# 45 Impedance 101

## What’s the Point of This Lecture?

One unfortunate aspect of Jackson’s book is that he barely
mentions, and does not really use, the concept of *wave impedance*. Yet as is made clear in Zangwill’s book, and in the
excellent online notes of Likharev, the most
natural parameters for describing waves (of any kind) going from one medium to
another are the *wave velocities* and
the *wave impedances* for the two media.
In the next few lectures, we’ll be discussing electromagnetic waves: in vacuum, in media and in various pipes
(waveguides, resonators, etc.). In
particular, we’ll be analyzing wave transmission from one medium to another.
We’ll use impedance a lot, so, since it’s not featured prominently in many
introductory physics texts, we’ll introduce it here with a simple example.

## Waves on a String

The simplest system with the essential ingredients is, I think, small amplitude waves on a taut uniform string: tension $\tau ,$ linear density $\rho .$ We need to find out what happens when this string is joined to another one having different density, or possibly tension, and a wave is sent towards the join.

We’ll review the relevant sections of *Waves*, *Berkeley Physics
Course*, Volume 3, by Frank Crawford.

Following Crawford, the string is taken to be along the $z$ axis (yes, not the $x$ -axis!) from the origin to a large positive distance, and is harmonically driven at the origin by a “transmitter”, oscillating harmonically in the $x$ direction. The wave amplitude is (using Crawford’s notation)

$$\psi \left(z,t\right)=A\mathrm{cos}\left(\omega t-kz\right),$$

from which $\partial \psi /\partial z=kA\mathrm{sin}\left(\omega t-kz\right)$ and $\partial \psi /\partial t=-\omega A\mathrm{sin}\left(\omega t-kz\right).$

(*Note*: Crawford distinguishes between the string
tension $\tau $ and its equilibrium tension ${\tau}_{0}.$ We won’t: we’ll consider only small amplitude
waves, and that difference is third order in the angle of slope.)

A natural question is: what transmitter power is necessary to produce a wave train of a given amplitude traveling down the string?

We assume the $\omega ,k$ relationship for waves on the string is
linear, $\omega ={v}_{\phi}k,\text{\hspace{0.33em}}$ ${v}_{\phi}=\sqrt{\tau /\rho}$ being the *phase*
$\phi $ velocity (if you’re rusty on where this comes from, look
at my notes here)
so

$$\partial \psi /\partial z=-\left(1/{v}_{\phi}\right)\partial \psi /\partial t.$$

To find how hard the transmitter has to work to generate a given wave $\psi \left(z,t\right),$ the (vertical) force the string exerts on the transmitter (see diagram) is

$${F}_{x}\text{(string)}=\tau \text{\hspace{0.05em}}\partial \psi \left(0,t\right)/\partial z=-\left(\tau /{v}_{\phi}\right)\partial \psi \left(0,t\right)/\partial t.$$

This is the resistance force the string presents to being made
to wave at this velocity, and the multiplying constant $\tau /{v}_{\phi}$ is called the *impedance* (or sometimes the *amplitude*
impedance) and written $Z.$

That is,

$${F}_{x}\text{(string)}=-\left(\tau /{v}_{\phi}\right)\partial \psi /\partial t=-Z\partial \psi /\partial t,$$

and using ${v}_{\phi}=\sqrt{\tau /\rho},$ the impedance

$$Z=\sqrt{\tau \rho}.$$

*Exercise*: Check the dimensions are the same as those of force/velocity.

Using $Z$ to express the *power* (rate of working of transmitter, force component times
velocity)* *generating the traveling
wave, with ${v}_{\phi}=\sqrt{\tau /\rho},$

$$P\left(t\right)={F}_{x}\text{(transmitter)}\left(\frac{\partial \psi}{\partial t}\right)=Z{\left(\frac{\partial \psi}{\partial t}\right)}^{2}=\frac{1}{Z}{\left(-\tau \frac{\partial \psi}{\partial z}\right)}^{2}.$$

Since there is no friction in the system, this power is traveling with the wave and, given the right receptor, it can be fully recovered at any later point along the $z$ -axis.

But what happens if the wave encounters a different medium?

## Transmission and Reflection

To be specific, we assume that at $z=0$ the string is joined to one of a different density, but they are pulled tight so the tension is the same throughout.

We know from experience that in general the wave is partially transmitted, and partially reflected. We can find the relevant amplitudes from the requirements that the wave amplitude and its slope must be the same for both strings at the join: the forces must balance there, or the infinitesimal bit of string at the join would have infinite acceleration (the tension is the same throughout).

Consider the situation after the (semi-infinite) wave train front has passed the join: there is still incoming wave

$${\psi}_{\text{in}}\left(z,t\right)=A\mathrm{cos}\left(\omega t-{k}_{1}z\right)$$

on the left, generating a force on the join:

$${F}_{\text{in}}=-{Z}_{1}\frac{\partial {\psi}_{\text{in}}\left(0,t\right)}{\partial t}.$$

This is balanced to some extent by the force from the string
on the right. But if ${Z}_{2}\ne {Z}_{1},$ since $\partial \psi /\partial t$ is the same for both, the forces can’t
balance: the change in impedance means that simple transmission doesn’t work,
it’s termed an *impedance mismatch*.

The solution (as you know) is that some of the wave is reflected, and the amount reflected is given by balancing out the impedance mismatch. The transmitted and reflected waves have necessarily the same $\omega $ as the initial wave (all must match at $z=0$ ), the reflected wave is in the same medium as the incoming wave, so has the same wave number ${k}_{1},$ but is going backwards, take its amplitude to be $RA$:

$${\psi}_{\text{ref}}\left(z,t\right)=RA\mathrm{cos}\left(\omega t+{k}_{1}z\right).$$

The transmitted wave will have a different wavenumber, since $\omega ={v}_{1}{k}_{1}={v}_{2}{k}_{2},$ so, in the obvious notation,

$${\psi}_{\text{trans}}\left(z,t\right)=TA\mathrm{cos}\left(\omega t-{k}_{2}z\right).$$

(This is why I’m not using $T$ for string tension here, although it’s standard.)

Matching amplitudes and slopes of ${\psi}_{\text{in}}+{\psi}_{\text{ref}}$ and ${\psi}_{\text{trans}}$ at $z=0,$

$$\begin{array}{l}1+R=T,\\ {k}_{1}\left(1-R\right)={k}_{2}T.\end{array}$$

From this, writing the first equation as ${k}_{1}\left(1+R\right)={k}_{1}T$ then adding and subtracting,

$$R=\frac{{k}_{1}-{k}_{2}}{{k}_{1}+{k}_{2}},\text{\hspace{1em}}T=\frac{2{k}_{1}}{{k}_{1}+{k}_{2}}.$$

*Exercise*: Open
the animation, and notice how the reflected wave is different depending on
which string is denser.

## Impedance Equations

Furthermore, from $\omega ={v}_{1}{k}_{1}={v}_{2}{k}_{2},$ we have ${k}_{1}/{k}_{2}={v}_{2}/{v}_{1}=\sqrt{{\rho}_{1}/{\rho}_{2}}={Z}_{1}/{Z}_{2},$ so we can write the reflection and transmission amplitudes in terms of the impedances:

$$R=\frac{{Z}_{1}-{Z}_{2}}{{Z}_{1}+{Z}_{2}},\text{\hspace{1em}}T=\frac{2{Z}_{1}}{{Z}_{1}+{Z}_{2}}.$$

**Important**: this
makes clear that for this linear system, the transmission coefficient depends *only* on material properties *not* on the wavelength of the wave.

If the impedances match, there is no reflection, the wave goes on. For ${Z}_{2}>{Z}_{1},$ the wave changes sign on reflection. For ${Z}_{2}<{Z}_{1},$ the transmitted amplitude is larger than the incoming wave (watch the animation), but of course having less energy.

### Different String Tensions

A more interesting situation rises if we allow the two strings to have different tensions. This could be accomplished by taking the join to be a very small ring, moving frictionlessly up and down a fixed rod parallel to the $x$ -axis. The requirement now is that the $x$ -direction forces on the ring must balance,

$${\tau}_{1}\left(\frac{\partial {\psi}_{\text{in}}}{\partial z}+\frac{\partial {\psi}_{\text{ref}}}{\partial z}\right)+{\tau}_{2}\frac{\partial {\psi}_{\text{trans}}}{\partial z}=0$$

at $z=0,$ so

$${k}_{1}{\tau}_{1}\left(1-R\right)={k}_{2}{\tau}_{2}T.$$

This time (since $\omega ={v}_{1}{k}_{1}={v}_{2}{k}_{2}$ )

$${k}_{1}/{k}_{2}={v}_{2}/{v}_{1}=\sqrt{{\rho}_{1}{\tau}_{2}/{\rho}_{2}{\tau}_{1}}$$

so

$${k}_{1}{\tau}_{1}/{k}_{2}{\tau}_{2}=\sqrt{{\rho}_{1}{\tau}_{2}/{\rho}_{2}{\tau}_{1}}\times {\tau}_{1}/{\tau}_{2}=\sqrt{{\rho}_{1}{\tau}_{1}/{\rho}_{2}{\tau}_{2}}={Z}_{1}/{Z}_{2}.$$

and *the above impedance
equations for transmission and reflection are still correct*.

It’s worth noting that if the strings have *different *densities, *but* *also*
different tensions in the opposite ratio, there will be *no reflection*: the wave train will go through, with altered speed
and wavelength but the same power transmission rate. We say the impedances are
“matched”, clearly desirable for signal transmission through a join in cables.

These same impedance equations appear in many branches of physics, for example light transmission normal to a surface, discussed below.

### Impedance in AC Circuits

Jackson spends no time on conventional AC circuit theory, assuming (I hope reasonably) that you’re fully familiar with it from undergraduate studies. If you want to brush up, you can’t do better than to read Feynman’s lecture on the subject: vintage Feynman, entertaining and with some very cute examples.

The reason I bring AC theory up here is that it affords
further insight into the nature of impedance.
AC impedance is the generalization of simple resistance in DC circuit
theory. Kirchhoff’s laws are still good for analyzing a general circuit, but
now the alternating current in a leg (meaning a component of a circuit with one
current in it, but it could include resistance, inductance, capacitance) is generally out of phase with the alternating
applied voltage, so both driving force $V$ and response $I$ are most naturally represented by rotating
complex numbers, the engineers call them *phasors*.
The impedance of a single leg of a
circuit can be written as $Z=R+iX$ where $R$ is the ordinary resistance, and $X$ is termed *reactance*,
generated by inductance and/or capacitance elements, for which current and
voltage are ninety degrees out of phase, hence they contribute terms $i\omega L$ and $1/i\omega C.$

That is to say, for an AC circuit Ohm’s (DC) Law $I=V/R$ with all numbers real of course, goes to

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

where $Z$ is a (non-time varying) complex impedance, and $I,V$ are complex numbers rotating at the same rate in the complex plane, with phase difference equal to the phase of the impedance.

If we have a
generator which is part of a circuit having inductance and capacitance, driving
a load which itself has resistance, inductance and capacitance, the power
transfer is at a peak when the *impedances
are matched*: to be precise, the load circuit impedance must be the complex
conjugate of the generator circuit impedance. Detailed proof available here.

## Impedance in the Reflection of Light

We’ll discuss reflection of light fully in the next lecture,
but we’ll find that for* normal incidence*
of light from one medium to another, the transmission and reflection
coefficients are *almost* the same as those for the string example above:

$$R=\frac{{Z}_{2}-{Z}_{1}}{{Z}_{1}+{Z}_{2}},\text{\hspace{1em}}T=\frac{2{Z}_{2}}{{Z}_{1}+{Z}_{2}},$$

where now the impedance is given in terms of the permittivity and permeability:

$$Z=\sqrt{\frac{\mu}{\epsilon}}.$$

It is instructive to note (Zangwill p 592) that if we assume uniform magnetic properties, and then rewrite these equations in terms of the refractive indexes, we find

$$R=\frac{{n}_{1}-{n}_{2}}{{n}_{1}+{n}_{2}},\text{\hspace{1em}}T=\frac{2{n}_{1}}{{n}_{1}+{n}_{2}},$$

precisely the same as the equations for the string impedances discussed earlier, so evidently the refractive index $n$ is analogous to the string impedance as we defined it: the denser string corresponds to the denser medium.

But $\epsilon ={n}^{2},$ so the electromagnetic parameter $n$ that apparently corresponds to the string
impedance is the *inverse* of the
usually defined electromagnetic wave impedance $\sqrt{\mu /\epsilon}$! How can they both make sense?

To go back to the string: if the string on the right is very
dense, it’s almost as if the wave coming in from the left meets a brick wall,
the wave is reflected. But if the string on the right is very *light*, a large amplitude wave *will* be transmitted, *but* it will carry very little energy,
and a wave coming in from the left will be almost
all reflected. It’s as if the left string ended
on a ring that could move up and down on a perpendicular rod. The most efficient transmission occurs when
the strings are *equally dense*, and
going away from equality in *either*
direction results in reflection (although with different phase shifts). So,
perhaps counterintuitively, a very *low*
impedance on the right is effectively a barrier: it impedes energy
transmission. A specific example is a mirror made of a highly conductive metal,
such as silver, which has a very low impedance at optical frequencies, so *reflects* efficiently.

Evidently, there is some flexibility in defining wave impedance: we’ve seen that given one expression, its inverse will lead to very similar equations for transmission and reflection at a boundary between media. It will become clear later that for the electromagnetic wave it is natural to use the ratio of the electric field to the (magnetic) $H$ field, from the Maxwell wave equation

${\overrightarrow{H}}_{0}=\widehat{\overrightarrow{k}}\times {\overrightarrow{E}}_{0}/Z,\text{\hspace{1em}}Z=\sqrt{\mu /\epsilon}.$

Finally, recall that the $H$ field from a current in a long straight wire is given by $2\pi rH=J,$ so the dimensions of $H$ are amps/meter; and the electric field has units volts/meter, so $Z$ ’s units are volts/amps, or, from Ohm’s Law, ohms, units of impedance. As we’ll see later, wave impedance is central to analyzing wave transmission through waveguides, etc.

In vacuum,

$$Z={Z}_{0}=\sqrt{{\mu}_{0}/{\epsilon}_{0}}=376.7\text{ohms}\text{.}$$