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This is the sixth and final blog in the series on electrical fields. Much material has been presented on the basics of electrofishing fields, the rationale for using applied current as the electrical measure for determining field size, a description of how to determine field size and specific examples for anodes made of Wisconsin rings or spider arrays, of spheres and of loops. This blog will simplify the calculations and summarize the information in a couple of tables. First, more will be explained about the field intensity (voltage gradient) profile decay equation and what it is telling us about the field shape.

An electrical field is a real entity. Fields are the fifth state of matter after solids, liquids, gasses and plasmas. An electrical field is defined in physics as force per charge. The field itself is not the same as electrical force, but the electric field exerts a force on other charges. Field vectors occur in three-dimensional space in the water. We try to visualize the field with field lines (think current lines) and with voltage or voltage gradient (field intensity) contours, in two-dimensional space, or to describe the field with graphs and equations of the field intensity by distance from the electrodes. We have made use of the latter multiple times in this series.

We have discussed and used the entire field intensity decay equation and its coefficient in certain calculations of field surface area and field intensity by distance from the electrode. Now let’s briefly focus on the equation exponent because it describes how the field intensity decreases with distance from the electrode center and thus informs us to some degree about the shape of the electrical field. Fields — think electrical charges or current — must exit or enter electrodes at right angles to the electrode surface. Further, the current must at all times be at right angles to the voltage and voltage gradient contours, those three-dimensional surfaces in the water which constitute the outer edge of an electrical net defined by the field intensity profile we choose. For example, we may choose a field intensity of 0.4 V/cm as the boundary of the field. The boundary has some shape and surface area. We can calculate the area of the surface by the methods presented in this blog series. We can estimate the shape, at least in two dimensions and at multiple depths, of the electrical net surface by measuring its field intensity at multiple points in a grid around the anodes and using those measurements to make maps with field intensity contours. This can be done by the kriging method or by spline interpolation in R, for example. Such maps are quite informative as regards the shape and size of the effective fishing field around anodes.

The simplest case of estimating electrical field shapes is the sphere. The surface area of a sphere is 4πr^2. From prior blogs in the series, E = (I x 1,000,000)/(SA x Ca). Therefore, E is a function of current/SA. In the case of a sphere, E is a function of 1/r^2 or is a function of r^-2. So, for a sphere, field intensity = Coefficient x r^-2, i.e. the exponent is -2. The field intensity is decreasing as a function of the distance from the sphere center squared. You may recall that Kolz (1993) sphere and loop exponents were near -2. He conducted his study in large, deep tanks. The energy is dispersed radially in all directions from a sphere, and evidently somewhat similarly for loops. For the smaller backpack loops used in the raceway, the exponents were -1.82 to -1.84. The energy was dispersed slightly more in the horizontal plane than the vertical direction. This is to be expected in the relatively shallow raceway with its boundary effects, especially its slight vertical compression of the field; the water depth was 53 cm, and the loop was placed 33 cm off the raceway bottom. Also, a loop is not a sphere; it is much larger in the horizontal axis than in the vertical axis.  Considering the shape and the likely boundary effects, the field intensity decreased quite rapidly with distance from the anode loop. That indicates substantial energy dispersion in more than the horizontal plane; evidently, much of it was dispersed vertically. Of course, the current must complete the circuit between anode and cathode, so it can only occur at right angles, or nearly so, from the anode surface for some distance before bending towards the cathode. By convention, electrical fields are shown as beginning from the anode and moving toward the cathode. We can perhaps thank Benjamin Franklin for that convention. Backpack electrofishers used in streams during class field trips have had their field intensities measured. Those loop anode field profiles typically have exponents of -1.4 to -1.8 or so. Those values are higher than for the tanks studies and likely indicate more boundary effects because the measurements are generally made in relatively shallow water and perhaps under the duress of a field trip with new equipment such as oscilloscopes to operate. Also, backpacks often use a rattail as a cathode, and the rattail is on the substrate. That may magnify the anode boundary effects to some degree as the electrical current flows between the anode and cathode.

Wisconsin rings and spider arrays — which function similarly if the electrode number, spacing and immersed size are about the same – produce local, intense fields around each dropper; these dropper fields for each array merge to form a larger diameter but less intense ring field which forms the fishing field of interest for our purposes. The dropper fields are much more intense and important for fish injury than is the ring field. If only a few droppers are used in an array, the applied current is concentrated in a small surface area at each dropper which increases the local intensity there and can increase the probability of fish injury. I should say here that the original Wisconsin ring design was composed of many (~20) dropper electrodes for each array. The objective, according to Dr. Donald Novotny, personal communication, was to simulate a sphere as for its electrical properties yet allow easier movement through the water and into obstructions such as brush, vegetation or rocks. Field profile maps for Wisconsin rings (at least the type we see) and spider arrays in class field trips typically have exponents of about -1.4 to -1.7. Much of the energy is dispersed radially in the horizontal plane but also there likely is a substantial vertical extent to the field before the current bends back toward the cathode, which often is the boat hull. The droppers around the ring are of like polarity, so their electrical charges repel one another. Thus, it is likely that the current is dispersed vertically down at that point before curving toward the boat hull or other cathode. The field bending likely occurs at less distance from the anode if the boat has an auxiliary set of cathode droppers at the bow called a cathode skirt. The field intensity measurements for boats generally are made in moderately shallow water but deeper than for backpack measurements. Any field intensity profile exponents less than -2 are suspect. That would indicate a high rate of field intensity decrease in the horizontal plane and therefore likely more energy dispersion in the vertical axis than expected from the electrodes commonly used. That is just my intuition.

In this series, the magnitude of the electrical field has been discussed primarily in terms of field intensity, E, in V/cm. In the background has been the power density, D, in μW/cc. Let me say now that it is most simple to think of the electrical field magnitude in terms of current, I, and current density, J. Current is in amps, and current density is in μA/cm^2. It is easy to measure E with a voltage gradient probe and oscilloscope, and then it is easy to convert E to J or D if the ambient conductivity is known. Some useful relationships are:

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

The target applied current, It = (Et x SA x Ca)/1,000,000 or, more simply, It = (Jt x SA)/1,000,000

The target current density, Jt, must be determined somehow. This can be determined from fish trials in tanks with uniform electrical fields or from published studies which provide thresholds in terms of field intensity or in terms of power density. Such thresholds at a known ambient conductivity from trials or from published articles need be converted to that at the water conductivity you will be sampling. The file EF Goals or the electrofishing phone app can be used for that conversion. A more general formula for converting from any current density, J1, at the initial water conductivity, C1, to a target current density, Jt, at a new conductivity, Ca, is Jt = J1 x (Ca+Cf)/(C1+Cf). That might be a good one to write down or remember.

Once the target current density is determined, the calculations for field size or target current are simple. Miranda (2005) concluded that fish 20 g or larger could be effectively sampled using a power density at match, Dm or D at 115 μS/cm, of 60 μW/cc. D = E^2 x Ca, so E = sqrt(D/Ca). In this case, Em = sqrt(60/115) = 0.722 V/cm. These values for Dm and Em have been used extensively in this series on electrical fields. Therefore, Jm = 0.722 x 115 = 83.0 μA/cm^2. If I = 1.0 amp, SA = (1.0 x 1,000,000)/83.0 = 12,048 cm^2 per amp of current applied to the electrode. Don’t worry about the extra significant figures just created; they will be removed later. One more example before providing a table. From Figure 1 of Miranda (2005), let me infer that a power density of 200 μW/cc at match may be needed for capturing fish of 3-5 grams. The associated field intensity is Em = sqrt(200/115) = 1.32 V/cm. Thus, Jm = 1.32 x 115 = 152 μA/cm^2 for the smaller fish. For I = 1.0 amp, SA = (1.0 x 1,000,000)/152 = 6,579 cm^2. The calculated field size is smaller because the larger threshold current density for the smaller fish is in the denominator. If the same field size as before were desired, then the output settings must be increased to compensate for the larger threshold to capture smaller fish. The table below shows this in terms of the field intensity at match, Em, but other electrical measures could have been used. The field intensity is directly proportional to the applied voltage, regardless of conductivity, and pulsators have a voltage control, so it should be easy to adjust the voltage to produce the desired field size for shocking the smaller fish.

The order of fish sizes in Table 1 above corresponds to the more likely used values at the top of the list. Values for the last two fish sizes are merely informed guesses at this point. I lack sufficient data to be definitive for the last two rows. However, the implication for them is that larger fish are stunned at lower thresholds than are smaller fish. Thus, if you are focused on large fish for pond management, for example, then the effective fishing field will be larger for large fish than small fish for a given applied voltage, current or power. The target Jm, Dm and Em values are less for larger fish. Conversely, producing a 120,000 cm^2 fishing field for large fish requires less voltage, current and power. The result is that the effective conductivity range for electrofishing is larger for large fish because the demands on the generator and pulsator are less. One may be able to capture sub-adult or adult fish well but miss the smaller fish when operating near the limits of electrofisher capability.

Let me now provide three formulas to use only for a short discussion.

Er = (I x 1,000,000)/(SAr x Ca). Note that E varies directly and proportionately with I and it varies inversely with Ca and SA with the current-based formula for E.

Er = (V x 10,000)/(SAr x R100). Note that E varies directly and proportionately with V and it varies inversely with R100 and SA with the voltage-based formula. The important point here is that, if voltage is held constant, the field intensity profile map is unaffected by a change in water conductivity. Unless the electrode geometry is changed, then only one profile map is needed! That is an important advantage. And a given profile map can be easily adjusted up and down with changes in applied voltage. The field intensity at any distance, or coordinate on a grid, from the anode center is changed proportionately with applied voltage. Now let’s investigate the formula for current density.

Jr = (I x 1,000,000)/SAr. Note that J varies directly and proportionately with I and it varies inversely with SA. It is unaffected by Ca and R100! Only one profile map of current density by distance from the anode center need be made. The profile map is scalable with applied current, and it is unaffected by electrode resistance though it is affected by electrode geometry changes.

Based upon the information in prior blogs, the following table is offered as a suggestion of applied current at match for a variety of anodes. The numbers have been rounded for simplicity. Such goal tables are guides, they are tools to help biologists and researchers better sample fish using electricity. There is nothing magic or immutable about such goals. One saying is, “All models are false, Some are useful.” Let these be useful guides for helping people more quickly find fishing thresholds. Another saying is, “People use tools, Tools don’t use people.” Use what is helpful; modify or disregard the rest.

To convert from Im to some other water conductivity, use I = Im x (Ca+115)/230 or, more generally, I = Im x (Ca+Cf)/2Cf if the effective fish conductivity, Cf, is not 115 μS/cm. Determination of Cf has been addressed in prior blog about Grass Carp. Let me just say that Cf may vary with fish species, fish size and the conditions of the trials used to determine Cf. The value of Cf is determined simultaneously with the determination of Dm, so they are linked. In fact, the coordinates on a graph at a particular point are (Cf, Dm). So, values of Cf may vary. That affects the conversions to ambient conductivity of E, D and J. That is another reason to use goal tables as a guide. The best instrument on the boat is not the conductivity meter, the oscilloscope, the current probe or the goal table. It is your brain. My flight instructor gave me some good advice long ago. He said, “Trust your instruments until proven wrong.” Let me again offer a quick estimate for calculating the applied current for a given conductivity. In most cases, this can be done in your head. I_hat = Im x (Ca+100)/200. All of these applied current figures are for peak current.

As another saying goes, “When all is said and done, there is usually more said than done.” My intention has been to provide in this electrofishing field blog series some information that people can use to do their job better. It is a compilation of information and observations from a variety of sources. Some has come from published articles, but much has been derived from classes or private work to make measurements. Drs. Alan Temple and Jim Reynolds and I have collaborated in classes, workshops and several shared experiences over the years. Some of these ideas may be collective ones, but any errors in logic or calculation are mine alone. We all owe a debt of gratitude to others including Larry Kolz, Dr. Steve Miranda and Dr. Donald Novotny for advancing the field of electrofishing and the science supporting it. May you find this blog series useful.

Miranda, L. E. 2005. Refining boat electrofishing equipment to improve consistency and reduce harm to fish. North American Journal of Fisheries Management 25:609-618.