Why complex numbers are used in electronics? [duplicate]

Question

The impedance of a capacitor or an inductor is imaginary. How do we know these quantities are imaginary?

Answer 1

Using imaginary numbers for current in reactive components just happens to make the maths a lot simpler. In AC circuits there is typically some phase difference between the voltage and the current. Manipulating these quantities without the use of complex numbers, but instead just keeping track of the phase differences (such as the power factor), is a right pain.

Answer 2

An RLC circuit satisfies$$L\ddot{I}+R\dot{I}+C^{-1}I=\dot{V}.$$To solve this with AC voltage such as $V=V_0\cos\omega t,\,V_0\in\Bbb R$, it's convenient to take the real part of a complex choice of $I$ for the case $V=V_0\exp j\omega t$. Substituting $I=I_0\exp j\omega t,\,I_0\in\Bbb C$ gives$$I_0=\frac{j\omega V_0}{C^{-1}-\omega^2L+j\omega R}.$$The special case $C^{-1}=L=0$ gives $I_0=\frac{V_0}{R}$. The general case gives capacitance (inductance) an effective resistance of $\frac{C^{-1}}{j\omega}$ ($j\omega L$), so that $I_0=\frac{V_0}{Z}$ with a complex impedance $Z=R+j(\omega L-\frac{1}{\omega C})$. If $\omega^2LC\ne1$, the phase of $Z$ causes the oscillating expressions for $V,\,I$ to have a phase difference, i.e. in the real case constants $A,\,\phi$ exist with $I=A\cos(\omega t-\phi)$. Since $\omega^2LC\ne1\implies\phi\ne0$, any definition of resistance we give for an RLC circuit has to use complex numbers to represent the LC parts' phasing effects. This is why in general impedance is complex.

Answer 3

A capacitor (with capacitance $C$) is fully described by the differential equation between current $I(t)$ and voltage $V(t)$: $$I(t)=C\frac{dV(t)}{dt} \tag{1}$$

Suppose you have an AC voltage with frequency $\omega$ connected to the capacitor. By using the complex calculus this is $$V(t)=V_0 e^{j\omega t} \tag{2}$$

Then, by plugging voltage (2) into differential equation (1), you get the current through the capacitor $$I(t)=C V_0 j\omega e^{j\omega t} \tag{3}$$

The impedance is defined to be $$Z=\frac{V(t)}{I(t)}.$$ From (2) and (3) you get the impedance of the capacitor $$Z=\frac{1}{j\omega C}.$$

From the $j$ you see, this is a purely imaginary value.

---

The impedance of a inductor (with inductance $L$) can be derived in a very similar way, except that here you begin with the differential equation $$V(t)=L\frac{dI(t)}{dt}.$$

From that you finally get the impedance of the inductor as $$Z=j\omega L.$$

Answer 4

Using complex numbers means you are trying to describe a value in a different domain and in complex number systems, the Imaginary number doesn't mean that the value of capacitor is imaginary. The imaginary number helps to signify the vector rotation when voltage is applied across it or when current flows through it. I would suggest you watch the series on complex numbers by Welch labs on youtube. This might help you to understand the number system better!

Answer 5

To study the frequency response of your system you use complex number.

Differential equation for RLC circuit

$$L\ddot{I}+R\dot{I}+C^{-1}I=\dot{V}\tag 1$$

Transfer equation (1) to Laplace domain. with $\frac{d}{dt}\mapsto S$ and $I(t)\mapsto I(S)$

$$L\,S^2\,I(S)+R\,S\,I(S)+C^{-1}I(S)=S\,V_0\tag 2$$

solve equation (2) for $I(S)$

$$I(S)=\underbrace{\frac{S}{L\,S^2+R\,S+C^{-1}}}_{G(S)}\,V_0$$

where $G(S)$ is the transfer function between the output signal $I(S)$ and the input signal $V_0$

with $S\mapsto i\,\omega$ we transfer $G(S)$ to frequency domain

$$G(i\,\omega)=\frac{i\,\omega}{-L\,\omega^2+i\,R\,\omega+C^{-1}}\tag 3$$

we can now obtain the amplitude

$\text{AMP}=|G(i\,\omega)|$

and the phase function

$\text{PH}=\arctan(\frac{\Im(G)}{\Re(G)})$

which are important to design a controller for example