# Circuit Resonance

When the frequency of the applied e.m.f. equals the natural frequency of the electrical circuit the current reaches the maximum value and the circuit is said to be in resonance with the in resonance with applied e.m.f.
I am confuse with the natural frequency of the electrical circuit. What is this mean?

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Simon Bridge
Homework Helper
The driving frequency at resonance is not the same as the natural frequency - though it is very close to it. The natural frequency is the frequency at which the circuit would oscillate without an oscillating source driving it.

CWatters
Homework Helper
Gold Member
Inductors and capacitors are energy storage devices. If you have a circuit comprising an Inductor and capacitor in series then it's possible for energy to flow back and forth between the capacitor and inductor. It does this most easily at the natural frequency because at the natural frequency the total impedance of the circuit is at a minimium. Therefore at this frequency the losses in the circuit are at a minimium.

You probably know that the impedance of an inductor and a capacitor varies with frequency. For an inductor the impedance increases with frequency. For a capacitor it reduces with frequency. At some frequency the sum of the two impedances is a minium. That's the natural frequency at which the circuit will prefer to resonate.

http://en.wikipedia.org/wiki/RLC_circuit#Natural_frequency

tiny-tim
Homework Helper

are you asking, i] what is the resonant frequency,

or are you asking, ii] what is the difference between the resonant frequency and the natural frequency?​

i] as you know, the impedance of the circuit is different at different frequencies,

and the resonant frequency is the one at which the impedance is real (no reactance)

ii] uart has explained it here …
The presenter was most likely referring to the damped natural frequency versus the undamped natural frequency. When the damping factor is small the two are very similar, but as the damping is increased the oscillation frequency decreases.

For example in a parallel LRC circuit the undamped natural frequency is :

$$\frac{1}{\sqrt{LC}}$$

whereas the actual oscillation frequency of the natural response's damped sinusoid is :

$$\sqrt{\frac1{LC} - \frac{1}{(2RC)^2}}$$
Resonance is a condition in which a vibrating system responds with maximum amplitude to a periodic driving force.
Mechanical systems (beams, pendula, springs, wine glasses, guitar strings etc) will have a number of possible frequencies at which this occurs. These are the system's natural frequencies of vibration. For example, a guitar string will have a series of possible frequencies where this happens, the lowest is called the fundamental frequency. The other frequencies are at values which are whole number multiples of the fundamental.
When resonance occurs, the frequency is often called a resonant frequency. This is just saying that resonance occurs when the driving force has the same value as one of the natural frequencies.
A beam can have more than one natural frequency, and therefore can be made to resonate at more than one frequency.
An (LC) series electrical circuit will resonate at a frequency given by f= (1/2π)√LC
This could be called its natural frequency or its resonant frequency. It doesn't really matter. (It's usually called its resonant frequency.)
The characteristic equation for a second order system is of the form :

$$s^2 + 2 \alpha s + w_0^2$$

For example in a parallel LRC circuit this would correspond to a function of the form :

$$s^2 + \frac{1}{RC} s + \frac{1}{LC}$$

If the damping factor (alpha) is zero then the roots are at

$$\pm j \sqrt{1/(LC)}$$

and it follows that the natural response is an undamped sinusoid of frequency 1/sqrt(LC).

When alpha is non zero the roots of the (quadratic) characteristic equation are

$$-\alpha \pm j \sqrt{(w_0^2 - \alpha^2)}$$

from which it follows that the natural response is a damped sinusoid of frequency $\sqrt{(w_0^2 - \alpha^2)}$.

The driving frequency at resonance is not the same as the natural frequency - though it is very close to it.
What do you mean by this? That confuses me. I almost want to disagree with you, but there could be something I'm confused about or I might not understand what you mean.

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AlephZero
Homework Helper
The driving frequency at resonance is not the same as the natural frequency - though it is very close to it. The natural frequency is the frequency at which the circuit would oscillate without an oscillating source driving it.
What do you mean by this? That confuses me. I almost want to disagree with you, but there could be something I'm confused about or I might not understand what you mean.
The first statement is a bit confusing. Let's try a longer version.

The frequency of the (damped) oscillations of a system with no exterrnal forcing depends on the amount of damping. "Natural frequency" usually means the theoretical frequency with zero damping, even though no real systems have zero damping. The change in frequency is very small if the damping is low. The relative frequency change is ##\sqrt{1 - \beta^2}## where ##\beta## is the damping factor, so even if ##\beta = 0.2## the change in frequency is only about 2%. But for high damping levels approaching critical damping the frequency change is NOT small, and if the damping is higher than critical (##\beta > 1##) there are no "oscillations" at all unless there is an external force.

There is also an issue about what you really mean by "at resonance". You can define that as the frequency when the applied force is exactly 90 degrees out of phase with the response. Or you can define it as the frequency where the response is a maximum, for a constant level of force input. The two are not necessarily the same.

For oscillating systems with small amounts of damping, none of this is very important in practice, so people often write as if "natural frequency" = "resonant frequency", rather than being precise about what they mean.

Mechanical systems (beams, pendula, springs, wine glasses, guitar strings etc) will have a number of possible frequencies at which this occurs. These are the system's natural frequencies of vibration. For example, a guitar string will have a series of possible frequencies where this happens, the lowest is called the fundamental frequency. The other frequencies are at values which are whole number multiples of the fundamental./QUOTE]

Hi Tim

Well now I am understanding this. As you talk about the string vibration it has ample number of natural frequencies at which it gives maximum sound. Same happens with the air pipes also like the vessels. If if we talk about an LC oscillator there we can find only one value of natural frequency. Is there only one value for which the current in the circuit becomes maximum or there can be a series of the frequencies?

Simon Bridge
Homework Helper
Thanks @AlephZero. I should have said that the resonant frequency is not usually the same as the natural frequency of the circuit.

tiny-tim
Homework Helper
… if we talk about an LC oscillator there we can find only one value of natural frequency. Is there only one value for which the current in the circuit becomes maximum or there can be a series of the frequencies?
for one LC oscillator, only one frequency (you can check that easily by solving the equations)

(for a complicated circuit with several oscillators, i don't know )

sophiecentaur
Gold Member

for one LC oscillator, only one frequency (you can check that easily by solving the equations)

(for a complicated circuit with several oscillators, i don't know )
A well designed oscillator circuit will ensure that the basic LC components (or crystal) have high Q, intrinsically) and that the 'coupling' between the resonator and the amplifier / feedback part is very loose. This ensures that the oscillator is affected as little as possible by other influences.
Mechanical clock mechanisms have the same requirement, with the pendulum / balance wheel being as isolated as possible from the source of energy (spring / weights) and the losses in the oscillating bit kept to a minimum (jeweled bearings / pallets etc). A good clock will keep going for many cycles if you remove the drive to it.

When crystals are used in a circuit, they may be driven at overtone frequencies and not at their fundamental - but the oscillation of a bar of quartz is not equivalent to a simple LC combination.