Download Voltage-Current Relationship for a Resistor:

Unit 13
Voltage-Current Relationship for a Resistor:
When an AC voltage v(t) is applied to a purely resistive circuit, Ohm's law and
Kirchhoff's laws apply. Since
v(t )  Vm sint   
Then by Ohm's law:
vt  Vm

sint    A
R
R
 I m sint    A
it  
p(t)
p
m
V
v(t)
m
Im
0
t
i(t)
T
It can be seen from the Fig above that when v(t) is zero then i(t) is zero. The maximum
positive and negative values also coincide, i.e. both waveforms follow each other. It can
be said that the voltage and current waveforms for a resistive element are in phase.
Since the power is
P  v i W
power is also a function of time, as indicated above.
Average value of AC Waveforms:
i(t)
Current
0
i1 i2 i3 i4

t
2
1
Unit 13
IAV is the mean value taken over one half cycle. Thus
I AV 
i1  i2  i3 ... in
n
Or
I AV 
area enclosed by half cycle
length of base of half cycle

I AV 
 I AV 
I max


0
i.d


 sin  . d 

0

I max


I max
0
 I AV 
1

I max sin  .d
 cos 0 
1 1 

 I max

cos 0
2 I max

or
 I AV  0. 637 I max
Effective (RMS) value of AC Waveforms:
+
i(t)
current
0
i1 i2 i3
in

2
t
Fig 1
Heating Effect
pm
2
i R
i12 R
0
i22 R
t
Fig 2
2
Unit 13
Consider that the current shown in Fig 1 is passed through a resistor R. The heating
effect of the current at the instant i1 is i12R, that of i2 is i22R, and so on. As shown in Fig
2, the variation of the heating effect during the second half cycle is exactly the same as
during the first half cycle. Thus we can write:
area enclosed by i 2 R

length of base
Average heating effect



0

i 2 R. d


I 2 max R



0
1


I 2 max R sin 2  . d 
0
I 2 max R



0
sin 2  . d
1  cos 2
I 2 max R 
. d 
1  cos 2 . d
2
2 0
I 2 max R 
1


  sin 2 

  2



0
I 2 max R
  0  0  0
2
I 2 max R

2
Now if I is the value of dc current through the same resistor R required to produce the
same heating effect, then
I 2 max R
I 2 max
I
2
I R
I 
 I  max
2
2
2
2
That is
I rms 
I max
or I rms  0.707 I max
2
Similarly it can be shown that for voltages
VRMS 
VMAX
2
or VRMS  0.707 VMAX
Form Factor: (by definition)
Form Factor =
rms value
average value
This for a sinusoidal wave the form factor is
0.707
 1.11
0.637
Form Factor =
3