# Can we define rms current for pure inductive and capacitive ac circuit?

For a purely inductive and capacitive ac(sinusoidally varying) circuit, average power dissipated across a capacitor or an inductor is zero. Then does that mean RMS current is zero?

No. The RMS current cannot (integral of a positive function).

The circuit receives power for half a period and returns it during the other half period so that the average power is zero.

• What about RLC circuits where inductor, capacitor and resistance are in series? Can we define RMS current for RLC circuits? – sheshin Jan 29 '19 at 7:34
• You can always define RMS current : $\sqrt{\left\langle {{i}^{2}} \right\rangle }=\sqrt{\frac{1}{T}\int\limits_{T}{{{i}^{2}}dt}}$ In the case of the RLC circuit, the circuit receive power for more than half a period and returns it during less than half period so that the average power is positive (Joule effect in the resistor). – Vincent Fraticelli Jan 29 '19 at 7:45
• But as mentioned by @Steeven in the other answer, mathematically speaking RMS current cannot be zero. Doesn't this lead to a contradiction? Or is it that RMS values are defined only for purely resistive circuits and their values are then used as references for the ac sources? – sheshin Jan 30 '19 at 7:14
• Maybe you are making a confusion between zero power and zero current ? You can have a non zero rms current and a non zero rms voltage but a zero mean power. It is important to note that we are speaking of mean power : averaged over a period. instantaneous power is non zero even for an inductor or a capacitor. – Vincent Fraticelli Jan 30 '19 at 7:19

No. The RMS value means root-mean-square. Read it from right to left and there is the recipe:

1. Square the values: $$x_i^2$$. Now, non are negative (what was negative has now been flipped to the positive side).
2. Take the arithmetic mean/average of these squared values: $$\sum x_i^2/n$$ ($$n$$ is the number of values). Since no values are negative, this average can't ever become zero.
3. Take the square root $$\sqrt{\sum x_i^2/n}$$. This sort-of "cancels out" the initial squaring, so to say. The final result is a number that is something like an average that ignores any negative signs there may have been to begin with.

$$RMS=\sqrt{\frac{\sum x_i^2}{n}}=\sqrt{\frac{x_1^2+x_2^2+\cdots +x_n^2}{n}}$$

This RMS value can never be zero unless all the values $$x_i$$ are zero. If just one of them is non-zero (regardless if positive or negative), the RMS becomes non-zero (and non-negative).

• You have given a mathematical explanation, I need an explanation in terms of physics. Thank you. – sheshin Jan 29 '19 at 7:33
• @shesin I'm not sure what you are looking for. RMS is a mathematical representation of something physical. There are moments of high current, zero current, high opposite current (negative). The variation is called a sine curve. It is not easy to calculate much with a sine curve; sometimes we need some kind of a single number that represents the sine curve. You can pick the arithmetic or geometric average, the RMS, the amplitude, the peak-to-peak (double amplitude) etc. No matter what you pick, you are now working with a mathematical representation and not with the original true behaviour. – Steeven Jan 29 '19 at 7:45
• thanks for the answer. RMS current in an AC circuit is defined as the equivalent dc current that leads to the same energy dissipation. For purely inductive or capacitive circuits, average power and hence average energy dissipated for a whole cycle is zero and hence if we equate the energies, it should mean RMS current is zero or undefined as there is no resistance in this case. But mathematically speaking RMS current cannot be zero if current at any instance is non zero (as mentioned by you). This leads to a contradiction. Can you kindly look into this. – sheshin Jan 30 '19 at 7:12
• @sheshin I think you are missing a small crucial detail in the definition. From Wikipedia citing A Dictionary of Physics: "For alternating electric current, RMS is equal to the value of the direct current that would produce the same average power dissipation in a resistive load." In a resistor, dissipated power adds up rather than cancel out for positive and negative currents. The average of AC power across a resistor is non-zero – Steeven Jan 30 '19 at 7:31