# Complex number in alternating current circuit

• IPhO' 2008
In summary, complex number can be used in AC circuits to replace time-derivative resistances. This makes things easier for circuit analysis because now the time derivatives are multiplicative factors of the frequency.
IPhO' 2008
How to use complex number in the alternating current circuit?

If we assume that we have time-harmonic currents and voltages, that is we can decompose our signals such that
$$\mathbf{I}(\mathbf{r},t}) = \sum \mbox{Re}\left[ \mathbf{I}_ne^{-i\omega_n t} \right]$$
Then we can work in the complex domain for ease of analysis. In this regard, the complex numbers retain information about the magnitude and phase of the signal. It also eases calculations because now the time derivatives are simply multiplicative factors of \omega, that is,
$$\frac{\partial}{\partial t} V = -i \omega V$$
where we are working with a single frequency. Since circuit components like capacitors and inductors work on the premise that the voltage and currents are related by time-derivatives, this means that we can subscribe a simple complex impedance to these circuit elements that fully entails their behavior. For example, a capacitor relates
$$I(t) = C\frac{d V(t)}{dt}$$
In the time-harmonic case,
$$I = -i\omega C V$$
So it is as if the capacitor is a resistor with a resistance of
$$Z_c = \frac{i}{\omega C}$$
But since it is complex number we call it an impedance. Specifically, real impedances are resistances, imaginary impedances are reactances.

Thus, we can now replace components that have a time-derivative dependence with effective impedances and use simple circuit analysis to analyze what would otherwise be difficult circuits.

In dc circuits we have resistence and it is always a real number.
In AC circuits we have three kind of "resistences".
1. A simple resistence. Expressed in ohms
2. A capacitor "resistence". It is imaginary resistence. A capacitor has capacitance that is changed to complex resistence using the formula 1/(2*pi*f*c) *(-i)

pi=3.1415...
f= AC frequency
i=sqrt(-1)

2. An inductor "resistence". It is imaginary resistence. A capacitor has capacitance that is changed to complex resistence using the formula (2*pi*f*L) *(i)

pi=3.1415...
f= AC frequency
i=sqrt(-1)
L= inductance in Henryies

mabs239 said:
In dc circuits we have resistence and it is always a real number.
In AC circuits we have three kind of "resistences".

We refer to it as Impedance because it describes how much a component impedes the current.

A simple way of saying what Born is saying is that inductors and capacitors produce phase shifts with AC signals (the voltage and current are 90O out of phase). This can be represented by using complex numbers and gives answers very conveniently.
You just have to grit your teeth and get into the basics of complex arithmetic and algebra, I'm afraid.

For a series RLC (resistor inductor capacitor) circuit, the complex impedance is

Z(w) = R + jwL - j/wC = (j/wC)(w2LC-1)

Not that the last two terms have opposite sign, and when these two terms are equal, the impedance is real, and you will have an LC resonance (LC=1/w2).
Bob S

Sophiecentaur ,

I tried to answer the "How" part with as much simplicity. I can not see the equations in the Born's post. Don't know why?
I were using "i" but it is "j" (as Bob has used it in his post) that is used by electrical engineers because "i" is reserved for the current.

Yes, I did all my Uni work with i then walked into electrical engineering and it became j. How confusing.
I think Born's post has embedded stuff that, maybe, your browser can't handle.

## What are complex numbers in alternating current circuits?

Complex numbers are mathematical quantities that have both a real and an imaginary component. In alternating current (AC) circuits, complex numbers are used to represent the amplitude and phase of the voltage and current.

## Why are complex numbers used in AC circuits?

Complex numbers are used in AC circuits because they allow for a more efficient and accurate way to analyze and solve complex circuit problems. They also allow for the representation of both magnitude and direction in a single quantity.

## How are complex numbers represented in AC circuits?

In AC circuits, complex numbers are typically represented using the notation of a real part (R) and an imaginary part (jX), where j represents the square root of -1. The combination of R and jX is known as the complex impedance (Z).

## What is the relationship between complex numbers and phasors in AC circuits?

Phasors are graphical representations of complex numbers in AC circuits. They show the amplitude and phase of a sinusoidal signal in a circuit. The magnitude of a phasor is equivalent to the absolute value of the complex number, while the angle represents the phase difference between the real and imaginary components.

## How are complex numbers used to solve AC circuit problems?

In AC circuit analysis, complex numbers are used to simplify and solve equations involving voltage, current, and impedance. By converting the circuit elements (such as resistors, capacitors, and inductors) into their corresponding complex impedance values, complex numbers can be used to find the total voltage and current in a circuit.

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