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PROFESSOR: Surface currents induced in a ground plane
by an overhead conductor illustrate
the method of images.
Also illustrated is the superposition
of fields due to line currents to satisfy boundary conditions.
Here is a metal cylinder mounted over a metal ground plane.
The cylinder is shorted to the ground plane at one end
and driven by a current source at the other.
This is familiar as an EQS situation, where
the same cylinder is insulated from the same ground plane
and driven by a voltage source.
Now, the situation is magnetoquasistatic, MQS.
Here's the metal cylinder mounted over the metal ground
plane.
The cylinder is driven by a current source at one end
and is shorted to the ground plane at the other end.
The ground plane consists of an aluminum sheet
having a thickness of about 1 millimeter.
It's resting on a plywood base.
Our magnetic field is the same as that due to a line current
here just below the center of the conductor and its image
here below the ground plane.
With the line currents at these positions,
the cylinder has no normal component of magnetic field.
The value of A is just that for which the cylinder of radius r
has no perpendicular component of magnetic field.
These are the field lines produced by the line currents.
The field is tangential to the surface of the cylinder
and to the midplane.
Thus, these surfaces can be replaced
by the metal surfaces of the cylinder and ground plane.
The fields are terminated by surface currents
in the cylinder and in the ground plane.
The image line current contributes
to the magnetic field above the ground plane
in exactly the same way as the actual surface current, which
is non-uniformly distributed along the ground plane surface.
When we drive a current from a three kilohertz source
through the circular conductor, the actual currents
are in the metal conductor whirls distributed in just such
a way that there is no magnetic flux normal to every point
on a given conductor surface.
Similarly, these currents are returned
in the ground plane in just such a way
that there is essentially no magnetic flux
density normal to the ground plane either.
We can measure the magnetic field using this induction
coil.
The induced voltage is related through Faraday's law
to the time rate of change of the linked magnetic flux.
Thus, the amplitude of the probe voltage
is proportional to the flux density, the average probe
area, the number of turns, and the angular frequency.
Our coil has an effective area of pi 0.85 centimeters squared,
200 turns, and an angular frequency
of 2 pi times our frequency, which is 3 kilohertz.
In this demonstration, a typical measured flux density
of 0.11 Gauss peak gives an induced voltage
of 9 and 1/2 millivolts peak.
With the coil oriented this way, the vertical field component
is recorded.
When oriented this way, the horizontal field component
is measured.
Here at the center line, the tangential field
is horizontal just because of symmetry.
Let's see if the flux density normal to the conducting
sheet does tend to be 0 off the center line.
Here to the side, the normal flux density
appears to remain small compared to that tangential.
So the current is returned in the sheet
with just that distribution required to make
the normal flux density small.
Our equivalent line currents can be
used to predict the flux density tangential to the sheet
as measured by the sensing coil.
l is the conductor height and r is its radius.
Then the height of the equivalent line current, a,
is defined like this.
y is the location of the probe on the ground plane.
Here is the y dependence of the tangential flux
density at the ground plane.
This is the predicted distribution
of normalized magnetic field with normalized probe position.
8 0 is the magnetic field at y equals 0.
The tangential magnetic field is largest directly
under the conductor, trailing off to 0
as the probe is pulled away.
As the sensing coil is moved out from under the conductor,
the tangential flux density decreases.
Because there is little normal flux density just
above the sheet, there is little of either
of the components of magnetic field below the sheet.
So what we see is the distribution
of surface current density returning in the sheet.
At high frequency, the current distributes itself
in the sheet in just such a way as to make the normal flux
density near 0.
Our theory predicts the flux density
at the center line, where y equals 0.
Our current is 3 amps.
The conductor height is 11 centimeters.
And its radius is 2.1 centimeters.
So a is 10.8 centimeters.
The predicted flux density is then 0.11 Gauss.
Directly under the conductor carrying a 3 amp current,
as shown in the upper scope trace,
we measure 9 and 1/2 millivolts peak on the lower trace.
This corresponds to 0.11 Gauss peak.
This is in better agreement with the theory
than we have a right to expect, given such factors as the size
of the sensing coil.
Here are some points that we have
taken with this probe at different positions
on the ground plane.
Remember, this is the distribution
of surface current density in the ground plane.
Our experiments have shown that the perpendicular
field at the surface of the ground plane
is much smaller than the tangential field.
This is true because the frequency of 3 kilohertz
is high enough that the ground plane behaves
as a perfect conductor.
Let's see what happens if we lower the frequency.
If we reduce the frequency by 10 to 300 hertz,
the induced voltage decreases by 10.
That's just because of Faraday's law.
So let's compensate by turning up the scope gain by 10.
At this lower frequency, we see that the ratio
of normal component to tangential component
is larger than at 3 kilohertz.
At 300 hertz, here's the tangential component.
And here's the normal component.
At 3 kilohertz, here's the normal component.
And here's the tangential component.
At 3 kilohertz, the ground plane is thick enough
to act as a perfect conductor, giving
the tangential feel predicted using the fields of a line
current and its image.