Thu., Jan. 24 - Fast and slow E field antennas

A field mill can be used to determine when a thunderstorm becomes electrified and monitor electrical activity in a thunderstorm. Later in the semester we will see that measurements at multiple locations of the lightning field changes can be used to determine the magnitude and location of the charge neutralized by the lightning discharge. Note that a fairly large dynamic range is needed (-12 kV/m to +12 kV/m) is needed to insure that the E field remains on scale.

We began a discussion of so-called slow electric field antenna systems last class. Before getting into the details of circuit design I want to show some examples of actual Fast E and Slow E signals.

A Slow E field system (frequency response of 0.1 Hz to 1 kHz or more) would be appropriate if you wanted to study one of the lightning discharges in the figure above on a faster time scale. A couple of actual Slow E field records are shown below (from the same source as the field mill record above).

Note first the much faster time scale, 0.4 seconds full scale in this case. The top example shows a 3 stroke cloud-to-ground discharge. The second discharge has 4 strikes to ground.

Because the Slow E antenna system does not have DC response the static E field (which can be several kV/m) is effectively filtered out (like switching from DC to AC coupling on an oscilloscope). Lightning field changes can be examined with more gain. Because the signals of interest last from 0.5 to 1 second, a decay time constant of 10 seconds would be appropriate here.

Recording systems were mentioned briefly in class. Just to give you some idea of how technology has changed, the signals above were (I believe) displayed on a storage oscilloscope and photographed with a Polaroid film camera!

Because of the long decay time constant, charged precipitation falling on a Slow E field antenna can drive the signal offscale. Inverted antennas are sometimes used to avoid that problem.

This antenna is mounted on the roof of the Penthouse atop the PAS Building. Because of its exposed position and the chance that it could be struck by lightning, signals are brought into the Penthouse on a fiber optic cable.

The E field variations that occur during a individual return stroke could be examined by displaying the Slow E field signals on a faster time scale. However, as sketched below, the long time decay would mean the signal would not decay back to zero in the interval between return strokes.

If the gain were turned up the second and any following return strokes could go offscale. The solution is to shorten the time decay constant with the Fast E field antenna.

A decay time of about 1 to 10 milliseconds would be long enough to accurately record the Fast E field variations but would allow the signal to decay back to zero in the interval between strokes. A short decay time constant would also mean charged precipitation would be less likely to drive the Fast E signal offscale.

We'll do some examples and calculate what value of integrating capacitor might be needed in a Fast or Slow E field antenna system.

Here's a passive integrator, a capacitor connects the antenna to ground. We assume the antenna area is 0.1 m2 (the actual area of the antenna brought to class), the E field change is 10 V/m (a typical 1st return stroke at 100 km range), and we want a 1 volt output signal.

C would need to be 10 pF. That's pretty small. Stray capacitance between the antenna plate and ground might be a few pF. We would need to connect some kind of a measuring or recording device like an oscilloscope. The impedance of the measuring device together with the small capacitance in the passive integrator gives a time decay constant that is too short (the time constant should be several times larger than the field variations that you are measuring).

There are situations, though, where a passive integrator circuit might be suitable.

A triggered lightning discharge at 100 meters range will produce a much larger E field change signal. You could make the integrating capacitor larger (with a correspondingly longer decay time constant) because you wouldn't need as much gain. Actually you would probably also need to reduce the area of the antenna plate so as to not have too large an output signal. In the case of triggered lightning, signals from antennas and other sensors are generally carried into a shielded metal trailer on fiber optics cables for safety reasons and to avoid noise pickup on the signal cables.

While a passive integrator is sometimes appropriate, it is more common to connect the flat plate antenna to an active integrator circuit like shown below.

When the E field antenna sensor plate is connected to an active integrator circuit like shown above, the input impedance of the measuring equipment won't affect the decay time constant. The decay time constant is determined by the R and C values in the operational amplifier feedback circuit.

A couple of typical circuits are shown below. The second circuit, in particular shows a technique that can be used to get the long decay time constants needed in Slow E field antenna systems.

An electric field antenna that is not flush with the ground surface will distort the surrounding electric field. This is an appropriate point to review a problem worked in most undergraduate electricity and magnetism courses (the notes that follow are based on a class handout prepared by Dr. E. Philip Krider). This portion of the notes was reproduced on a class handout.

Conducting Sphere in a Uniform Electric Field

Notes on the distortion of an initially uniform electric field Eo by the presence of an uncharged conducting sphere. The sphere distorts the field such that the field lines are everywhere normal to the surface of the conductor. Choosing the origin of our coordinate system at the center of the sphere,

Spherical polar coordinates are used, there is azimuthal symmetry, so the potential and the electric field depend on r and θ only. We can proceed to solve Laplace's equation for the potential, Φ, subject to the boundary conditions that

In the spherical coordinate system

Assuming the variables are separable, we try a general solution of the form:

Just looking at the boundary condition (Eqn. (ii)), we simply try a T(θ) function of the form

Now, inserting this into our original differential equation, we find

Some of the missing details are appended at the end of this section.
Now, let's try a solution to the R equation of the form

When we substitute this into the differential equation above we get

Now a general R solution will be of the form

and

Where A and B must be determined from the boundary conditions.

Now,

applying boundary condition (ii) as r goes to infinity

since

and our potential function is now

with this Φ our electric field components are:

and the surface charge density is

Note: The presence of the sphere increases the value of the ambient field at the top and bottom surface of the sphere by a factor of 3.

This figure gives you a rough idea of how the field is changed in the vicinity of the sphere. E field lines must intersect the sphere perpendicularly. The field is enhanced (amplified) by a factor of three at the top and bottom of the sphere.

Some of the details of one of the earlier calculations are shown below:

Enhancement of fields by conducting objects is an important concern. In some cases (we'll look at an example or two later) the enhanced field is strong enough to initiate or trigger a lightning discharge.

The following handout gives a rough, back-of-the-envelope kind of estimate of the factor of enhancement.

This might require a little explanation.

First you write down the potential at the surface of two conducting spheres of radius a and b, carrying charges Q and q (really just the potential a distance a or b from a point charge Q or q)

Then you connect the two spheres with a wire which forces the two potentials to be equal (this would of course cause the charge to rearrange themselves and turn this into a much more complex problem, but we will ignore that).

Finally we write down expressions for the relative strengths of the electric fields at the surfaces of the two spheres (we assume Q and q would be uniformly spread out over the two spheres which wouldn't be true). We see that the field at the surface of the smaller sphere is a/b times larger than the field at the surface of the bigger sphere.

Here is a real example of field enhancement that lead to triggering of a lightning strike and subsequent loss of a launch vehicle (you'll find the entire article here)

In this case the rocket body together with the exhaust plume created a long pointed conducting object. Enhanced fields at the top and bottom triggered lightning.

We'll talk about triggered lightning later in the course. I'm referring to lightning that is purposely trigger so that it can strike instrumentation on the ground and studied at close range.

The basic idea is to launch a small rocket (about 3 feet tall) in a high electric field under a thunderstorm. A spool of wire is mounted on the tail fins of the rocket. One end of the wire is connected to ground and the other end runs up to the nose of the rocket. Wire un-spools (probably the hardest part is to keep the wire from breaking) once the rocket is launched forming a narrow tall conducting object. Field enhancement at the top of the rocket is enough often times to initiate an upward leader discharge that then triggers lightning.

Enhancement of the E field at the top of a mountain (or tall building or structure) is sometimes high enough to trigger lightning also.

Note the direction of the branching. This indicates that this discharge began with a leader process that traveled upward from the mountain. Most cloud to ground lightning discharges begin with a leader that propagates from the cloud downward toward the ground. We will of course look at the events that occur during lightning discharges in a lot more detail later in the semester.

And finally (will this lecture ever end). The ability of a point to draw off or throw off electrical charged that so interested Benjamin Franklin involves enhancement of the E field.

A pointed conductor brought near a Van de Graaff generator enhances the field enough to ionize air and create charge carries in the air. A weak current flows between the Van de Graaff and the point. Charge on the generator is not able to build to the point where a large bright spark occurs.