02 March, 2007

Parallel and Series Resonance

Parallel Resonance :
A condition of resonance will be experienced in a tank circuit when the reactances of the capacitor and inductor are equal to each other. Because inductive reactance increases with increasing frequency and capacitive reactance decreases with increasing frequency, there will only be one frequency where these two reactances will be equal.



In the above circuit, we have a 10 µF capacitor and a 100 mH inductor. Since we know the equations for determining the reactance of each at a given frequency, and we're looking for that point where the two reactances are equal to each other, we can set the two reactance formulae equal to each other and solve for frequency algebraically:
XL = 2*pi*f*L Xc = 1/2*pi*f*c

Equating these two and solving for frequency f gives

a formula to tell us the resonant frequency of a tank circuit, given the values of inductance (L) in henrys and capacitance (C) in Farads. Plugging in the values of L and C in our example circuit, we arrive at a resonant frequency of 159.155 Hz.

What happens at resonance is quite interesting. With capacitive and inductive reactances equal to each other, the total impedance increases to infinity, meaning that the tank circuit draws no current from the AC power source! We can calculate the individual impedances of the 10 µF capacitor and the 100 mH inductor and work through the parallel impedance formula to demonstrate this mathematically:

The resonant frequency we obtain calculating with above formula is 159.155 Hz.
Calculating XL and Xc with this frequency values are XL = Xc = 100 Ω
Now, we use the parallel impedance formula to see what happens to total Z:
ZParallel = 1 /(1/ZL + 1/Zc)

Calculating impedance with above formula we get
ZParallel = 1/0 Undefined!
We can't divide any number by zero and arrive at a meaningful result, but we can say that the result approaches a value of infinity as the two parallel impedances get closer to each other. What this means in practical terms is that, the total impedance of a tank circuit is infinite (behaving as an open circuit) at resonance.
Series Resonance :
A similar effect happens in series inductive/capacitive circuits. When a state of resonance is reached (capacitive and inductive reactances equal), the two impedances cancel each other out and the total impedance drops to zero!

At f = 159.155Hz ZSeries = ZL + Zc = 0Ω.
With the total series impedance equal to 0 Ω at the resonant frequency of 159.155 Hz, the result is a short circuit across the AC power source at resonance

For detail calculations visit the following site:
http://www.allaboutcircuits.com/vol_2/chpt_6/2.html

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