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In my text book, I came across a problem where I needed to determine the equivalent reactance of a capacitor and an inductor (its ohmic resistance is negligible). To solve my problem I started to determine the equivalent reactance of both of them and to neglect the ohmic resistance of the inductor, and I thought of doing this:

Z = √Xl²−Xc². (Where: Z is the impedance, Xl is the inductive reactance and Xc is the capacitive reactance.)

But in the problem statement, Xc is 3Xl. So, putting them in the above equation is like:

Z =√Xl²-9Xl² =√-8Xl²

Which has no sense because of the negative sign. So my question is: What would be the equivalent reactance of an inductive and capacitive reactance? Is my approach correct but missing something, or it is wrong?

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    \$\begingroup\$ The impedance is a complex number. So you should use complex math with addition and subtraction, that's it. \$\endgroup\$ – Marko Buršič Mar 23 '17 at 14:25
  • \$\begingroup\$ You should read into what is called "complex numbers". \$\endgroup\$ – 12Lappie Mar 23 '17 at 14:26
  • \$\begingroup\$ \$X_C\$ = 3\$X_L\$. So Z = \$X_{NET}\$ = \$X_C\$ - \$X_L\$ = 3\$X_L\$ - \$X_L\$ = 2\$X_L\$. As for the complex numbers, go there if you understand how it applies. But most circuits can be solved using trigonometry and pythagoras. \$\endgroup\$ – StainlessSteelRat Mar 23 '17 at 15:23
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Basically the magnitude of the net impedance is the difference between the two impedances and you can ignore the sign if it becomes negative. Taking the square root of the difference of the squares is fundamentally wrong because capacitive and inductive reactances share the same axis: -

enter image description here

The above picture shows an inductor (reactance 300 ohms) in series with a resistor of 400 ohms. If you were trying to calculate the net impedance of R and XL then square rooting the sum of the squares is appropriate.

If the reactance (vertical axis) becomes negative then this is due to the dominant impedance being capacitive rather than inductive. An inductor in series with a capacitor produces an impedance somewhere on the vertical axis and, when the two impedances have identical magnitudes they cancel to zero. this is called series resonance. You might have heard of it.

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  • \$\begingroup\$ OK, do you say that I shouldn't take the square root?! Should I simply add them to each other, as this: Z= 3Xl-Xl =2Xl \$\endgroup\$ – Asmaa Mar 23 '17 at 17:23
  • \$\begingroup\$ @Asmaa yes that is what you do with reactances. If you had two capacitors in series then you add the reactances. Ditto two inductors in series but because inductive reactance and capacitive reactance share the same graphical axis, simple straight subtraction is all you need to do. You need to use Pythagoras only when you have quadrature impedance like R and L or R and C. \$\endgroup\$ – Andy aka Mar 23 '17 at 18:21

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