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Resonance in AC Circuits

concept
Imagine a circuit with a capacitor and an inductor where all of the reactive energy stored and released by one is then absorbed and released by the other back and forth forever. We call this "resonance" and a similar phenomenon occurs in all fields of science and engineering. When a circuit is resonating our voltage or our current reaches peak levels, this can be good in some cases and expensive or dangerous in others. For a given circuit resonance only occurs at a single frequency. This topic will cover how resonance comes about, how to find the resonant frequency, and just how large things will get when a circuit is resonating.
You can get resonance in either a series capacitor and inductor combination or a parallel combination. In both cases the reactive energy is passed back and forth between the two indefinitely however the two different forms have different effects on the circuit's voltages, currents and impedances.
fact
For series resonance our capacitor and inductor are connected in series like below: The impedance of a series RLC combination is $$Z = R + j(\omega L - \frac{1}{\omega C})$$ If we plot the magnitude of this using nominal values for the components ($$R = L = C = 1$$) we get: And we can see that the impedance reaches a minimum value of R, this happens when the impedance of the capacitor cancels the impedance of the inductor. The frequency of this minimum we call the "resonant frequency". When the impedance is at a minimum the current will be at a maximum.
fact
The resonant frequency of a series LC circuit is given by: $$f = \frac{1}{2\pi \sqrt{LC}}$$Hz
derivation
Since we know resonance occurs when $$Z = R + j(\omega L - \frac{1}{\omega C})$$ is at a minimum we'll use some good old fashioned calculus to find the $$\omega$$ that minimises $$Z$$. $$\frac{d}{d\omega}Z = j(L - \frac{-C}{\omega^2 C^2}) = j(L + \frac{C}{\omega^2 C^2})$$ Setting this equal to zero: $$L\omega^2 = -\frac{1}{C} \implies \omega^2 = -\frac{1}{LC}$$ Now a negative frequency is the same as a positive frequency here so we can drop that pesky negative sign and continue: $$\omega = \frac{1}{\sqrt{LC}}$$ And finally we convert the radian frequency to hertz to find: $$f = \frac{1}{2\pi \sqrt{LC}}$$
example

Find the resonant frequency of the following circuit: Plugging these values into our trusty formual we find: $$f = \frac{1}{2\pi \sqrt{470\cdot 10^{-6}\cdot 1\cdot 10^{-3}}}$$ $$f = 232$$Hz At resonance the impedance is $$Z = R$$ since the capacitor's and inductor's impedances cancel. When $$Z = R$$ our current is: $$I = 0.1\sin(2\pi \cdot 232t)$$
fact
In a series RLC circuit at resonance the voltages across the inductor and capacitor can become many times higher than the source voltage.
fact
In a parallel resonant circuit the capacitor and inductor are in parallel as below: For resonance we'll have the impedance becoming a maximum. The impedance of the parallel resonant circuit is given by: \begin{align} Z & = R + \frac{\frac{j\omega L}{j\omega C}}{j(\omega L - \frac{1}{\omega C})} \\ & = R - j \frac{L}{C}\frac{1}{C[\omega L - \frac{1}{\omega C}]} \\ & = R - j \frac{L}{C}\frac{\omega C}{\omega^2 LC - 1} \end{align} We need to find where this impedance maximises so setting the derivative equal to zero: \begin{align} \frac{d}{d\omega}Z & = -j\frac{L}{C}\frac{\omega^2LC^2-C-2\omega^2LC^2}{(\omega^2LC-1)^2} \\ & = 0 \\ -\omega^2LC^2-C & = 0 \\ \omega^2 = -\frac{1}{LC} \\ \end{align} As we said above we can drop the negative sign since in our discussions the difference between positive and negative frequencies is irrelevant. So $$\omega = \frac{1}{\sqrt{LC}}$$. Which is exactly what we got for the series case
fact
The resonance frequency of a parallel resonant circuit is given by: $$f = \frac{1}{2\pi \sqrt{LC}}$$ Just like in a series resonant circuit.
fact
In a parallel resonant circuit the impedance has a maximum at the resonant frequency which maximises the voltage across the components.
For $$R = L = C$$ a plot of the impedance is: fact
In a parallel RLC circuit at resonance the magnitude of the currents through the capacitor and inductor can be many times higher than the rest of the circuit's current.
Summary
• Resonance occurs when the reactance of both the inductor and capacitor are equal: $$\omega L = \frac{1}{\omega C}$$
• Resonant frequency is the same for both series and parallel LC combinations: $$f = \frac{1}{2\pi \sqrt{LC}}$$Hz
• Series resonant circuits have their impedance at a minimum at resonance and produce a peak voltage across the reactive elements at this frequency
• Parallel resonant circuits have their impedance at a maximum at resonance and produce a peak current across their reactive elements at this frequency
practice problems