromagnetic induction states that the
current
i(t)

in the coil
sets up a magne
tic field, whose magne
tic flux
Φ
B

is proportional to the field strength

B
, which
in turn
is proportional to the current
.

Therefore, the
s
elf 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 m
agnetic flux







According to Kirchhoff’s First Law the

emf is

a voltage rise; therefore,
the voltage drop v
L

across the inductor

should be:







v
L
(
t
)


e
L

d
Φ
B
d
t

d
d
t
L

i
(
t
)



L

d
d
t
I
m

s
i
n
(
ϖ
t
)



ϖ
L

I
m

c
o
s
(
ϖ
t
)

V
m

s
i
n
(
ϖ
t

π
2
)


As in the case of the capacitor, we

define the
inductive reactance

X
L

as the ratio

of the
magnitudes of the voltage and
the
current, and from the equation above we see that

X
L

= 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







V
m

= X
L
.I
m





AC  Circuits  

 
page  
6
 
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12
 







Figure 6: V and I: Sinusoidal variation for Solenoid



Figure 7: Solenoid Phaso
r 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. The
refore, 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

Resistor

Capacitor

Inductor

Difference of
Phase between
Voltage and
Current

Voltage and
Current are in
phase

Voltage lags
behind Current by
π
/2

Current
lags
behind Voltage
by
π
/2

Ohm’s Law

R

V
R
I

X
c

V
C
I

1
ϖ
C

X
L

V
L
I

ϖ
L


V(t),
 
 
i
(t)
 
t
 
AC  Circuits  

 
page  
7
 
of  
12
 
R
esistor and
C
apacitor connected in Series

When we connect components together, Kirc
h
hoff's laws apply at any instant. So the voltage
v
s
(t) across
a
resistor and capacitor

in series is just


v
s
(t) = v
R
(t) + v
C
(t)


However
,

the addition is
more
complicated be
cause the two are not in phase:
they add to give
a new sinusoidal voltage, but the amplitude
V
S

is
less

than V
R

+ V
C

as it can be seen from
figur
e 11.








Figure 10
:
Resistor and Capacitor in Series




Figure 11
:
R
-
C Series

Phasor Diagram

However, using

Pythagoras' theorem

in figure 11 we have
:






V
S
2

V
R
2

V
C
2


Using Ohm’s Law and expressing the three v
oltages and substituting in the equation above,
we obtain the impedance Z
RCS

and the phase difference
ϕ

between voltage V
RCS

and the
current I:





I

Z
R
C
S


2

I

R


2

I

X
C


2





Z
R
C
S

R
2

1
ϖ
C
!
"
#
$
%
&
2


and




t
a
n
ϕ


V
C
V
R


X
C
R


1
ϖ
R
C



R
 
V
S
 
i
 
C









v
s
er
ie
s

=
v
R

+
v
C







b
u
t

V
s
er
ie
s

>
V
R

+
V
C
.




V
C









v
seri
es

=
v
R

+
v
C





but

V
seri
es

>
V
R

+
V
C
.










The
am
plit
ude
s
and
the
RM
S
volt
age
V
R









v
ser
i
es

=
v
R

+
v
C





but

V
seri
es

>
V
R

+
V
C
.










The
am
plit
ude
s
I
 
V
R
 
V
C
 
V
RCS
 
ϕ
 
AC  Circuits  

 
page  
8
 
of  
12
 
R
esistor and Inductor co
nnected in Series

Similarly, when

we connect
a solenoid in series with a resistor,

the instantaneous
voltage v
s
(t)
acros
s the

resistor and
inductor

in series
will be:


v
s
(t) = v
R
(t) + v
L
(t)


And again,

the amplitude
V
RLS

will be always less

than V
R

+ V
L

as

it can be seen from figure
13.









Figure 12
:
Resistor and Inductor in Series




Figure 13
:
R
-
L Series

Phasor Diagram


Applying again

Pythagoras' theorem
, from figure 13 we have
:






V
R
L
S
2

V
R
2

V
L
2


Using Ohm’
s Law to express the three voltages in the equation above, we obtain the
impedance Z
RLS

and the phase difference
ϕ

between voltage V
RLS

and the current I:





I

Z
R
C
S


2

I

R


2

I

X
L


2





Z
R
C
S

R
2

ϖ
L


2


and




t
a
n
ϕ

V
L
V
R

X
L
R

ϖ
L
R





R
 
V
S
 
i
 
L









v
s
eri
es

=
v
R

+
v
C






b
ut

V
s
eri
es

>
V
R

+
V
C
.










T
h
e
a
m
pli
V
L









v
seri
es

=
v
R

+
v
C





but

V
seri
es

>
V
R

+
V
C
.










The
am
plit
ude
s
and
the
RM
S
volt
age
s V
do
not
V
R









v
seri
es

=
v
R

+
v
C





but

V
seri
es

>
V
R

+
V
C
.










The
am
plit
ude
s
and
the
RM
I
 
V
R
 
V
L
 
V
RLS
 
ϕ
 
AC  Circuits  

 
page  
9
 
of  
12
 
R
esistor, C
apacitor and Inductor connected in Series

Now

we connect
a capacitor, and solenoid in series with a resistor,

the instantaneous
voltage
v
s
(t) acros
s the

resistor, capacitor
and
inductor

in series
will be:


V
RCLS
(t) = v
R
(t) + v
C
(t)

+
v
L
(t)



But, t
he amplit
ude
V
RCLS

will be always less

than V
R

+ V
C

+
V
L

because of the same reason
explained before.







Figure 14
:
Resistor, Capacitor and Inductor in Series

Figure 15
:
R
-
C
-
L Series

Phasor Diagram


Applying again

Pythagor
as' theorem
, from figure 15 we have
:






V
R
C
L
S
2

V
R
2

V
L

V
C


2


Using Ohm’s Law to express the four voltages in the equation above, we obtain the
impedance Z
RCLS

and the phase difference
ϕ

between voltage V
RCLS

and the current I:





I

Z
R
C
L
S


2

I

R


2

I

X
L

I

X
C


2





Z
R
C
S

R
2

ϖ
L

1
ϖ
C
"
#
$
%
&
'
2


and




t
a
n
ϕ

V
L

V
C
V
R

X
L

X
C
R

ϖ
L

1
ϖ
C
R

Since

the inductive and capacitive phasors are 180° out of phase, their reactances tend to
cancel

each other
.

This happens at
resonance

when X
L

= X
C
. At resonance
ϕ

= 0, the
impedance Z = R has
a minimum and the current through the circuit can reach very large
I
 
V
R
 
V
L
 
V
RLS
 
ϕ
 
R
 
V
S
 
i
 
V
C









v
ser
i
es

=
v
R

+
v
C





but

V
seri
es

>
V
R

+
V
C
.










The
am
plit
ude
s
and
the
RM
S
volt
age
s V
do
V
L









v
ser
i
es

=
v
R

+
v
C





but

V
seri
es

>
V
R

+
V
C
.










The
am
plit
V
C









v
seri
es

=
v
R

+
v
C





but

V
seri
es

>
V
R

+
V
C
.










The
am
plit
ude
AC  Circuits  

 
page  
1
0
 
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 sinuso
idal 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 cu
rrent 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 g
ive you the input signal) and t
he other end going
to the points of the
circuit where the source is going to be connected. The voltage collected acro
ss 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 o
scilloscope 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 acr
oss 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 oscilloscop
e 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 X
C

and the capacitance of the capacitor C.


Table # 1: Data for Resistor and Capacitor in
Series

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


(

s)




ϕ


(rad)




tan
ϕ





R


(
Ω

)




f



(Khz)


ϖ

=
2
π
f

(
S
-
1
)




C


(F)


Capacitive
Reactance

X
C


(
Ω

)

Resistor









Capacitor










Component




ϕ


(

s)




ϕ


(rad)




tan
ϕ





R


(
Ω

)


Inductive

Reactance

X
L

(
Ω

)




f



(Khz)


ϖ

=
2
π
f

(
S
-
1
)




L


(H)

Resistor









Inductor









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
X
L

-

X
C

and the impedance Z of the circuit.



Table # 3: Data for Resist
or, Capacitor and Inductor in Series




Component




ϕ


(

s)




ϕ


(rad)




tan
ϕ





R


(
Ω

)




Overall

Reactance


X
L

-

X
C


(
Ω

)




f



(Khz)


ϖ

=
2
π
f

(
S
-
1
)




Z


(
Ω

)



Resistor









Capacitor &
Inductor