# Derivation of Ground Impedance

Did you ever wonder why we use 0.1 ohm (and sometimes 0.5 ohm) for the ground impedance value for plug-and-socket connected equipment? For years I wondered where that number came from. I asked all the experts I knew. I was referred here and there, but I never found the answer.

So, I started studying the grounding circuit. As with any such problem, I needed to put some bounds on the problem, state some operating parameters, and make some assumptions. For the purposes of this analysis, I assumed that the ground circuit of the equipment was truly connected to the ground system of the building installation. (This discussion does not consider the situation of the open ground.)

The grounding circuit, for the purpose of analysis, has three operating modes. The first mode is normal operation. In this mode, current through the body is prevented by the grounding circuit returning the leakage current directly to its source, thereby making the leakage current “inaccessible.”

The second mode is the first fault condition. In this mode, we presume a direct, zero-impedance short from the “live” conductor to the grounded parts of the equipment. The grounding circuit returns the fault current to its source, thereby causing the operation of an overcurrent device such as a fuse or circuit-breaker. Note that during the fault, the voltage on the grounded parts with respect to the local ground is one-half the mains voltage. This value of voltage is hazardous, either 60 volts for 120-volt systems, or 115 volts for 230-volt systems. For this mode, electric shock is prevented in the same manner as a GFCI, namely by very fast operation of the fuse or circuit-breaker to disconnect the voltage.

The third mode is the second fault condition. In this mode, we presume a finite-impedance short from the “live” conductor to the grounded parts of the equipment. The impedance of the fault is just slightly less than the maximum-time trip-current of the fuse or circuit-breaker. Recall that at twice rated current, fuses can take up to one minute to operate, and at four times rated current, circuit-breakers can take up to two minutes to operate. For this mode, electric shock is prevented by limiting the voltage on the grounded parts to less than 30 volts with respect to the local ground.

So, I stated this mode as a rule:

The IMPEDANCE

of the protective grounding circuit

shall be of such value that,

in the event of a fault,

the VOLTAGE

at any accessible part

with respect to the supply-circuit ground point

SHALL NOT EXCEED

THE VOLTAGE LIMIT VALUE (30 volts)

for longer than

the maximum time duration trip current of the overcurrent device.

The key to solving the circuit equations is knowing the maximum allowable voltage drop in the system.

Discussing this with an electrical consulting engineer, I learned that electrical distribution systems are designed for a maximum 6 percent voltage drop at the equipment (i.e., at the socket-outlet) at maximum load. He went on to say that typical voltage drop is more like 2 or 3 percent.

So, now we know all of the parameters necessary to solve the problem. Knowing the allowable voltage drop in the system, we can calculate the resistance of the distribution system. For a 120-volt, 15-amp system, with a maximum voltage drop of 6 percent (i.e., 7.2 volts), the resistance of the system is the voltage (7.2 volts) divided by the current (15 amps). This is 0.48 ohms.

Next, we assume that half the resistance is in the “line” conductor, and half the resistance in the neutral conductor. So, each wire has a resistance of 0.24 ohms.

Furthermore, since the building grounding system is wired exactly the same as the line and neutral conductors, we can assume the ground wire is
0.24 ohms.

Now, we can calculate the impedance of the equipment ground. We know that the maximum voltage under a fault condition that doesn’t immediately blow the fuse is 30 volts. We will assume that the maximum-time trip-current for a 15-amp branch circuit is 30 amps. The resistance of the grounded part of the equipment to “real” ground must be 30 volts divided by 30 amps, or 1 ohm. Since the ground wire is 0.24 ohm, the equipment ground impedance must be 0.76 ohm.

If we repeat the same calculations for a 15-amp branch circuit but with a maximum-time trip-current of 60 amps, then the equipment ground impedance must be 0.26 ohm.

I will leave it to you to calculate the equipment ground impedance for other trip-currents, other system voltage drops (e.g., 3%, 2%), and other voltages (e.g., 230 volts).

If you go through the calculations, you will find that:

1. As the overcurrent trip current increases, the equipment ground impedance must decrease to satisfy the 30-volt criterion.
2. As the system voltage drop decreases, the equipment ground impedance may increase and still satisfy the 30-volt criterion.
3. As the system nominal voltage goes up, the equipment ground impedance must decrease to satisfy the 30-volt criterion.

Now, will these equipment ground impedances satisfy the short-circuit fault to ground? That is, is the impedance sufficiently low to quickly operate the overcurrent device so as to limit the duration of voltage on the equipment ground?

Let’s look at the case of the greatest value of grounding resistance, 0.76 ohms. In this case, the maximum circuit current is the system voltage, 120, divided by the total circuit resistance, 1.24 (the sum of 0.24 + 0.76 + 0.24). The maximum current is 96.8 amps.

As the distribution impedance decreases, the short-circuit current increases. In 120-volt, 20-amp systems, with 3 percent system voltage drop, the short-circuit current will be about 250 amps.

I prepared a spread-sheet with all the variables and looked for the worst-case (least-value equipment grounding impedance) situation. The value of equipment grounding impedance is most critical when the system percent voltage drop is high. For example, for a 120-volt, 20-amp system, with 6% voltage drop, and 80-amp trip-current, the equipment grounding impedance must be no more than 0.2 ohms to hold the voltage to 30 volts.

For a 230-volt, 16-amp system, with 6% voltage drop and 64-amp trip-current, the equipment grounding impedance must be no more than 0.04 ohms to hold the voltage to 30 volts.

So, the value of 0.1 ohm is acceptable for virtually all 120-volt systems, and for all 230-volt systems where the system percent voltage drop at maximum load does not exceed 5 percent.

However, note that at higher fault currents, the voltage on accessible parts always exceeds 30 volts, and, at short circuit, always exceeds one-half the mains voltage.

So, the equipment grounding impedance is important, and its value, 0.1 ohm, is reasonable. But, in the event of a fault, and until the overcurrent device operates, an electric shock can occur from the grounded parts. Richard Nute is a product safety consultant engaged in safety design, safety manufacturing, safety certification, safety standards, and forensic investigations.

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