# Voltage reflected wave in electric circuits

Why when analyzing lower frequency circuits does not appear a reflected voltage wave, and yet in radio frequency circuits it does? From what i know, the module of the reflection coefficient does not depend on the frequency, so, why do we only take into account this phenomenon of reflection in radiofrequency circuits?

The reason for not using nor needing the concept of "reflected wave" is not that they do not exist but rather the circuit itself is so much smaller than a wavelength that the sum of the reflected and forward waves is essentially constant between and within the circuit elements, hence their sum voltage is enough to describe what is happening in the circuit.

If the wavelength is short enough that you can see spatial variation along the connecting wires between the elements then you must take their lengths into account, and the easiest way of doing it is by analyzing the forward and reflected waves separately. Note that over a transmission line the amplitudes of the forward and reflected waves, resp., are constant individually, only their sum when properly phased along the length is position dependent. That position dependence of the sum is neglected in low frequency circuit work, so you might as well just deal with the sum voltage alone.

There is a reflected wave at all frequencies.

The circuit picture with scattering parameters (and forward and backward wave) is fully compatible to simple ohms law. In fact both are equivalent at low frequencies.

Let $$a$$ and $$b$$ be forward and backward propagating waves. Then current and voltage are given by

$$U = \sqrt{Z_0} (a + b)$$

$$I = 1/\sqrt{Z_0}(a-b)$$

also we have for the reflection factor

$$r = \frac{b}{a} = \frac{Z - Z_0}{Z + Z_0}$$

These relations hold at all frequencies. At DC frequency the phase of the $$a$$ and $$b$$ parameters can be assumed to be constant and be neglected in the analysis (except for sign). That is all the difference. Also at DC the value of the wave impedance $$Z_0$$ does not matter. We always obtain the same result.

So $$a$$ and $$b$$ are really another way to describe voltage and current. The advantage is that at high frequency the propagation along a wire is easier predictable for $$a$$ and $$b$$ paramaters than it is for current and voltage. There is only an exponential factor for the former and more complicated for the latter. However there is no fundamental difference, both descriptions are capturing the same physics.