# Parallel and series AC question

#### 2nowman

Joined Jan 14, 2017
17
Hi,

I wonder if anyone can help me out with this problem, as i'm not sure where to start, the circuit is:

The questions i've got to calculate are:

a) the resonant frequency
b) the dynamic resistance
c) the current at resonance
d) the circuit 'Q' factor
e) the voltages across all 4 components

My thought was to initially ignore the 990k resistor and work on the parallel part, but the formula I know for resonant frequency is:

and would apply to a resistor in series with an inductor which are then both in parallel with a capacitor, so i'm not sure how to adapt the equation - or can i just ignore the 470ohm resistor in the resonant frequency equation?

Thanks for looking, and any help/advice with this is much appreciated, as i'm not getting anywhere.

#### The Electrician

Joined Oct 9, 2007
2,953
Are you aware that in a circuit exhibiting parallel resonance, there is more than one way to define resonance? See: http://www.hyperphysics.de/hyperphysics/hbase/electric/parres.html

Which definition are you using?

With respect to question b), dynamic resistance of what part of the circuit?

With respect to question c), current in what part of the circuit?

#### 2nowman

Joined Jan 14, 2017
17
Hi Electrician, and thanks for getting back to me.

I honestly don't know the answer, I guess it will be the third definition 'The frequency at which the current is in phase with the voltage, unity power factor' but i'm not sure ... i've written the questions (a to e) exactly as they are written on my assignment, with all other information shown on the circuit diagram.

The only other information i've been given is a course handout written exactly as this:
'Resonance occurs when the frequency of oscillation oscillates without loosing energy. This is also the frequency that produces highest amplitude.
It's the systems most natural frequency of vibration.
For electronics it is when the waveform is at it's highest. So when you are looking for resonance its when the waveform is at it's biggest'.

I guess for question c) it will mean the total current, but again i'm not sure.

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#### 2nowman

Joined Jan 14, 2017
17
For dynamic resistance, the only information i've got states:

'Since the current at resonance is in phase with the voltage, the impedance of the circuit acts as a resistance.
This resistance is known as the dynamic resistance or dynamic impedance.
ZD = RD = L / CR'

This also seems to tie in with the third definition from the website you linked, and in my (simplified) mind, i need to specify a frequency for the supply that would make the reactances cancel each other out and thus the only impedance/resistance would be the resistor itself.

#### The Electrician

Joined Oct 9, 2007
2,953
If you perform an analysis of the circuit, determining all the currents and voltages as a function of frequency, the answers to all the questions will fall out.

What analysis methods do you know? Do you know all the standard methods such as mesh, nodal, superposition, etc.?

I would think that a mesh analysis would work out nicely. If you know that method, show mesh currents on your schematic and set up the equations in your next post.

#### 2nowman

Joined Jan 14, 2017
17
I'll try mesh analysis next, but before i do, can i just confirm that the following example (we did in class), can't be modified? the circuit was a slightly different configuration, but i tend to think this will be along the lines of what we should do:

a 100uF cap is connected in parallel with a coil of resistance 10ohms and inductance 0.1H. If the circuit is connected to a 240V variable frequency supply, calculate:
a) the resonant frequency

this is the formula and answer that we were shown:

Thanks

#### The Electrician

Joined Oct 9, 2007
2,953
This formula is the correct one for the case of a parallel resonant circuit where the inductor has some series resistance but the capacitor doesn't. It results when the definition of resonance is the zero phase definition, and it's somewhat tedious to derive. This formula doesn't work for the circuit you show in post #1.

The circuit of post #1 is much simpler to work with. All you have to do is find the frequency where the voltage across the RLC parallel combination is in phase with the current through the 990k resistor.

#### MrAl

Joined Jun 17, 2014
10,900
Hi,

I wonder if anyone can help me out with this problem, as i'm not sure where to start, the circuit is:

The questions i've got to calculate are:

a) the resonant frequency
b) the dynamic resistance
c) the current at resonance
d) the circuit 'Q' factor
e) the voltages across all 4 components

My thought was to initially ignore the 990k resistor and work on the parallel part, but the formula I know for resonant frequency is:

and would apply to a resistor in series with an inductor which are then both in parallel with a capacitor, so i'm not sure how to adapt the equation - or can i just ignore the 470ohm resistor in the resonant frequency equation?

Thanks for looking, and any help/advice with this is much appreciated, as i'm not getting anywhere.

Hi,

a. Since they ask for the dynamic resistance they probably want you to calculate the physical resonant frequency.
b. The dynamic resistance would probably just be the equivalent resistance once in resonance, because the circuit will look all resistive with physical resonance.
c. The current comes after you calculate the resonant frequency.
d. There are different ways to calculate the Q factor. You should mention which way you have done it in the past.
e. The voltages come from solving the netowrk, whatever way you learned in the past. One way is to use Nodal Analysis.

You can lump the parallel part into one circuit element and go from there. The impedances are all in parallel there. You can then use the complex voltage divider formula.

#### 2nowman

Joined Jan 14, 2017
17
Thanks again for the help, and yes i agree that once i have the resonant frequency everything else will follow, but i've still no idea where to start, and i'm very close to failing and giving up.

#### BR-549

Joined Sep 22, 2013
4,928
What is the formula for resonance? Plug C and L in and solve.

#### The Electrician

Joined Oct 9, 2007
2,953
Let's relabel the 990k resistor Rs (for source resistance) and leave the resistance in parallel with L and C as just R. Now derive an expression for the impedance of the RLC combination. The impedance of an inductor is j*ω*L and for a capacitor its 1/(j*ω*C); for the 470Ω resistor the impedance is just R.

If the impedance of a particular component is Z, then the admittance is 1/Z. The way to derive the impedance of the RLC combination is to add the admittances of R, L and C, then take the reciprocal of that admittance and you will have the impedance.

Give it a try. Write an expression consisting of the sum of the 3 admittances and post it here. Then take the reciprocal of that expression and do the algebra to simplify it and post it here.

#### MrAl

Joined Jun 17, 2014
10,900
Thanks again for the help, and yes i agree that once i have the resonant frequency everything else will follow, but i've still no idea where to start, and i'm very close to failing and giving up.
Hi,

[see shortcut below]

Well actually you could analyze the circuit using some circuit analysis method like Nodal and you can get all the information you need from that just by looking for what you know to be true about resonance. For this circuit you would want to try to set the imaginary part of the response to zero and see if you can find w0 which is the angular resonant frequency. We can do that with physical resonance.

There is a shortcut here however which maybe they wanted you to know, which is a sort of hint.

We are looking for physical resonance not some peak, so that means the circuit MUST be all resistive at the resonant point. This means that capacitive reactance must cancel any inductive reactance. We also know that the L and C are in parallel, and we know that the response to a parallel LC circuit at resonance is infinite impedance and the resonant point is a very simple calculation. So we end up with a simple circuit which i dont want to outline in complete form just yet, but think of what happens when you combine an infinite impedance in parallel with a finite resistance, and that is the key to the shortcut.

Can you figure out the solution to at least some parts of the problem now?

Dont give up as you will see how simple this problem really is.

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#### 2nowman

Joined Jan 14, 2017
17
It's probably too late now as my assignment needs handing in tomorrow and it's 23:20 now... unfortunately i've had to finish another assignment today (which i've just managed to complete) - i will ask for more time to complete this, but i'm not hopeful.

Just looking at your last paragraph MrAI, i guess the infinite impedance will act like an open circuit and no current will flow through the inductor or capacitor, meaning all the current will flow through the resistor?

Thanks again though, i really appreciate you people getting back to me and trying to help.

#### The Electrician

Joined Oct 9, 2007
2,953
It's probably too late now as my assignment needs handing in tomorrow and it's 23:20 now... unfortunately i've had to finish another assignment today (which i've just managed to complete) - i will ask for more time to complete this, but i'm not hopeful.

Just looking at your last paragraph MrAI, i guess the infinite impedance will act like an open circuit and no current will flow through the inductor or capacitor, meaning all the current will flow through the resistor?

Thanks again though, i really appreciate you people getting back to me and trying to help.
It's not true at all that no current will flow through the inductor or capacitor. When parallel resonant circuits are driven at the resonance frequency, there is a circulating current in the inductor and capacitor which does not appear in the external circuit. However it is true that the current in the 470 ohm resistor will be the same as the current in the 990k resistor.

#### MrAl

Joined Jun 17, 2014
10,900
It's probably too late now as my assignment needs handing in tomorrow and it's 23:20 now... unfortunately i've had to finish another assignment today (which i've just managed to complete) - i will ask for more time to complete this, but i'm not hopeful.

Just looking at your last paragraph MrAI, i guess the infinite impedance will act like an open circuit and no current will flow through the inductor or capacitor, meaning all the current will flow through the resistor?

Thanks again though, i really appreciate you people getting back to me and trying to help.
Hi,

Yes, and so you can remove both the cap and inductor and guess what that leaves you with (only at physical resonance though)?

This kind of trick works now and then, but it's not really a trick it is the result of thinking about what happens with a cap in parallel with an inductor. In the analysis, we find the resonant point as given to you already in another post by someone and is 1/sqrt(LC).
That does not always work so you should really learn how to analyze networks.

We can look at this more if you are interested.

Edit:
It's not that there is "no current" through the inductor and capacitor, they could have current flow between them, but it does not affect the rest of the circuit. So as a whole there is no net current into or out of both elements, but the two may share current themselves. This is of course for ideal elements L and C only.

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#### 2nowman

Joined Jan 14, 2017
17
Funilly enough, 1/sqrt(LC) was the first formula i tried, but when i tested it in multisim it was reading out of phase in the scope:

that reading made me believe it was the wrong formula.... and on Monday when i spoke to our tutor, he spotted my notes and laughed, realising i'd got it right then wasted a few days banging my head against the wall.

Upon trying the formula again, i moved the osc probe to include the 990k resistor and it was exactly as i originally wanted/expected:

I didn't think the resistor on it's own would affect the phase angles, so again, this has me confused.

#### The Electrician

Joined Oct 9, 2007
2,953
Your result for the second setup in post #16 is misleading. The 990k resistor is so large that its influence on the circuit dominates. Your current probe is measuring a current that is essentially determined by the 990k resistor no matter what the resonant tank does. And, of course, the current through the 990k is in phase with the voltage across it, so naturally the phase difference between voltage and current in the second setup is zero.

Go back to the setup in the first image of post #16, but remove L and C. See if there is any phase shift. Perhaps some component in your simulation is not ideal. L may have some parasitic resistance. Maybe the current probe has phase shift at low frequency.

#### 2nowman

Joined Jan 14, 2017
17
with C and L removed:

I can't see any parameters of the cap or inductor that are causing issues, tolerance and leakage are both set to 0%, which i guess is basically off.

#### 2nowman

Joined Jan 14, 2017
17
Same circuit but the 990k is replaced with a 1k:

#### The Electrician

Joined Oct 9, 2007
2,953
Try tweaking the frequency a little plus and minus to see if you can find a frequency where the displayed phase shift is zero.