Energy gain in a resonant system

In summary, the conversation discusses the concept of passive systems with resonance, particularly the gain of a second-order system near resonance. The individual is confused about how a passive system, such as an RLC circuit, can have a gain greater than 1. The expert explains that this is possible due to impedance changes and compares it to the amplification of voltage in a transformer. They also mention the use of amplifiers to maintain the gain in resonant circuits.
  • #1
maldata
3
0
Something is bothering me about passive systems in resonance. Consider the Bode plot of a second-order system, for example:

http://en.wikipedia.org/wiki/File:Harmonic_oscillator_gain.svg

Let's say we excite this system near its resonance, so the system has a gain > 1. I don't understand how this can be the case for a passive system like an RLC circuit. If the output amplitude is greater than the input amplitude, that energy has to come from somewhere.

What is it that I'm misunderstanding here? It seems like in a passive circuit, the output amplitude should, at most, be the same as the input amplitude. So how is there a gain of greater than 1?

Thanks.
 
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  • #2
Remember the power (and therefore energy) is not determined by one quantity alone.

Your graph is for the amplitude of a single quantity.

Look at what happens to the other quantities in one of the energy or power equations, particularly the impedance.
 
  • #3
Hi Studiot, thanks for the reply.

I'm still not sure how that matters, though. In that plot, the gain at resonance is 100. So, say this is an underdamped RLC circuit. We put in a sinusoidal voltage with amplitude 1 at the resonant frequency, and we get out a sinusoidal voltage with amplitude 100. Regardless of what any of the other quantities are, we've just amplified a signal passively. I don't see how that's possible.

Other passive circuits make sense to me... an RC circuit or an overdamped RLC circuit never have gain greater than 1. So how can an underdamped RLC circuit do this?

Obviously there's still some concept I'm missing. Thoughts?

Thanks!
 
  • #4
Think of a (perfect) transformer.

24 volts in 240 volts out.

Does it amplify the voltage?
Well yes.

Does it amplify the energy or power
Well no, because the current available is less, by the same factor (1/10 in this case)

Furthermore this may be viewed as an impedance transformation, from low impedance to high impedance, hence the lower current.

Your resonant circuit is the same. You will get out a higher voltage, but at higher impedance, so the minute you apply it to a load you will not be able to draw the same level of current as the original signal could support.

It is difficult to be more specific without specific values, so work a few out to see.
 
  • #5
Ah. There's the analogy I was looking for... that seems a little more intuitive to me. Thanks for the insight!
 
  • #6
In fact with resonant systems the impedance change usually so great that we have to use an amplifier to prevent the rest of the circuit loading (damping) the resonance completely.
We often place the resonant circuit in the feedback path of the amplifier for this purrpose, alternatively we buffer the resonant circuit with an amplifier. Thus we can inject the extra energy to maintain the amplification at lower impedance.
 

1. How does energy gain occur in a resonant system?

In a resonant system, energy gain occurs through the phenomenon of resonance, where the system's natural frequency matches the frequency of an external force. This causes the system to vibrate with a larger amplitude, resulting in an increase in energy.

2. What factors affect the amount of energy gained in a resonant system?

The amount of energy gained in a resonant system depends on several factors, including the system's natural frequency, the frequency of the external force, and the quality factor (Q-factor) of the system, which represents the amount of energy that can be stored and released by the system.

3. Can energy gain in a resonant system be controlled?

Yes, energy gain in a resonant system can be controlled by adjusting the frequency of the external force or by changing the system's parameters, such as its natural frequency or Q-factor. This can be achieved through the use of feedback control systems or by designing the system with specific materials and dimensions.

4. Is energy gain always beneficial in a resonant system?

Not necessarily. While energy gain can be useful in some applications, such as in musical instruments or radio receivers, it can also lead to destructive effects in others. For example, bridge collapses and mechanical failures can occur due to resonance, where energy gain causes excessive vibrations that weaken the structure over time.

5. How is energy gain in a resonant system measured?

The amount of energy gained in a resonant system can be measured through various methods, such as using instruments to measure the amplitude of vibrations or the amount of energy stored and released by the system. Additionally, mathematical models and simulations can also be used to analyze and predict the energy gain in a resonant system.

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