Resonant frequency of L - R||C circuit

[phantomvommand]

Homework Statement

See picture below

Relevant Equations

XL = jwL
Xc = 1/jwc

I am confused as to why 2 different approaches to finding the resonant frequency of the above circuit contradict; below are the 2 approaches:

Approach 1:
Total Impedance Z = $j \omega L + (\frac {1} {R} + j \omega C)^{-1}$
At resonance, impedance is purely resistive, i.e. imaginary term = 0.
Imaginary term = $j \omega L - \frac {j \omega CR^2} {1 + (\omega C R)^2} (= 0)$
Solving, $\omega = \sqrt {\frac {1} {LC} - \frac {1} {(RC)^2}}$

Approach 2:
Take the Norton across the inductor and voltage source, thereby giving an equivalent circuit with current source equal to $\frac {v} {j \omega L}$ and R, L, C in parallel. This is the standard setup of a parallel RLC circuit with resonant frequency $\omega = \sqrt {\frac {1} {LC}}$.

Therefore, the 2 resonant frequencies differ slightly. Why is this the case, and which approach is correct?

 

[marcusl]

#1 is correct. I don't think your voltage is correct for #2. The equivalent voltage must be $v-L\frac{di}{dt}$, but the current i depends on the load, so your expression looks wrong.

 

[phantomvommand]

#1 is correct. I don't think your voltage is correct for #2. The equivalent voltage must be $v-L\frac{di}{dt}$, but the current i depends on the load, so your expression looks wrong.

I don't think Norton's theorem was applied erroneously. The short circuit current across the voltage source and inductor is indeed v/jwL.

 

[marcusl]

I'm not an EE so I'm not really familiar with Norton equivalence (or Thevenin, for that matter), but I can say that your circuit is not a parallel LCR, so you've made some type of topological error. For example, the classic LCR analysis assumes a voltage excitation v, but your effective voltage looks very different.

 

[QuarkyMeson]

I'm not an EE so I'm not really familiar with Norton equivalence (or Thevenin, for that matter), but I can say that your circuit is not a parallel LCR, so you've made some type of topological error. For example, the classic LCR analysis assumes a voltage excitation v, but your effective voltage looks very different.

Thevenins theorm is that any two terminal network who's internal circuitry consists solely of resistors, batteries and current sources, no matter how so connected, is indistinguishable from a two terminal network consisting of a single battery $V_{thev}$ in series with a single resistor $R_{thev}$. One of the better proofs is in an appendix in "The Art of Electronics".

Nortons Theorem is in the same vein, but swaps out the voltage source for a current source $I_{norton}$ in parallel with a resistor $R_{norton}$.

I've never seen anyone try to apply the theorms to RCL circuits this way, but maybe? I'm unsure.

 

[DaveE]

#2 is correct. But this is by definition of resonance. Sorry, I don't have time right now to do the algebra required for a good answer.

It is not true in general that the impedance of the entire network is real at resonance. This is a great question that is often misunderstood, at least according to the way I learned analog EE. Check out my responses in this thread for now.

 

[renormalize]

#2 is correct. But this is by definition of resonance. Sorry, I don't have time right now to do the algebra required for a good answer.

If $\omega = \sqrt {\frac {1} {LC}}$ is correct it should hold for any value of $R$. So set $R=0$ in the circuit so that it "shorts-out" the capacitor $C$. Then the circuit consists only of the voltage source $V$ and the inductor $L$ which doesn't resonate at all. Thus #2 can't be true.

 

[phantomvommand]

It is not true in general that the impedance of the entire network is real at resonance.

How should resonance be defined then?

 

[QuarkyMeson]

For an ideal parallel tank circuit $\omega = \sqrt{\frac{1}{LC}}$ is whats often quoted, but it's just an approximation assuming all the circuits resistance is indeed parallel. So I believe method two is giving an approximation of the true resonance, while method one is telling you the complete story when your circuit isn't idealized.

Obviously there are also problems at the limiting case of $R \to \infty$ and $R \to 0$ for method two. Method one would still be valid (in the sense it lets you know that something is wrong in the $R \to 0$ case and in the case of $R \to \infty$ as long as the resistors placement just doesn't break the LC circuit).

Could be wrong.

 

[marcusl]

Different topologies have different resonant frequencies. The most realistic circuit model in practical cases places a resistance in series with the inductor, since coils are lossy and capacitors are usually virtually lossless. The L+R series combination can then be either in series or in parallel with a capacitance, producing different resonance frequencies. You can take a look at various topologies here
https://en.wikipedia.org/wiki/RLC_circuit

 

[marcusl]

We can refer to the golden bible of radio engineering, Fred Terman's book "Radio Engineer's Handbook." The resonance frequency in the high Q (small R) limit approximates to $\omega_0=\frac{1}{\sqrt{LC}}$, which is generally true in the high Q limit for all topologies. At low Q, Terman quotes the resonant frequency as
$$\omega=\frac{\omega_0}{\sqrt{1-\frac{1}{Q^2}}}$$
The more common model places a capacitance in parallel with an R+L series combination. This is a better model for practical tank circuits since coils are lossy while capacitors are typically nearly lossless. In this case,
$$\omega=\omega_0\sqrt{1-\frac{1}{Q^2}}$$ where $Q=\frac{\omega_{0} L}{R}$

I've attached screen shots of the pages that deal with this circuit.

 

[DaveE]

We can refer to the golden bible of radio engineering, Fred Terman's book "Radio Engineer's Handbook." The resonance frequency in the high Q (small R) limit approximates to ω0=1LC, which is generally true in the high Q limit for all topologies. At low Q, Terman quotes the resonant frequency as
ω=ω01−1Q2

This is a great illustration of the point I started with in the previous thread. There isn't universal agreement about these words.

Terman is defining resonance as the maximum gain (Ok, really something equivalent like maximum or inflection point, etc.). I very much prefer the more consistent mathematically based definition of when the poles/zeros of the transfer function polynomial(s) change from real to complex roots/zeros. This is more generally applicable to complex networks and is not ambiguous. In my experience it is the definition you will find in advanced linear system analysis (control systems, filter design, etc.). I won't dispute that Terman could help you build a radio, but I prefer textbooks to handbooks.

In any case, the underlying point is to be aware of what you really care about and describe it precisely. Memorizing words or blindly following someone else's formula (even if it's golden) will never be as correct as doing the math yourself to get the answer you need. In my experience "resonant frequency" isn't a term you will hear much in grad school except when people are hand waving. In this case the difference between the resonance and the peak gain isn't significant. They will be talking about pole and zero locations in the transfer function, in which case there really is a significant difference between only the real axis and the rest of the complex plane.

 

[phantomvommand]

This is a great illustration of the point I started with in the previous thread. There isn't universal agreement about these words.

Terman is defining resonance as the maximum gain (Ok, really something equivalent like maximum or inflection point, etc.). I very much prefer the more consistent mathematically based definition of when the poles/zeros of the transfer function polynomial(s) change from real to complex roots/zeros. This is more generally applicable to complex networks and is not ambiguous. In my experience it is the definition you will find in advanced linear system analysis (control systems, filter design, etc.). I won't dispute that Terman could help you build a radio, but I prefer textbooks to handbooks.

In any case, the underlying point is to be aware of what you really care about and describe it precisely. Memorizing words or blindly following someone else's formula (even if it's golden) will never be as correct as doing the math yourself to get the answer you need. In my experience "resonant frequency" isn't a term you will hear much in grad school except when people are hand waving. In this case the difference between the resonance and the peak gain isn't significant. They will be talking about pole and zero locations in the transfer function, in which case there really is a significant difference between only the real axis and the rest of the complex plane.

Thanks for this, however, as an undergrad, could you explain the significance of the resonant frequency as calculated in the 2nd approach? I understand that the 1st approach gives the frequency where current is in phase with voltage, but what does the 2nd approach give, and why do you feel that it is more appropriate?

BTW, the suggested answer employs approach 2, so I am quite interested in what you have to say, since most here seem to lean towards approach 1.

 

[DaveE]

Thanks for this, however, as an undergrad, could you explain the significance of the resonant frequency as calculated in the 2nd approach? I understand that the 1st approach gives the frequency where current is in phase with voltage, but what does the 2nd approach give, and why do you feel that it is more appropriate?

BTW, the suggested answer employs approach 2, so I am quite interested in what you have to say, since most here seem to lean towards approach 1.

Approach #1 is the "lab" version; maximum response is what you will see on your oscilloscope or spectrum analyzer. It is often what the radio designers want.

Approach #2 will require some study of linear systems analysis, and specifically Laplace Transforms. This is an approach that characterizes a system by the mathematical representation in the frequency domain. It may require you to study a bit more to get to these foundational concepts of advanced linear systems. I'm not sure I can explain it online. It's really textbook stuff initially.

But... read the other thread too. I had more to say there.

 

[marcusl]

This is a great illustration of the point I started with in the previous thread. There isn't universal agreement about these words.

Terman is defining resonance as the maximum gain (Ok, really something equivalent like maximum or inflection point, etc.). I very much prefer the more consistent mathematically based definition of when the poles/zeros of the transfer function polynomial(s) change from real to complex roots/zeros. This is more generally applicable to complex networks and is not ambiguous. In my experience it is the definition you will find in advanced linear system analysis (control systems, filter design, etc.). I won't dispute that Terman could help you build a radio, but I prefer textbooks to handbooks.

In any case, the underlying point is to be aware of what you really care about and describe it precisely. Memorizing words or blindly following someone else's formula (even if it's golden) will never be as correct as doing the math yourself to get the answer you need. In my experience "resonant frequency" isn't a term you will hear much in grad school except when people are hand waving. In this case the difference between the resonance and the peak gain isn't significant. They will be talking about pole and zero locations in the transfer function, in which case there really is a significant difference between only the real axis and the rest of the complex plane.

Ah well, I'm not an engineer. I yield to the more sophisticated analysis.

Do real poles only exist for DC response?

 

[DaveE]

Do real poles only exist for DC response?

DC analysis has no poles or zeros. Inductors are short circuits and capacitors are open circuits. In a passive network only the resistors count for DC. DC has no dynamics, no transient response (beside the trivial case of the applied transients).

In a real network transient response, real poles correspond to simple exponential solutions, like $e^{\frac{-t}{\tau}}$. Complex poles/zeros have an oscillatory component, like decaying (or not) sinusoids, like $e^{\frac{-t}{\tau}} sin(\omega_o t)$ . This is why the pole/zero definition of resonance is generally applicable and not arbitrary.

PS: Also, a resonant network must have at least two reactive components (inductors or capacitors). If a network has none it has no transient response (also the DC case). If it has only one it will have a simple exponential response. If it has two, or more, it may or may not be resonant, depending on the pole locations. This all relates back to the fundamental theorem of algebra and polynomial factoring. All of this understanding is related to Laplace Transforms and linear systems analysis.

 

[marcusl]

Ha, I started to write a damped exponential and then got cold feet. I haven’t thought about poles since taking a complex variables class a half century ago, so it’s time to get reacquainted with all that.

 

[DaveE]

Ha, I started to write a damped exponential and then got cold feet.

It's a mess IRL. That's why analog EEs tend to work in the frequency domain, like 99.99% of the time.

 

[DaveE]

I've never seen anyone try to apply the theorms to RCL circuits this way, but maybe? I'm unsure.

I do it all the time. It's usually my first attempt at network analysis. Graphically simplify all of the stuff you aren't measuring first. The math is much easier for lower dimensional systems.

 

[QuarkyMeson]

I do it all the time. It's usually my first attempt at network analysis. Graphically simplify all of the stuff you aren't measuring first. The math is much easier for lower dimensional systems.

Interesting! I've also gone down a bit of a rabbit hole with what you've said so far in both threads.

If we look at say a mass-spring system in the homogenous case in canonical form $$\ddot{x}(t) + 2\zeta \omega_0 \dot{x}(t) + \omega_0^2 x(t) = 0 $$ And then we take the laplace transform $$ s^2 + 2\zeta \omega_0 s + \omega_0^2 = 0$$
The poles for this are $$ -\zeta \omega_0 \pm \omega_0 \sqrt{\zeta^2 -1} $$ So what you're saying is that $\omega_0$ is the resonance, and that how damped that resonance is equals the maximum gain? Aka what you would actually measure on an oscilloscope?

I was trying to do KCL in the s-domain on the above circuit, set up the transfer function, and got something like $$s^2 +\frac{s}{RC} + \frac{1}{LC} = 0$$ So here $\omega_0 = \sqrt{\frac{1}{LC}}$ and the damping ratio is $\frac{1}{2R}\sqrt{\frac{C}{L}}$ if $\zeta < 1$ the poles will be complex conjugates so you will have a maximum gain around $\omega = \omega_0 \sqrt{1 - \zeta^2}$ that tends to the resonance as $\zeta \to 0$? The poles themselves are $$\frac{-\frac{1}{RC} \pm \sqrt{\frac{1}{RC}^2 - \frac{4}{LC}}}{2}$$

$\zeta > 1$ there are only real roots, so nothing there $\zeta < 1$ we have complex conguate roots, so we have some resonance. $\zeta = 1$ is a transition and $\zeta = 0$ is the undamped case?

Am I totally misunderstanding? Is it basically like you're saying a definitional argument? Or have I missed something subtle or just way off base? If we set $\zeta = 0$ ($R \to \infty$ in the OP so we just have an ideal LC circuit) then the poles are of course just $\pm i\omega_0$.

 

[DaveE]

So what you're saying is that ω0 is the resonance, and that how damped that resonance is equals the maximum gain? Aka what you would actually measure on an oscilloscope?

Yes, the peak response for a damped oscillator is not at the resonant frequency. It is pulled slightly lower as the damping increases. It isn't always obvious since the peak is also more rounded at higher damping levels. There are some good graphs here for the RLC LPF case. But in that link the author choses the opposite definition and adds in "pole frequency" as the description for $\omega_o$. I think he's wrong, he probably doesn't. His definition makes more sense in the lab, mine makes more sense to a mathematician. Ultimately I don't care since we both can do the analysis correctly.

Is it basically like you're saying a definitional argument?

Yes, a definition of resonance that is independent of the damping as long as ζ<1, otherwise there is no resonance, the poles are on the real axis.

 

[DaveE]

If $\omega = \sqrt {\frac {1} {LC}}$ is correct it should hold for any value of $R$. So set $R=0$ in the circuit so that it "shorts-out" the capacitor $C$. Then the circuit consists only of the voltage source $V$ and the inductor $L$ which doesn't resonate at all. Thus #2 can't be true.

There is no resonance if $Q < 0.5$. No definition exists in this case, as you said.