RELATION BETWEEN R.M.S. VALUE AND PEAK VALUE OF A.C
Let the alternating current be represented by
I 
If this alternating current flows through a resistance R for a small time
, then small amount of heat produced is given.
In one complete cycle 
for time 0 to T of alternating current the total amount of heat produced in R is given by
If Irms is the virtual or 
value of the alternating current, then heat produced in R in the same time 
(0 to T) is given by
From equation (i) and (ii), we have
Hence the 
value of effective value or virtual value of alternating current is 0.707 times the peak value of alternating current.
The 
value is the same whether calculated for half cycle or full cycle.
Alternative Method
Let the alternating current be represented by
I 
If this current is passed through a resistance R, then power delivered at any instant is given by
Because the current is squared, power is always positive since the value of 
varies
between 0 and 1
Average power delivered
If 
is the virtual or 
value of the alternating current then by definition
Power delivery P
From equation (I) and (ii)
ROOT MEAN SQUARE VALUE OF ALTERNATING E.M.F
The root mean square (r.m.s) value of alternating voltage is that steady voltage (d.c voltage when applied to a given resistance for a given time produces the same amount of heat as is produced by the alternative 
when applied to the same resistance for the same time.
It is also called virtual value of alternating e.m.f and is donated by 
of 
or 
The instantaneous value of alternating 
is given by
E
When this alternating voltage is applied to a resistance R, small amount of heat produced in a small time 
is
But
Then,
In one complete cycle 
for time 0 to T the total amount of heat produced in resistance R is
If Erms is the 
value of the alternating
, then heat produced in the same resistance R for the same time T is
From the equation (i) and equation (ii)
Hence the 
value of alternating 
is 0.707 times the peak value of the alternating 
Therefore, 
value is the same whether calculated for half cycle or full cycle.
Importance of r.m.s values
An alternating voltage or current always specified in terms of 
values.
Thus an alternating current of 10A is the one which has the same heat effect as 10A d.c under similar conditions. The following points may be noted carefully.
i) The domestic a.c supply is 230V, 50H. It is the 
or effectively value. It means that the alternative voltage available has the same effect as 230 V d.c under similar conditions.
The equation of this alternative voltage is
ii) When we say that alternating current in a circuit is 5A, we specifying the 
value.
– It means that the alternating current flowing in the circuit has the same heating effect as 5A d.c under similar conditions.
iii) A.C ammeters and voltmeters record 
values of current and voltage respectively.
The alternating voltage/current can be measured by utilizing the heating effect of electric current.
Such Instruments are called hot wire instruments and measure the 
value of the voltage/current since r.m.s value is the same for half cycle or complete cycle.
WORKED EXAMPLES
1. An a.c main supply is given to be 220V what is the average 
during a positive half cycle?
Solution
Average 
during positive half is given by
2. An alternating current I is given by
Find
(i) The maximum value
(ii) Frequency
(iii) Time period
(iv) The instantaneous value when t= 3ms
Solution
Comparing the given equation of the alternating current with the standard form
(i) Maximum value, 
(ii) Frequency f
f = 50 HZ
(iii) Time period T
(iv) 
When 
3. Calculate the 
value of the current shown in figure below
Solution
4. An a.c voltmeter records 50V when connected across the terminals of sinusoidal power source with frequency 50Hz. Write down the equation for the instantaneous voltage provided by the source.
Solution
An a.c voltmeter records 
values
Now
Here
5. An alternating voltage of 50 Hz has maximum value of 200
volts. At what time measured from a positive maximum value will the instantaneous voltage be 141.4 volts.
Solution
This equation is valid when time is measured from the instant the voltage is zeroi.e point O. Since the time is measured from the positive maximum value at point A.
The above equation is modified to
Let the value of voltage become 141.4 volts t second after passing then the maximum positive value.
6. An alternating current of frequency 60 Hz has a maximum value of 120 A.
i) Write down the equation for the instantaneous value.
ii) Recording time from the instant the current is zero and becoming positive. Find the instantaneous value after 1/360 second.
iii) Time taken to reach 96 A for the first time.
Solution
i) The instantaneous value of alternating current is given by
ii) Since point 0 has been taken as the reference the current equation is
When
iii) To reach the current 96A for the first time
7. Find the r.m.s value of the current after current I shown in figure below
Solution
Consider the current variation over one complete cycle.
