Electrical circuits lecture notes b. Tech

So,
V
avg
= V
pk
Root-Mean-Square Voltage (V
rms
)
As the name implies, V
rms
is calculated by taking the square root of the mean 
average of the square of the voltage in an appropriately chosen interval. In the case of 
symmetrical waveforms like the square wave, a quarter cycle faithfully represents all four 
quarter cycles of the waveform. Therefore, it is acceptable to choose the first quarter cycle, 
which goes from 0 radians (0°) through π/2 radians (90°). 
V
rms
is the value indicated by the vast majority of AC voltmeters. It is the 
value that, when applied across a resistance, produces that same amount of heat that a direct 
current (DC) voltage of the same magnitude would produce. For example, 1 V applied across 
a 1 Ω resistor produces 1 W of heat. A 1 V
rms
square wave applied across a 1 Ω resistor also 
produces 1 W of heat. That 1 V
rms
square wave has a peak voltage of 1 V, and a peak-to-peak 
voltage of 2 V. 
Since finding a full derivation of the formulas for root-mean-square (V
rms
) 
voltage is difficult, it is done here for you. 

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So,
V
rms
= V
pk
Form factor: 
Form factor of a triangular signal = V
rms
/V
av 
=Vpk/Vpk 
=1 
Peak Factor (Or Crest factor):
 
Peak factor of a triangular signal=V
pk
/V
rms 
=Vpk/Vpk 
J notation: 
 
The mathematics used in Electrical Engineering to add together resistances, currents or DC 
voltages use what are called “real numbers” either as integers or as fractions
.
But real numbers 
are not the only kind of numbers we need to use especially when dealing with frequency 
dependent sinusoidal sources and vectors. As well as using normal or real numbers, Complex 
Numbers were introduced to allow complex equations to be solved with numbers that are the 
square roots of negative numbers, √-1. 
In electrical engineering this type of number is called an “imaginary number” and to 
distinguish an imaginary number from a real number the letter “ j ” known commonly in 
electrical engineering as the j-operator, is used. The letter j is placed in front of a real number 
to signify its imaginary number operation.
Examples of imaginary numbers are: j3, j12, j100 etc. Then a complex number consists of 
two distinct but very much related parts, a “ Real Number ” plus an “ Imaginary Number 
”.Complex Numbers represent points in a two dimensional complex or s-plane that are 
referenced to two distinct axes. The horizontal axis is called the “real axis” while the vertical 
axis is called the “imaginary axis”. The real and imaginary parts of a complex number are 
abbreviated as Re(z) and Im(z), respectively. 
Complex numbers that are made up of real (the active component) and imaginary (the 
reactive component) numbers can be added, subtracted and used in exactly the same way as 
elementary algebra is used to analyse dc circuitsThe rules and laws used in mathematics for 
the addition or subtraction of imaginary numbers are the same as for real numbers, 
j2 + j4 = j6 etc. The only difference is in multiplication because two imaginary numbers 
multiplied together becomes a negative real number. Real numbers can also be thought of as 
a complex number but with a zero imaginary part labelled j0. 
The j-operator has a value exactly equal to √-1, so successive multiplication of “ j “, ( j x j ) 
will result in j having the following values of, -1, -j and +1. As the j-operator is commonly 
used to indicate the anticlockwise rotation of a vector, each successive multiplication or 
power of “ j “, j
2
, j
3
etc, will force the vector to rotate through an angle of 90
o
anticlockwise 

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as shown below. Likewise, if the multiplication of the vector results in a -j operator then the 
phase shift will be -90
o
, i.e. a clockwise rotation.