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The R.M.S. Value of an Alternating Current (or Voltage)
Consider the two circuits shown below.  
 
   
In the circuit on the left, there is a d.c. supply.  
This supply delivers a constant strength current, Idc  
In the circuit on the right we have an a.c. supply delivering a current which is constantly changing its magnitude and sense.  
We imagine that we have found two identical light bulbs and that the a.c. supply has been carefully adjusted so as to illuminate the bulb to exactly the same brightness as in the other circuit.  
   
It would then seem reasonable to suggest that the current, iac has the same effective value as the current Idc  
Note that, when referring to alternating currents we usually use lower case letters for the instantaneous values of current (or voltage) and capitals for the maximum values.  
   
   
   
However, if the current in the circuit varies as shown above then the simple mathematical average is equal to zero, not a very suitable figure to give for the effective value of the current!  
We get out of this embarrassing situation as follows:   
1. remember that the power dissipated in the bulb (or in any component possessing resistance) is proportional to the current squared (see here for proof)  
2. remember that the square of a negative number has a positive value  
With these ideas in mind, first draw a graph of the current squared. All the values are positive.  
   
   
   
Now find the average value of the current squared.  
As the variation is symmetrical, this is obviously half of the maximum value, as shown in the next graph.  
   
   
   
Now we take the square root of that average value to give us a meaningful effective value for the current called, rather logically, the root mean square value or r.m.s.  
   
   
   
Therefore, the effective or root mean square value of the current is given by  
 
and, although current and voltage are, of course, totally different quantities, as the discussion above is purely mathematical, we can use the same logic for an alternating voltage, so  
 
where V is the maximum voltage.  
   
Note that these equations apply when the variation is sinusoidal as shown in the graphs above.  
In situation where the variation is as shown below,  
   
   
the r.m.s value is equal to the maximum value as there is virtually zero time when the current has any other magnitude than I.    
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