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Why is the impedance maximized at resonance in a parallel LRC circuit, whereas it is minimized in a series LRC circuit?

In both cases, is the impedance purely resistive on resonance?

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In a series circuit, the current that flows through the inductor ends up having to flow in the rest of the circuit- so you will see a finite current flowing when there is (almost) no voltage. this looks like a low impedance.

When you have the parallel circuit, the current can go around the LC loop without ever flowing into the rest of the circuit. this means no current flows in the rest of the circuit- so it looks like a large impedance.

Mathematically you can write the series resistance as

$$Z=j\omega L + \frac {1}{j\omega C} \\ \frac {1-\omega^2LC}{j\omega C}$$

using the complex impedance equation. it is easy to see this equals zero at resonance. if you add an additional series resistance (from the inductor) then that will be the resistance at resonance - so it will be real.

I will leave it as an exercise for you to use the expression for the impedance of inductor (first term) and capacitor (second term) in parallel and prove you end up with a 1/0 expression at resonance (remembering that $1/Z = 1/Z_1+1/Z_2$)

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In a series RLC circuit, you need to add the impedance of the resistor, capacitor and inductor. At high frequencies the inductor's impedance is dominant and the impedance of the circuit becomes very large, whereas at low frequencies the capacitor is the dominant one which increases the total impedance. Somewhere in the middle is a resonant frequency where the impedance is minimized to the resistor's impedance.

In a parallel RLC circuit, as a rule of thumb the lowest impedance determines the response of the circuit (more precisely, the "conductance" is the sum of the 3 components' conductance, so its exactly the opposite). At low frequencies the inductor's low impedance will influence the whole circuit's impedance, and at high frequencies the capacitor will let current run free. At some frequency in the middle, the resonant frequency, the impedance will be maximized.

Hope it helped!

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