Physical Interpretation of Reflection and Transmission in Lumped Systems

In lumped systems, traveling waves do not occur, in principle, because lumped elements are characterized as one-ports interconnected by ``wires'' having no time delay associated with them. It may therefore seem strange that a scattering theory formulation exists for lumped networks.

There does exist, however, a physical interpretation of reflection and transmission in lumped networks [33]. Suppose we have a ``force source'' $f(t)$ which drives a ``load impedance'' $R_1$ in series with a ``source impedance'' $R_0$. For simplicity, let the load and source impedances be real (dashpots) as shown in Fig. J.15.

Figure J.15: A series connection of two dashpots $R_0$ and $R_1$ driven by a force $f(t)$. Dashpot $R_0$ models the source impedance, while $R_1$ models a load impedance.

An equivalent electrical circuit is shown in Fig. J.16.

Figure J.16: Equivalent circuit of a dashpot $R_1$ driven by a force source $f(t)$ with internal impedance $R_0$.

Then the velocity is given by $v(t) = f(t)/(R_0+R_1)$, and the ``force drop'' across the load is

$$f_1(t) = R_1v(t) = f(t) \frac{R_1}{R_0+R_1}$$

The instantaneous power delivered to the load is therefore

$${\cal P}_1(t) = f_1(t) v(t) = \frac{v^2(t)}{R_1} = f^2(t) \frac{R_1}{(R_0+R_1)^2}$$

If this expression is differentiated with respect to $R_1$ and set to zero to find its maximizer, we find that maximum power is delivered when $R_1=R_0$, i.e., in the matched impedance case.^J.5 The force on the load at matched impedance is $f(t)/2$, and the power delivered is

$${\cal P}_{\mbox{max}} = \frac{f^2(t)}{4R_0}$$

which is called the maximum available power from a force $f(t)$ through a source impedance $R_0$. Define the ``matched velocity'' in the matched impedance case as

$$v_0(t) = \frac{f(t)}{R_0+R_0} = \frac{1}{2} \frac{f(t)}{R_0}.$$

The relative difference between the matched velocity $v_0(t)$ and any other velocity $v_1(t) = f(t)/(R_0+R_1)$ delivered to an unmatched load $R_1$ is given by

$$\frac{v_0 - v_1}{v_0} = \frac{R_1- R_0}{R_1+ R_0} = \rho$$

This is the formula for the reflection coefficient seen at the junction of two waveguides (or transmission lines), where one waveguide has impedance $R_0$ and the other has impedance $R_1$. In this context, multiplying the maximum available power velocity $v_0$ by the reflection coefficient $\rho$ gives the difference between $v_0$ and the velocity actually delivered. This difference can be interpreted as reflected power. Conceptually, the driving force ``sends'' maximum available power, and the load ``reflects back'' some of it, unless the load impedance matches the source impedance, and this transaction occurs instantaneously. Thus, the wave variable reflection coefficient in lumped systems can be thought of as the coefficient of velocity reflection relative to the velocity at maximum available power. A similar calculation shows that the force reflection coefficient is $-\rho$, and the reflection coefficient for signal power itself is $-\rho^2$.