Electricity and electrical components are a major part of our lives today. Our lives pretty much come to a halt if there is a power outage or if our electrical devices stop working, sometimes we even start panicking because we are so dependent on these components that we cannot afford to lose them and their importance and use only increases as time passes. We all use these electrical devices and also electricity itself but most of us do not think about the math and physics that works behind all of these things.
In this paper, I will talk about how AC circuits can be described by considering voltage and current using complex numbers. An AC circuit requires two separate numbers to be able to completely describe it. This is because it takes into account the amplitude and the phase of the current. The fact that complex numbers can be easily added, subtracted, multiplied or divided with each other makes them ideal for this operation where both amplitude and phase have to work together.
For a DC circuit, Ohm’s Law is used. According to the Ohm’s law, v=iR Where, v= voltage i= current flowing through the circuit
R= resistance offered by the resistors attached in the circuit.
Whereas, there is an AC analogy for the Ohm’s that is used. Here, the equation for the Ohm’s law changes to: v=iZ Where, v= voltage i= current flowing through the circuit
Z= impedance
Here, all the three quantities can be used as complex numbers. Z defines a quantity called impedance. Impedance acts in a similar way as resistance in a DC circuit. Also, if the frequency of the AC circuit is allowed to go to 0, impedance will go down to 0 as the imaginary part of it will disappear, hence, making it equal to the resistance in a DC circuit.
In the paper, the followi...
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...he original vector.
P*=A-jB
The modulus of a complex number is the square root of the sum of squares of each real and imaginary part of the number. It is denoted by |P| if P is the complex number.
|P|=√(A^2+B^2 )
A complex number may also be expressed in a polar form using the Euler’s Formula:
P=A+jB=exp(jθ)=cosθ+jsinθ
Also,
P=|P|exp(jθ)
And tanθ=B/A Relation of the AC Circuit and Complex numbers
Complex numbers provide a structure to describe sinusoidal functions of time, especially the current and voltage functions that are observed in AC circuits.
The exponential for of the complex numbers is the most useful because the relationship between voltage and current is about multiples. In sinusoidal voltages, different types of impedances/resistances exist. Here, I am particularly talking about capacitors and inductors, excluding the non-linear devices.
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where I is the current through the conductor in units of amperes, V is the potential difference