# Electronic – complex numbers in linear circuits

circuit analysis

I’m a little confused on how complex numbers are used in circuit analysis . On one hand, complex numbers can only be used in linear circuits which do not involve any squares, square roots, and so forth of the voltage or current, or multiplication of one voltage or current by another. This way the real and imaginary parts don’t get mixed up. On the other hand, we do divide complex numbers by each other (e.g. when calculating complex impedance we divide voltage (in complex form) by current (also in complex form)) and the numbers still don’t get messed up. Why is it so?
Any help will be much appreciated.

On the other hand, we do divide complex numbers by each other

That's true. The impedance of a circuit element is the phasor voltage divided the phasor current

$$Z = \frac{\vec V}{\vec I}$$

But note that the impedance is not a phasor - it does not represent the amplitude and phase of a sinusoid like the voltage and current phasors.

Similarly, the complex power is the product of the (rms) phasor voltage and (rms) conjugate phasor current

$$S = \tilde V \cdot \tilde I^*$$

And again, the complex power is not a phasor, it is just a complex number.

That fact is that products and ratios of phasors are not phasors. Thus, we can't apply phasor analysis to non-linear circuits.

For example, let a circuit element voltage be proportional to the current squared:

$$v = ki^2$$

If the current is a sinusoid of frequency \$\omega\$, the voltage is a constant plus a sinusoid of frequency \$2\omega\$.

$$v = k(I\cos\omega t)^2 = \frac{kI^2}{2}(1 + \cos2\omega t)$$

But, for phasor analysis, we depend on the fact that all the circuit voltages and currents are of the same form, i.e., are sinusoids of the same frequency, only differing in amplitude and phase.

Moreover, as pointed earlier, the square of a phasor is not a phasor thus we cannot square the current phasor and hope to get a voltage phasor.