Complex number in alternating current circuit

[IPhO' 2008]

How to use complex number in the alternating current circuit?

 

[Born2bwire]

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.

 

[mabs239]

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

pi=3.1415...
f= AC frequency
i=sqrt(-1)
C=Capacitance in Ferrads

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

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

 

[sophiecentaur]

In dc circuits we have resistance 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 90^O 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.

 

[Bob S]

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

 

[mabs239]

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.

 

[sophiecentaur]

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.