Thank you for your question regarding resistance-to-ground calculations, it is our pleasure to help.
It sounds like you are trying to conduct hand calculations which is indeed a very limited and frustrating process. It is also highly inaccurate. In particular, hand calculations cannot take into account three (3) key factors: Complex ground designs, multi-layer soil, and frequency.
As you point out in your email, there are no readily available formulas to hand calculate for ground rings with ground rods. This is because such a formula would be a multi-staged process and would require a white paper just to explain how to conduct the formula. But that really isn’t the issue with hand calculations you should be worried about. Your concern for hand calculations is that the formulas found in the Green Book assume a uniform soil resistivity, which simply does not exist in nature. Therefore, you can never accurately calculate even the most simple of electrode structures using these formulas, let alone trying to hand calculate a complex grounding system.
Let’s consider the calculation for a single (1) 10-ft ground rod. Hand calculations almost never get the actual resistance right because the formulas don’t account for changes in soil resistivity with depth. There have been several studies conducted that show that the typical soil is best modeled using 3 to 5 layers, with the top 3 layers generally occurring within 10-ft of the surface. We often see 3 to 4 layer models, so as to leave the 5th layer available for changes in soil resistivity due to frost line issues during winter conditions (when soil freezes, the resistivity of the soil increases by a factor of at least 10x). This means that your 10-ft ground rod will often have 3 to 4 different soil resistivities along its length, and will have a higher resistance-to-ground during winter. That said, you would need to calculate the change in resistance of the electrode as the soil resistivity changes along its length at depth. This means that you really need to account for leakage current rates and voltage drops across the length of the electrode.
Of course, keep in mind that you are probably not even dealing with Direct-Current (0 Hz), your ground system is probably being used for a 60 Hz system, which means that you actually should be concerned with impedance-to-ground. Of course frequency based calculations are not discussed in the Green Book at all. And yes, impedance makes a huge difference in the final calculations. You also need to account for the material properties of your grounding system. The horizontal conductors of the ring will almost certainly be made of copper, while the ground rods will be made of a low-grade steel. Copper is 12 to 17 times (based on the steel) more conductive than steel, and is at least 250 times less magnetic. Proper calculations will take into account these two dissimilar materials and the change that the conductivity and magnetism will have on the ground system.
The point of the above information is to demonstrate the futility of hand calculations. The formulas in the green book are for theoretical use only; they assume direct-current electrical systems in imaginary uniform soil conditions. The only accurate way to calculate even a simple ground electrode, is by using a computer program designed for this task. We recommend and use the CDEGS computer program from Safe Engineering Services www.sestech.com
Now, if you are simply insistent on hand calculations, you would need to calculate the ground ring by hand, then calculate the ground rods separately, and then combine the two by adding resistances in parallel. That is the best you can do. The problem with adding resistances in parallel is that it does not take into account resistance loss due to overlapping spheres of influence, so we wouldn’t recommend that you show the results to your customer. But then again, we wouldn’t show any calculation from the Green Book to a customer.
There is really a lot to know about this subject. If you have further questions, please feel free to call us at 310-318-7151 and someone will be happy to speak with you about your project, free of charge.
The Engineering Team at E&S Grounding Solutions
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