A scale model is a physical representation of an object which maintains general relationships between its constituent aspects. Alternately, it is “a small copy of something” or “a miniature representation of something.” This blog describes electrical measurements made on a scale model, yet functional, electrofishing boat and how to scale those up or down to a full size boat. Warning, this will be a long blog, and it may contain errors of logic or calculation. You can use the MythBusters TV show approach and determine if you believe the assumptions are confirmed, plausible or busted.

The story began when biologists from the Missouri Department of Conservation asked Jim Reynolds and me to put on an electrofishing workshop at the American Fisheries Society meeting in Kansas City this August. My initial plan for a portion of that workshop was to demonstrate electrical fields around an electrofishing boat during a field trip. The workshop became limited to four hours, so a field trip was not possible. My next idea was to demonstrate electrical fields around a model electrofishing boat in a large round tank. For that, we needed a model boat – a functioning model boat. So, I approached Shawn Banks of Midwest Lake Electrofishing Systems (MLES) about the possibility of building one. From a past visit to his headquarters, it was clear that Shawn had the facilities, ability and drive to build a functioning model boat. To my delight, he said that he had been thinking for years about building one. He had it mostly built by that weekend! Some final details and wiring came later, and it was completed before the workshop.

Side view of the scale model electrofishing boat in a display aquarium. It was powered by the MLES Infinity Xstream backpack electrofisher.

Front view of the model electrofishing boat in display aquarium showing detail of the anode arrays and of the wiring block.

We took a few electrical measurements during the workshop while it was in an aquarium, but the boundary effects of such a close confinement affected the results versus being in a larger body of water. So, we went back to MLES headquarters in nearby Polo, Missouri and made measurements in Shawn’s fishing pond. The question then became how to scale up the electrical parameters from the model boat to a full size boat. This is where you, as a reader, can play a part in the whole process. I made some assumptions from the start and then just worked out the algebra to calculate the associated resistance, power, voltage and current values for a full size electrofishing boat. So there may be mistakes in either logic or math. I invite you to question both and to correct me where needed. My email address is dean.efishing@gmail.com.

The first assumption is that resistance scales down from the model to the full size as to the scale value itself. The second assumption is that power scales up from the model to the full size as to the square of the scale value. I’m most concerned about the latter assumption. A respected colleague suggested that power would scale up as to the cube of the scale value; and that makes sense in terms of the model increasing in three dimensions to that of the full size boat. The idea of scaling up power as to the square of the scale value came to me as intuition, and I can’t fully explain why it did. Reportedly, Albert Einstein said that “intuition is nothing but the outcome of earlier intellectual experience.” Perhaps the basis of my intuition was my thought that voltage and current should both scale up as some function of the scale factor. Therefore, power – the product of voltage and current in a completely resistive circuit such as for an electrofisher – should scale up as some function of the scale factor squared. Now it is time to look at some results.

First, we measured some dimensions of the model boat to determine the scale. Its length was 79 cm without the anode booms, so the scale was 1:6 in relation to a 16 ft. full size boat. Next, we measured ambient conductivity and temperature of the pond water with a Hanna DiST 5 EC meter. The results were 275 µS/cm at 28.6 C. After placing the model boat in the pond with both anode arrays deployed as in a typical electrofishing configuration on a full size boat, we applied 151 volts of pulsed direct current to the model boat with an MLES Infinity Xstream backpack electrofisher. The peak current was 2.2 amperes and the peak power was 332 watts. We used the power and voltage values to calculate current to three significant figures. Doing so, current was 332 watts divided by 151 volts = 2.20 amperes, perfect agreement with the peak ammeter reading. All of these measurements were done with the Xstream peak metering. The resistance in this typical fishing configuration was 68.6 ohms for the model boat. If we let SC be the scaling factor, 6 in this case, the resistance for a full size boat, R = Rsc/SC. Dividing 68.6 ohms by 6 equals 11.4 ohms for a full size boat in 275 µS/cm water. Multiplying that by 275 µS/cm and dividing by 100 µS/cm provides the estimated resistance of a full size boat in 100 µS/cm water, a value called the R100. In this case, it was 31.4 ohms; and that is a realistic R100 value for a 16 ft. electrofishing boat with a clean hull.

Model boat in actual fishing pond during actual electrical measurements. Left to right are Jan Dean, Alan Temple, Jim Reynolds (standing), Shawn Banks and Seth Stonum. Paul Horner took the photo.

We then measured resistance (voltage divided by current) of the model boat with only one anode array in the water. By doing so, and then using a few calculations, we estimated the resistance values for an anode array, for the boat hull cathode and the percentage resistance (therefore power) to the anode for this model boat. These values were adjusted as above to estimate the values for a full size boat in 100 µS/cm water. The A100 and C100 values were 33.9 and 14.5 ohms, respectively, and we estimated 54% power to the anode. All of these values are consistent with expectations for a full size electrofishing boat using this configuration and relative sizes of anodes and cathode. So far, so good.

Now for the question of how to scale up the power, voltage and current. The assumption is that the power to a full size boat, P = Psc x SC^{2}. The power applied to the model boat, which we thought was higher than needed to immobilize small fish, was 332 watts. Therefore, the corresponding power for a full size boat would be 332 x 6^{2} = 11,952 watts. The high voltage used for the resistance determination was used to reduce round-off error in those calculations. In the aquarium, it was known that 80 volts was enough to immobilize fish near the bow and anodes of the model boat. That equates to a model power of 93 watts which scales up to about 3350 watts for a full size boat using the square of the scaling factor, and that amount of power is plausible for electrofishing. Using the scaling factor cubed, the 80 volts and 93 watts for the model boat equates to an unbelievably high value of 20,088 watts for a full size boat. Whereas this proves nothing, it gives credence to the idea of scaling up power using the square of the scaling factor.

Rearranging Joule’s formula for power, voltage *V* = sqrt (P x R) and current *I* = sqrt (P/R), where P is power and R is resistance. However, it is assumed that R = Rsc/SC. Therefore, to scale up voltage and current requires more than just multiplying the model values by the scaling factor SC. If the power scaling assumption is correct, then voltage *V* = sqrt(Psc x SC^{2} x Rsc/SC) = sqrt(Psc x Rsc x SC). Similarly, current *I* = sqrt(Psc x SC^{2}/(Rsc/SC)) = sqrt(Psc /Rsc x SC^{3}). Here is where I most need verification in both logic and calculation. For our example in the pond with the high voltage applied for the resistance determination, *V* = sqrt(332 watts x 68.6 ohms x 6) = 370 volts and *I* = sqrt(332/68.6 x 6^{3}) = 32.3 amps.

A check on the calculations is P = 370 volts x 32.3 amps = 11,951 watts, the same answer as just scaling up the power according to the square of the scaling factor. Again, that is no proof of the power scaling assumption, but it confirms the algebra for scaling up the voltage and the current.

Now let’s make an estimate of the settings for fishing with a full size boat in 275 µS/cm water. Based upon some empirical findings from multiple electrofishing classes and some relationships derived from those, let me assert that one should be successful fishing with a full size boat having two anode booms and with a square pulse shape and with an effective frequency and duty cycle when using 17.0 amps in 275 µS/cm water. For the scaled resistance in this example, that equates to 195 volts and 3312 watts for a full size boat. Those values equate to scale model values of 1.16 amps, 79.3 volts and 92 watts.

Above is a graph of the maximum output of peak power for both the model boat and for a full size boat versus the resistance of each over a range of water conductivity. From left to right, the ascending lines are for voltage and the descending lines are for current. The curved lines are power goal lines for each boat over a wide range of water conductivity. The upper curved red line is for the full size boat, and the lower curved black line is for the model boat. The lowest point on each curve, with their associated dots, represent 115 µS/cm water for each boat whereas the dots to the right represent 275 µS/cm water. Both the resistance and power values are on a log scale. Between 2 and 20 ohm, each vertical line represents 2 ohms; between 20 and 200 ohms, each vertical line represents 20 ohms. On the upper curved line for the full size boat, the rightmost dot represents 11.4 ohms and 3312 watts. On the lower curved line for the model boat, the rightmost dot represents 68.6 ohms and 92 watts.

We also measured the voltage profile at 1 cm intervals lateral to the starboard anode array with a scopemeter when 150 volts of direct current was applied. The volts per distance data were fit to a power regression with vertical offset. The first derivative of that equation was used as the estimate of field intensity or voltage gradient in V/cm per distance from the anode center. Those results were adjusted down linearly from the actual 150 volts applied for the voltage profile to the 79.3 volt goal from the equation above for this model boat. I estimated that the target voltage gradient for small (think scale model) fish at 275 µS/cm is 0.94 V/cm. The voltage gradient profile crosses the target voltage gradient line at 15.7 cm for the model boat. For a full size boat, we often find that the voltage gradient profile crosses the target voltage gradient line at about 90 cm from the anode array center. Scaling the distance up from the model boat to the full size boat equaled 15.7 cm x 6 = 94 cm. I can’t be certain about the target voltage gradient of 0.94 V/cm for the scale model fish, but it is based on a power density at match of 200 µW/cc, a value we typically use for small (two inch) stream fish, so I thought it may apply here. There was close agreement between the typical 90 cm distance for a full size anode array and the scaled up 94 cm found here.

Measurement of the voltage profiles of the model boat in the pond. Left to right are Jan Dean, Jim Reynolds and Alan Temple.

Another view of Alan Temple and Jan Dean taking voltage profile data for the model boat in the pond.

Voltage profile data for the model boat with 150 volts applied and with a fitted power regression model.

Voltage gradient profile as calculated from the first derivative of the voltage profile shown above and adjusted for the voltage goal of 79.3 volts. The red line is an assumed voltage gradient goal of 0.94 V/cm for the model boat in the pond. The lines crossed at 15.7 cm from the center of the model boat anode array.

Well, there you have it. The results seem plausible to me. I hope there is enough information provided for you to assess both the logic and the calculations. I would prefer to be shown the error of my ways and to learn something, so let me know if you are aware of the correct way to scale up voltage, current and power.

Let me thank Jim Reynolds for recording the data and Alan Temple for helping measure the voltage by distance data. Addison Banks assisted with those lateral voltage by distance measurements by running home and providing a non-conductive metric ruler. Paul Horner took the photos at the pond and helped in many ways at the workshop. My special thanks to Shawn Banks for building this remarkable scale model boat for use in the workshop and for allowing us to make these measurements at his pond.