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Electrical Net Size for Spheres

A series of three blogs about electrofishing fields has just been posted. The first one dealt with basic physics of fields around spheres. Let’s now build upon that one in this blog. Time to review some formulas.

J = I/SA, J is current density in A/cm2, I is current in A, SA is surface area in cm2

Therefore, J = I x 1,000,000/SA where J is current density in μA/cm2

J = E x Ca, E is field intensity (or voltage gradient) in V/cm, Ca is ambient conductivity in μS/cm

Therefore, E = J/Ca or E = (I x 1,000,000)/(SA x Ca)

For a sphere, SA = 4πr2 where r is the radius from the sphere center to some point in the water surrounding the sphere. Think of a watery sphere around the actual electrode.

Thus, Er = (I x 1,000,000)/(4πr2 x Ca) or Er = (I x 1,000,000)/(4π x Ca) x r-2, Er is the field intensity at some radius r from the sphere center.

Whew, that is enough theory for now. Let’s look at some real data. Kolz (1993) described electrical fields around several types and sizes of electrodes, including spheres. The above equations are for ideal conditions in deep water so that there are no boundary effects from the substrate or the water surface to distort the field around the electrodes. Kolz (1993) used large, deep tanks for his research and carefully measured volts by distance from the electrodes. I re-created his data by using digital images of his graphs and digital image measurement software to calculate the coordinates of the points on his graphs. The points were fit to a shifted power regression with offset, and the first derivatives of those equations were used to make graphs of field intensity by distance. Kolz (1993) converted his values to an ambient water conductivity of 100 μS/cm, and I followed that convention.

If the applied current at match, Im = 115 μS/cm, is 5.0 amps, then the current at 100 μS/cm, I100, is calculated as follows: I100 = 5.0 x (100+115)/230 = 4.67 amps

Assume that the target field intensity at match, Em, is 0.722 V/cm. This can be converted to 100 μS/cm by: E100 = 0.722 x (100+115)/230 x 115/100 = 0.776 V/cm

For the 15-cm diameter sphere, the decay function of field intensity by distance was Er = 3489 x r-2.032. If the target field intensity, Et, is 0.776 V/cm, then r = 62.76 cm from the center of the sphere.

For the 28-cm diameter sphere, the decay function was Er = 3371 x r-1.988. The distance r was 67.60 cm.

So how do those values compare to the theoretical formulas for spheres energized with 5.0 amps?

Theoretically, Er = (4.67 x 1,000,000)/(4π x 100) x r-2 = 3716 x r-2. Distance r = 69.20 cm.

Those experimental and theoretical equations and distances are similar, so that increases our confidence in the data and in the relationships expressed in these formulas. Now let’s develop equations for SA for each sphere field intensity and use those to calculate the electrofishing net sizes for each.

The SA equation coefficient is (I x 1,000,000)/Ca divided by the decay function coefficient. Then just multiply that by r to the positive exponent, i.e. change the sign of the decay function exponent.

For each experimental sphere, (4.67 x 1,000,000)/100 = 46,700.

For the 15-cm sphere, SA = 13.38 x r2.032 = 60,166 cm2.

For the 28-cm sphere, SA = 13.85 x r1.988 = 60,171 cm2.

For the theoretical sphere, use the equation for the SA of a sphere of radius 69.20 cm.

Thus, SA = 12.57 r2 = 60,192 cm2.

The calculated electrofishing net sizes for the two test spheres agree closely with that for the theoretical sphere. In an earlier blog, the net size for a theoretical Wisconsin ring or a spider array with 5 amps applied at match was 60,245 cm2. The target field intensity distance for that electrode configuration was about 90 cm from the center of the ring or spider array. The 5 amps at match, 115 μS/cm, produce the same size electrical net – effective fishing zone — as 4.67 amps at 100 μS/cm as used here.

What are some implications of these findings? The net size was the same for the two sphere sizes if the same applied current was used. In contrast, the corresponding voltages for the 15- and 28-cm spheres were 416 and 257 volts, respectively. Even more surprising may be the finding that the electrical net sizes were basically the same for spheres and Wisconsin rings or spider arrays if the same current was applied to each. The dissipation of the electrical energy from the two types of electrodes certainly differed in shape. The field around the sphere is spherical for some distance radial from the electrode surface until the current must bend back to the boat hull or other cathode to complete the circuit. The field from cylindrical droppers associated with a Wisconsin ring or a spider array is largely radial in the horizontal plane, but there is substantial current in a vertical downward direction before bending back to the cathode. Given that, 5 amps is 5 amps. The same quantity of electrical charges are dissipated by the electrodes, regardless of the electrode size, shape and geometry.

As in a prior blog, the 5 amps per electrode seems reasonable but needs more work for confirmation. We need to make accurate fishing threshold measurements using spheres with rectangular pulsed direct current of an “effective” waveform such as 60-120 Hz with 15-40 % duty cycle. I do have one set of data for spheres using rectangular pulses which may be useful in exploring the validity of the 5-amp per sphere assumption.

In August 2014, Jim Reynolds and I conducted a workshop in Grand Junction, Colorado for the Upper Colorado River Endangered Fish Recovery Program. The purpose was to evaluate the amp goal table for rafts and boats as used in the Standard Operating Procedures manual. Based upon a fishing threshold study in the Colorado River, the Im value suggested for both rafts and boats was 2.3 amps per half immersed spheres. Because the spheres were half immersed, the effective SA of the sphere is half that of a totally immersed sphere, so the applied current should be about half of the 5 amps expected for a fully immersed sphere. The calculated electrical field SA = (2.3 x 1,000,000)/(0.722 x 115) = 27,701 cm2. Based on half of 5 amps (2.5 amps) for half of the sphere immersed, the expected SA was half of that above, or about 30,100 cm2. The value above was 92% of that expected for a theoretical sphere if 2.5 amps were applied; or 2.3/2.5 is 92%. The conductivity readings varied in the Colorado River among the four raft and five boat crews in the stretches of the river sampled by each, and the spheres bobbed some vertically which caused variation in the electrical field intensity. The slightly lower than expected applied current values may have been due to another reason, a psychological one. Determination of fishing thresholds is subjective and is based on the perceptions of each individual electrofishing crew. There are four endangered species in the river, and the crews try to accomplish multiple objectives while out sampling fish. One important concern is to avoid harming or killing an endangered species. That concern may have caused a slightly conservative current setting on the river. The day prior to the study on the Colorado River, the same crews using the same rafts and boats also conducted a fishing threshold study at nearby Highline Lake where the conductivity was about 750 μS/cm. The resultant Im values for rafts and boats at Highline Lake averaged 2.54 and 2.86 amps, respectively. Those were somewhat higher than the 2.3 amps per sphere recommended for the SOP manual used on the river. It was the first threshold study of the workshop, and there was substantial recreational boating on the lake that day which created waves. The boating activity and the first study may have caused some variation in the results, but they approximated the 2.5 amp per sphere expected. The calculated electrical net sizes at Highline Lake for rafts and boats were 30,591 and 34,445 cm2 per sphere. These are somewhat higher than the approximate 30,100 cm2 expected based on the assumption of 5 amps per sphere used in the above calculations. Overall, the Im values per sphere were 2.3, 2.54 and 2.86 amps versus the 2.5 amps expected for a half-immersed sphere.

These findings are limited and are not presented here as totally definitive. Given that, the values of 5 amps per electrode and electrical net surface areas of about 60,000 cm2 per electrode appear reasonable starting points for further investigation. A future blog will explore applied currents and associated electrical field sizes for loops.

Kolz, A. L. 1993. In-water electrical measurements for evaluating electrofishing systems. U. S. Fish and Wildlife Service Biological Report 11.

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