# Why do inductive and capacitive reactances have opposite signs and why don't they cancel out each other?

For example, in a series RLC circuit with an AC source why is total reactance inductive reactance minus capacitive reactance? I don't need any mathematical proof. What is a logical answer for that?

• In the capacitor, the current is proportional to how fast the voltage across the capacitor is changing. We have opposite situation in the inductor. The voltage across the inductor is proportional to how fast the current is changing
– G36
Jun 11, 2017 at 9:17
• All this means that the capacitor is the "inverse" of the inductor. When the capacitor is fully charged we have 0 current and "full" voltage. In the inductor, we have the opposite situation. When "fully energize" the voltage is 0V but the current is at his max. Hance the capacitor and the inductor can exchange energy with one another without any additional current from the source. Please study the parallel resonant circuit. en.wikipedia.org/wiki/LC_circuit#Operation
– G36
Jun 11, 2017 at 9:28
• It is not totally exact say they cancel. They cancel only at one frequency. Recall total impedance (circuit in serial) imaginary part is not L minus 1/C, it is -1/Cw+Lw. Jun 11, 2017 at 10:26
• Total impedance equals the difference impedance of the reactances plus the resistance summed as Pythagoras would. You don't lose resistance along the way. Jun 11, 2017 at 11:25

If you applied a sine wave voltage across an inductor you would see that the current lags the voltage by 90 degrees. If you did the same with a capacitor, the current would lead by 90 degrees.

The picture below is a particular example of where the inductor and capacitor currents have the same magnitude (electrical resonance) and, are opposite by 180 degrees (one leads by 90 degress and the other lags by 90 degrees): - The values in the above example produce equal magnitude currents at 1kHz.

If you put them in parallel across the same 1 kHz supply, the currents would exactly cancel and there would be no current taken from the supply. This is the AC analysis and is called parallel resonance. Just in case you cannot see that, here is a picture that makes things a little easier on the eye: - Hopefully, if you look at the upper and lower red sinewaves you should see that the two currents are exactly antiphase (180 degrees apart) and the net current is zero.

If you now put them in series they share a common current but, the voltage on the inductor will lead by 90 degrees whilst the voltage on the capacitor will lag by 90 degrees i.e. the two voltages are exactly 180 degrees out of phase.

When you have two series voltages that are exactly out of phase by 180 degrees the net voltage across the pair is zero. Can you see this?

It's the same with two batteries - if you have two 9 volt batteries connected in series but one is the wrong way round, the net voltage produced is exactly zero volts.

If you could make a resistor that had a negative resistance of "R" and put it in series with a conventional resistance of "R" then, the total series impedance is R + (-R) = 0. If you put those two resistors in parallel you would get infinite resistance. You can check this easily by using the "product over sum" calculation for parallel impedances/resistances.

• Could be you must clarify that these draws are only valid for the frequence w=sqrt(1/LC) Jun 11, 2017 at 10:33
• @pasabaporaqui yes, in the answer above, I focused on the resonant frequency where XC equals the magnitude of XL. I did mention "resonance" in my answer. Jun 11, 2017 at 11:21
• Yes, I known, but I think some reader can appreciate an explicit reference. Otherwise, values 1 kHz, 470uF and 54mH seems irrelevant and unrelated Jun 11, 2017 at 11:28
• Ok added a couple of extra words. Jun 11, 2017 at 12:01