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Maxwells Equations and Electrical Circuits

Maxwells Equations and Electrical Circuits.


Electrical circuit analysis is usually presented as a theory unto itself. However, the basis of electrical circuit analysis actually comes from electromagnetics, i.e., Maxwell’s equations. It is important to recognize this since electrical circuit theory is really only an approximation and under the “right” conditions, it can fail. We will illustrate how electrical circuit analysis is derived from Maxwell’s equations by considering the following physical circuit (i.e., an electrical circuit that has physical dimensions):

First, we will apply Faraday’s law to the contour c

Between any two adjacent terminals a and b, we will define voltage as

To make the LHS of (1) fit (2), we will move the minus sign in so that

As we apply (3) to the physical circuit in the figure, we will choose to ignore the effects of the leads and the wires connecting the elements. (We can come back later and add these effects in, if we wish.) The integral on the LHS of (3) will be broken up into four subsections as

We will consider separately each of the five terms in
Source Voltage
V21 – This is the source voltage (or emf):

Resistor Voltage
V32 – This is the resistor voltage:

You considered resistors supporting direct current previously in EE 381. For sinusoidal steady state, we will assume here that the resistors are electrically small. That is, their dimensions are much smaller than a wavelength (l=c / f ). Inside a conductive material, by Ohm’s law` J((t)= s `E((t).
Therefore,

Assuming the frequency f is small enough so that `J is nearly uniform over the cross section (which is not true at high f), then

Where

This is the typical formula for compute the DC resistance of a conductor that has a uniform current density over its cross section. Why the minus sign in (6)? Because of the assumed polarity:

Capacitor Voltage
V43 – This is the capacitor voltage:

We considered the capacitor in detail earlier in Lecture 5 in connection with displacement current. Under the quasi-static assumption Q((t) ) » CV( (t) ) then

Forming the antiderivative of this expression gives

In terms of V43, we can write this result as

We see here once again that the basic circuit operation of the capacitor arises because of displacement current, as we discussed earlier in
Inductor Voltage
V14 – This is the inductor voltage:

Consider the contour c

And apply Faraday’s law to this contour

We can separate the line integral into two parts

The second term on the LHS equals zero if there is no resistance in the wires of the inductor. Otherwise, there would be an RL term similar to V32 earlier. For the RHS of (8)
where l (t) is the flux linkage through the surface formed by the coil of wire. Substituting (9) into (8) gives

For magnetostatic fields, you saw in EE 381 that l = LI We will assume here that the frequency is small enough (i.e., is “quasi-static”) that

Substituting (12) into (10) gives

Or

We see that the basic operation of the inductor arises directly from Faraday’s law!
Collecting the Results
Now, we’ll pull all of this together. Substituting (13), (7), (6), and (5) into (4) gives

This result was derived directly from Maxwell’s equations! In (14), the surface s is the open surface bounded by c, minus the surface bounded by the inductor. The effect of the magnetic flux through the inductor surface is accounted for in the inductor term in (14). Next, let’s apply KVL to the lumped element representation for this physical circuit:


This result was derived directly from circuit theory. Comparing (14) and (15) we see that if the RHS of (14) is negligible, then these two equations are equal! We have derived the terminal (V-I) characteristics for the lumped element circuit from Maxwell’s equations. This is a BIG accomplishment. In effect, every electrical circuit is a little electromagnetic “test bed:”
. Resistor: behavior governed by Ohm’s law,
. Capacitor: behavior governed by “Maxwell’s law,”
. Inductor: behavior governed by Faraday’s law.
“Missing” Term in KVL?
contribution from the magnetic flux linking the surface s of the entire circuit (i.e., the entire surface s minus the inductor):
. At low f, we can often safely ignore this term.
. At high f and overall physical circuit dimensions approaching tens of centimeters or larger, this term may not be negligible