There are several approaches to understanding reflections, but the relationship of reflections to the
conservation laws is particularly enlightening. A simple example is a step voltage, V\,u(t) (where V is the height of the step and u(t) is the
unit step function with time t), applied to one end of a lossless line, and consider what happens when the line is terminated in various ways. The step will be propagated down the line according to the
telegrapher's equation at some velocity \kappa and the incident voltage, v_\mathrm i, at some point x on the line is given by :v_\mathrm i = V\,u(\kappa\,t - x)\,\! The incident current, i_\mathrm i, can be found by dividing by the characteristic impedance, Z_0 :i_\mathrm i = \frac{v_\mathrm i}{Z_0} = I\,u(\kappa\,t-x)
Open circuit line The incident wave travelling down the line is not affected in any way by the open circuit at the end of the line. It cannot have any effect until the step actually reaches that point. The signal cannot have any foreknowledge of what is at the end of the line and is only affected by the local characteristics of the line. However, if the line is of length \ell the step will arrive at the open circuit at time t = \ell/\kappa, at which point the current in the line is zero (by the definition of an open circuit). Since charge continues to arrive at the end of the line through the incident current, but no current is leaving the line, then conservation of electric charge requires that there must be an equal and opposite current into the end of the line. Essentially, this is
Kirchhoff's current law in operation. This equal and opposite current is the reflected current, i_\mathrm r, and since :i_\mathrm r = \frac{v_\mathrm r}{Z_0} there must also be a reflected voltage, v_\mathrm r, to drive the reflected current down the line. This reflected voltage must exist by reason of conservation of energy. The source is supplying energy to the line at a rate of v_\mathrm i i_\mathrm i. None of this energy is dissipated in the line or its termination and it must go somewhere. The only available direction is back up the line. Since the reflected current is equal in magnitude to the incident current, it must also be so that :v_\mathrm r = v_\mathrm i \,\! These two voltages will
add to each other so that after the step has been reflected, twice the incident voltage appears across the output terminals of the line. As the reflection proceeds back up the line the reflected voltage continues to add to the incident voltage and the reflected current continues to subtract from the incident current. After a further interval of t = \ell/\kappa the reflected step arrives at the generator end and the condition of double voltage and zero current will pertain there also as well as all along the length of the line. If the generator is matched to the line with an impedance of Z_0 the step transient will be absorbed in the generator internal impedance and there will be no further reflections. This counter-intuitive doubling of voltage may become clearer if the circuit voltages are considered when the line is so short that it can be ignored for the purposes of analysis. The
equivalent circuit of a generator matched to a load Z_0 to which it is delivering a voltage V can be represented as in figure 2. That is, the generator can be represented as an ideal voltage generator of twice the voltage it is to deliver and an internal impedance of Z_0. :V_\mathrm{o} = 2\,V_\mathrm{i} \frac {Z_\mathrm{L}}{Z_\mathrm{0} + Z_\mathrm{L}} The reflection, V_\mathrm{r} must be the exact amount required to make V_\mathrm{i} + V_\mathrm{r} = V_\mathrm{o}, :V_\mathrm{r} = V_\mathrm{o} - V_\mathrm{i} = 2\,V_\mathrm{i} \frac {Z_\mathrm{L}}{Z_\mathrm{0} + Z_\mathrm{L}} - V_\mathrm{i} = V_\mathrm{i} \frac {Z_\mathrm{L} - Z_\mathrm{0}}{Z_\mathrm{L} + Z_\mathrm{0}} The reflection coefficient, \mathit{\Gamma}, is defined as :\mathit{\Gamma} = \frac{\,V_\mathrm{r}\,}{V_\mathrm{i}} and substituting in the expression for V_\mathrm{r}, :\mathit{\Gamma} = \frac{V_\mathrm{r}}{V_\mathrm{i}} = \frac{I_\mathrm{r}}{I_\mathrm{i}} = \frac {Z_\mathrm{L} - Z_\mathrm{0}}{Z_\mathrm{L} + Z_\mathrm{0}} In general \mathit{\Gamma} is a complex function but the above expression shows that the magnitude is limited to :\left|\mathit{\Gamma}\,\right| \le 1 when \operatorname{Re}(Z_\mathrm{L}), \operatorname{Re}(Z_0) > 0 The physical interpretation of this is that the reflection cannot be greater than the incident wave when only passive elements are involved (but see
negative resistance amplifier for an example where this condition does not hold). For the special cases described above, When both Z_0 and Z_\mathrm{L} are purely resistive then \mathit{\Gamma} must be purely real. In the general case when \mathit{\Gamma} is complex, this is to be interpreted as a shift in
phase of the reflected wave relative to the incident wave.
Reactive termination Another special case occurs when Z_0 is purely real ( R_0) and Z_\mathrm L is purely imaginary ( j\,X_\mathrm L), that is, it is a
reactance. In this case, :\mathit \Gamma = \frac {j\,X_\mathrm L - R_\mathrm 0}{j\,X_\mathrm L + R_\mathrm 0} Since :|j X_\mathrm L - R_\mathrm 0| = |j X_\mathrm L+R_\mathrm 0|\, then :|\mathit \Gamma| = 1\, showing that all the incident wave is reflected, and none of it is absorbed in the termination, as is to be expected from a pure
reactance. There is, however, a change of phase, \theta, in the reflection given by :\theta = \begin{cases} \pi - 2\,\arctan\frac{X_\mathrm L}{R_\mathrm 0} & \mbox{if } {X_\mathrm L} > 0 \\ -\pi - 2\,\arctan\frac{X_\mathrm L}{R_\mathrm 0} & \mbox{if } {X_\mathrm L}
Discontinuity along line A discontinuity, or mismatch, somewhere along the length of the line results in part of the incident wave being reflected and part being transmitted onward in the second section of line as shown in figure 5. The reflection coefficient in this case is given by :\mathit \Gamma = \frac {\,Z_{02} - Z_{01}\,\,}{Z_{02} + Z_{01}} In a similar manner, a
transmission coefficient, T, can be defined to describe the portion of the wave, V_\mathrm t, that it is transmitted in the forward direction: :T = \frac {V_\mathrm t}{V_\mathrm i} = \frac {2\,Z_{02}}{\,Z_{02}+Z_{01}\,\,} Another kind of discontinuity is caused when both sections of line have an identical characteristic impedance but there is a lumped element, Z_\mathrm L, at the discontinuity. For the example shown (figure 6) of a shunt lumped element, :\mathit \Gamma = \frac {-Z_0}{\,Z_0 + 2\,Z_\mathrm L\,} :T = \frac {2\,Z_\mathrm L}{\,Z_0 + 2\,Z_\mathrm L\,} Similar expressions can be developed for a series element, or any electrical network for that matter.
Networks Reflections in more complex scenarios, such as found on a network of cables, can result in very complicated and long lasting waveforms on the cable. Even a simple overvoltage pulse entering a cable system as uncomplicated as the power wiring found in a typical private home can result in an oscillatory disturbance as the pulse is reflected to and from multiple circuit ends. These
ring waves as they are known persist for far longer than the original pulse and their waveforms bears little obvious resemblance to the original disturbance, containing high frequency components in the tens of MHz range. ==Standing waves==