# Impedance of an RLC series circuit when in resonance

• fluidistic
In summary, the impedance of an RLC series circuit is given by \frac{1}{\sqrt{R^2+\left ( \omega L-\frac{1}{\omega C}\right )^2 }} and in resonance, Z=\frac{1}{R}. For an RLC parallel circuit, the impedance is approximated by \frac{1}{\sqrt{\frac{1}{R^2}+\left ( \omega C-\frac{1}{\omega L}\right )^2 }} with Z=R in resonance. However, it is possible to define the impedance as a real quantity, Z = \sqrt{R^2 + \left(\omega L - \frac{1}{\omega
fluidistic
Gold Member
I don't find my notes right now, I'll try to use my memory on this.
According to my notes, the impedance of an RLC series circuit is given by $$\frac{1}{\sqrt{R^2+\left ( \omega L-\frac{1}{\omega C}\right )^2 }}$$.
So when in resonance, $$Z=\frac{1}{R}$$ instead of $$Z=R$$.

Also if I recall well, for an RLC parallel circuit, $$Z=\frac{1}{\sqrt { \frac{1}{R^2} } +\left ( \omega C-\frac{1}{\omega L}\right )^2 }$$ or something close to this, meaning that in resonance Z=R instead of Z=1/R.
Are my notes wrong? Or am I doing something wrong?

Your notes have to be wrong because all your impedances are real. The reactance caused by the inductive and capacitive elements give rise to imaginary impedances. The wikipedia article gives the results but they are trivial to work out yourself too.

Series: $$Z = R-i\omega L+\frac{i}{\omega C}$$
Parallel: $$\frac{1}{Z} = \frac{1}{R}+\frac{i}{\omega L}-i\omega C$$

where the time dependence is $$e^{-i\omega t}$$.

Actually, it is possible to define the impedance as a real quantity, the magnitude of the complex impedance,
$$Z = \sqrt{R^2 + \left(\omega L - \frac{1}{\omega C}\right)^2}$$
(for series circuits) This is sometimes done in introductory EM courses. It's not as powerful a concept as the complex impedance but it still allows you to do calculations.

fluidistic, I would imagine you should know that impedance is a quantity analogous to resistance, and specifically, it has the same units as resistance, which should have told you that
$$Z = \frac{1}{\sqrt{R^2 + \left(\omega L - \frac{1}{\omega C}\right)^2}}$$
couldn't be right.

Ok thank you both.
I'll have to check out my notes.
Indeed I know that in resonance the circuit is purely resistive and the impedance have ohm's units. That's why I doubted about my notes.

## 1. What is the formula for calculating the impedance of an RLC series circuit when in resonance?

The formula for calculating the impedance of an RLC series circuit when in resonance is Z = R, where Z is the impedance, R is the resistance, and L and C are the inductance and capacitance, respectively.

## 2. How does the impedance of an RLC series circuit change when it is in resonance?

When an RLC series circuit is in resonance, the impedance decreases to its minimum value, which is equal to the resistance of the circuit. This is because at resonance, the reactance of the inductor and capacitor cancel each other out, leaving only the resistance in the circuit.

## 3. What is the significance of resonance in an RLC series circuit?

Resonance in an RLC series circuit is significant because it allows the circuit to have maximum current flow and minimum impedance. This makes it ideal for use in various applications, such as in filters and oscillators.

## 4. How does the frequency affect the impedance of an RLC series circuit when in resonance?

The frequency has a direct relationship with the impedance of an RLC series circuit when in resonance. At resonance, the frequency is equal to the resonant frequency of the circuit, which is determined by the values of the inductance and capacitance. Any changes in frequency will cause the impedance to deviate from its minimum value.

## 5. Can an RLC series circuit be in resonance for different frequencies?

Yes, an RLC series circuit can be in resonance for different frequencies as long as the frequency is equal to the resonant frequency of the circuit. This means that by changing the values of the inductance and capacitance, the resonant frequency can be adjusted, allowing the circuit to be in resonance at different frequencies.

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