Finding Capacitance Required To Produce A given Reactance

Jsw123

Joined Jun 20, 2009
46
I am confused on a textbook I have. It suggests that if I know the frequency of a circuit and the required capacitive reactance, that I use the same formula to calculate capacative reactance but use instead the capacitive reactance instead of the capacitance. For instance, the formula for capacitive reactance is 1/2*pi*f*c. If I need 33.3 ohms of reactance at 60 hz, the book says I should calculate it as C=1/6.28*60*33.3. I was never taught to swap the ohms in the formula and don't see how that would work. I had thought the correct way to calculate this would be the equation 33.3 = 1/6.28*60*C. What am I missing?

ZCochran98

Joined Jul 24, 2018
192
The formula for (the magnitude of) capacitive reactance is, as you said:
$X_c = \frac{1}{2\pi fC}$
If you know the reactance you need at the appropriate frequency, all you have to do is solve for C - it just boils down to a little algebra. In this case, multiply both sides by $$C/X_c$$:
$X_c\cdot\frac{C}{X_c} = \frac{1}{2\pi fC}\cdot\frac{C}{X_c}$
$\rightarrow C = \frac{1}{2\pi f X_c}$
You can do the same thing with any other equation to solve for any of the variables if you know the others.

• Papabravo

Papabravo

Joined Feb 24, 2006
17,580
As a suggestion, it might be instructive for the TS to do a dimensional analysis to see if the agebraic swapping preserves the identity of the units. That would be a confirmation of the validity of the algebra.

• ZCochran98

Papabravo

Joined Feb 24, 2006
17,580
It may also be hepful to notice that for inductive reactance it is directly proprtional to inductance and frequency. If you graph it for a constant value of inductance you get a straight line that goes off to infinity. Capacitive reatance is inversely proportional to both capacitance and frequency. If you graph it you get a hyperbola with asymptotes of the positive x and y axes.

So much for theory. If you ever get a look at what real componets will do as you sweep the frequency on a VNA you will see that they are far from ideal in their behavior. Inductors at high frequency start to look like they have small capacitors across each winding and capacitors start to look like the have small inductors around the edges. It is hard to imagine how wierd things can get until you see it with you own lyin' eyes. They actually switch their behavior back and forth as the frewquency continues to increase.