# Fancy math on a derivation

#### KevinEamon

Joined Apr 9, 2017
281
Hey guys struggling with this equation in the picture.
I'm just not seeing what is happening from eq1 to eq2.

Where is the w^2 gone? And hows he rearranging the rest of it?

I wonder if anyone would be so kind as to break this up into more steps so I can see it...? Looks like he' done a few things at once.

Thx guys

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#### Hymie

Joined Mar 30, 2018
976
Hey guys struggling with this equation in the picture.
I'm just not seeing what is happening from eq1 to eq2.

Where is the w^2 gone? And hows he rearranging the rest of it?

I wonder if anyone would be so kind as to break this up into more steps so I can see it...? Looks like he' done a few things at once.

Thx guys
View attachment 152071
Lower case omega is used to indicate angular frequency in radians/second (2.pi.f); whereas Z indicates impedance, so I would assume Zab is referring to the impedance between terminals A and B.

#### KevinEamon

Joined Apr 9, 2017
281
Yeh I get that... but it not as if he defined omega^2 or anything. Its just the next stage of the equation. Perhaps hes sqrt-ing both sides then doing a bit of rearranging. But damned if I know... what I really need is the intermediate steps

Joined Mar 10, 2018
4,057
This is a solution for a basic series resonance of complex impedance's.

https://www.electronics-tutorials.ws/accircuits/series-resonance.html

The circuit you show is more involved because of the non ideal addition
of R.

Series resonance occurs when the impedance between the two endpoints
of the circuit is minimum in magnitude. Z here is magnitude (sum of Zl and
{Zc || R}).

Regards, Dana.

#### WBahn

Joined Mar 31, 2012
26,398
Hey guys struggling with this equation in the picture.
I'm just not seeing what is happening from eq1 to eq2.

Where is the w^2 gone? And hows he rearranging the rest of it?

I wonder if anyone would be so kind as to break this up into more steps so I can see it...? Looks like he' done a few things at once.

Thx guys
The second equation is simply the impedance (which is purely resistive) at the resonant frequency.

So they aren't going FROM equation 1 TO equation 2, but more likely there is an earlier equation giving the impedance at ANY frequency (so it would have ω in it) and then they ask what the resonant frequency is (probably defined as the frequency at which the reactance goes to zero) and come up with equation 1. They then plug that back into the prior equation to come up with equation 2, which is only good at that one frequency.

But why didn't they simplify equation 2 to just L/(RC)?

#### WBahn

Joined Mar 31, 2012
26,398
Yeh I get that... but it not as if he defined omega^2 or anything. Its just the next stage of the equation. Perhaps hes sqrt-ing both sides then doing a bit of rearranging. But damned if I know... what I really need is the intermediate steps
So do the intermediate steps!

What is the complex impedance of that circuit from 'a' to 'b'?

Zab = Z_L + (Z_R || Z_C)

Z_L = jωL
Z_R = R
Z_C = 1/(jωC)

Express the result as

Zab = Req + j·Zeq

At what frequency is Zeq = 0?

What is Zab at that frequency?

#### KevinEamon

Joined Apr 9, 2017
281
This is the question that he was doing:-

For the circuit below show that the impedance between the terminals A and B, at the resonance frequency, is given by

Zab = L/RC.

Then it has a circuit like the one i've drawn above.

The stuff I've showed in the picture is just the last 2 steps of the lecturers notes. There's no definition of w^2. I'm going to answer the questions now Wbahn. If I can... and upload a pic of the thing.

#### KevinEamon

Joined Apr 9, 2017
281

#### KevinEamon

Joined Apr 9, 2017
281
Please excuse the random head at the bottom. I've installed some VHDL editor that seems to have possessed my laptop and haven't time to fix it atm. It's killed my photo editor package.

#### KevinEamon

Joined Apr 9, 2017
281
So I've been doing a bit of researching on this resonance frequency Wbahn. It occurs at

XL_max and XC_max when phase angle of XL = - phase angle of XC
and if (angle XC = angle XL).

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#### WBahn

Joined Mar 31, 2012
26,398
Do you understand that the impedance of the three basic passive elements, as a function of frequency is
This is the question that he was doing:-

For the circuit below show that the impedance between the terminals A and B, at the resonance frequency, is given by

Zab = L/RC.

Then it has a circuit like the one i've drawn above.

The stuff I've showed in the picture is just the last 2 steps of the lecturers notes. There's no definition of w^2. I'm going to answer the questions now Wbahn. If I can... and upload a pic of the thing.
Do you understand that the impedance of the three basic passive elements, as a function of frequency, are the three equations I gave earlier?

If so, do you understand the meaning of the ω in those equations?

If so, then ω² is just ω·ω.

#### WBahn

Joined Mar 31, 2012
26,398
So I've been doing a bit of researching on this resonance frequency Wbahn. It occurs at

XL_max and XC_max when phase angle of XL = - phase angle of XC
and if (angle XC = angle XL).
I don't know what you mean by XL_max or XC_max. The inductive reactance increases towards infinity, without bound, as the frequency increases and the capacitive reactance increases toward infinity, without bound, as the frequency decreases.

There are a few definitions of resonance depending on the situation. The one that is relevant here is that the circuit it as resonance when the impedance is purely resistive.

#### KevinEamon

Joined Apr 9, 2017
281
Not entirely sure what you meant Wbahn. So I did something stupid to see if it would work out. But it didn't obviously as you can see

Joined Mar 10, 2018
4,057
Here is a sim of series resonance, attached, low Q. Low Q because of the 1 ohm R.

Plot is normalized to 1 F, 1 Ohm, 1 Henry, 1A on stim = Y axis directly in Z

Below min Z network is R dominated, then C kicks in, lastly above L dominates.

With no R w = 1 /2pi, in this network the R effectively lowers "reflected" C.

At resonance in graph Xl = 2pifres, Xc= 1/2pifres

Unlike series network with low R one with no R (high Q) looks like -

Regards, Dana.

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#### KevinEamon

Joined Apr 9, 2017
281
Thx Dana. I'm guessing this is the resonance frequency at 800Hz? It helps conceptually to see it like this, but all I need to do is get from one equation to the other. I mean I could just learn it off for the test, but I want to understand the math behind it.
See the steps from W^2=
to Zab =
Can anyone put something between that?
Just one intermediate step would probably allow me to pull it apart. Thx

Joined Mar 10, 2018
4,057

#### KevinEamon

Joined Apr 9, 2017
281
Liked the interactive graph there. Interesting to see it all working together. None of the settings seem to affect the voltage curve, I'm guessing it's held steady for reference? Maybe...
Anyways... procrastinating is what I'm doing at this point. I'm just gonna learn the thing, like a trained monkey.

#### WBahn

Joined Mar 31, 2012
26,398
Not entirely sure what you meant Wbahn. So I did something stupid to see if it would work out. But it didn't obviously as you can see

View attachment 152086
I have NO idea what you are doing.

The symbol ω is just a variable, namely the frequency. What does R² mean? Nothing more than the value of R multiplied by itself, or R·R. What does ω² mean? Nothing more than the value of ω multiplied by itself, or ω·ω.

Let's take it one step at a time.

You have a circuit consisting of a resistor R, a capacitor C, and an inductor L.

What is the impedance of each of these three elements?

#### KevinEamon

Joined Apr 9, 2017
281
Thanks Wbahn I do tend to miss the most simplest things. I only truly learned what ground was, on electronic schematics, very recently. How terrible is that? Sigh...
Of course I always had an idea. But anyways... let's do this please:-

(I'm at work so please excuse if I take a while to respond).

(R + Xl + Xc) = Zt
Depending on the arrangement.

#### WBahn

Joined Mar 31, 2012
26,398
Thanks Wbahn I do tend to miss the most simplest things. I only truly learned what ground was, on electronic schematics, very recently. How terrible is that? Sigh...
Of course I always had an idea. But anyways... let's do this please:-

(I'm at work so please excuse if I take a while to respond).

(R + Xl + Xc) = Zt
Depending on the arrangement.
This is NOT depending on your arrangement. That is ONLY for when the three components are in series.

But you are jumping ahead. Let's consider each component by itself.

What is the impedance of a resistor?

What is the impedance of a capacitor?

What is the impedance of an inductor?

Remember that impedance can be written in the form

Z = A + jB

where either A or B might end up being zero.