RMS value of Alternating
Current and current
05.06.09
By R. S. Saini
(M.Sc. Physics, M.Ed.)
Kendriya Vidyalaya, Sector 47, Chandigarh
9417071540
Punjab EDUSAT Society (PES)
RMS value of Alternating
current
It is a comparison between heat produced by
Alternating Current and Direct Current
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RMS value of Alternating
Voltage
Root mean square or R.M.S. value of
Alternating voltage is defined as that value of
steady Potential Difference, which would
generate the same amount of heat in a given
resistance is given time, as is done by A.C.
voltage , when maintained across the same
resistance for the same time.
The R.M.S. value is also called effective value or
virtual value of alternating voltage. It is
represented by Vrms
, Veff orSociety
Vv(PES)
.
Punjab EDUSAT
RMS value of Alternating
current
Root mean square or R.M.S. value of
Alternating current is defined as that value of
steady current, which would generate the same
amount of heat in a given resistance is given
time, as is done by A.C. current , when
maintained across the same resistance for the
same time.
The R.M.S. value is also called effective value or
virtual value of alternating Current. It is
represented by Irms
, Ieff orSociety
Iv. (PES)
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Expression for RMS value of
Alternating current
Let I = I oSin ωt
Let this current on flowing through R for a
small time 'dt' produces a heat dH = I 2 Rdt.
Hence in one complete cycle (zero → T )
the heat produced by A.C. is
H =
∫
T
0
T
T
dH = ∫ I Rdt = ∫ I 02 R Sin 2ωt. dt
0
T
2
= I Rò
0
2
0
é1- Cos 2ωt ù
ê
ú.dt [ As 2Sin 2 θ = 1- Cos θ]
êë
ú
2
û Society (PES)
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Expression for RMS value of
Alternating current
2
0
2
0
T
I Ré
ù I R é Sin 2ωt ù
êt ú
which is =
êò (1- Cos 2ωt).dt ú =
ú
2 ëê 0
2 êë
2ω ú
û
û0
éæ
ù
ö
2. 2π ÷
ç
ê
Sin
.T ÷
÷ æ Sin0 öú I 2 R
I 0 R 2 êçç
ú
0
T
÷
÷
ç
ç
=
T
0
=
[T].....(1)
÷
÷
êç
ú
ç
÷
÷
ç
2 êçç
2ω
2ω øú
2
÷ è
÷
÷
êçè
ú
ø
ë
û
[ Sin4π = Sin0 = 0]
T
If Iv is the virtual value of A.C., then heat produced by this steady
current in time T is H=I 2V R.T
…….(2)
From (2) and (3) we get
2
2
I
R
I
I
I 2v R.T = 0 . T or I 2V = 0 or I v = 0 or I v = 0.707 I 0
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Society (PES)
2
2
2
Expression for RMS value of
Alternating Voltage
On the similar lines one can calculate the
RMS value of Alternating Voltage
Small heat produced by Alternating Voltage
in small time dt is = (E0Sinωt)2 dt
R
on integrating it over 0 to T and equating
with heat produced by DC in the same
time i.e. Ev2T/R we get
Eo
Ev =
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2