Inductive Reactance:

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rattus

Senior Member
Then why do you continue to test or tempt US?...

Not many people are running around in your ball park... Most of this crowd is suffering through mixed fractions, and the A=B/C math.

The substitution is not fair to the jest of the thought. That's not how I learned to present a proof.

I'm glad you can enjoy that level of thought!

Cad, you don't need the fancy math to understand the not so hidden message. In this case it is:

Since reactance, and hence impedance, are derived from sinusoidal equations, they should be used only with sinusoidal waveforms.
 

__dan

Senior Member
analogues

analogues

The circuit where R, L, and C elements are arranged as a filter section, a Laplace transform is written and solved for steady state and transient frequency response.

The analogue in mechanics is a spring and shock absorber mechanical suspension system. The RLC electronic filter and the mechanical analogue, spring + shock absorber, elements can be arranged so that the differential equation and the Laplace transform are identical mathematically in every way. Given only the equation one cannot distinguish if the origin was an electronic or mechanical system. The algorithm for solving and results are identical.

Rather than visualizing the purpose of reactance in the circuit, look at its perfect analogue, the spring shock absorber.

You're in the truck and drive over railroad tracks at 35 mph, perpendicular to the tracks. What happens.

The tire hits the first track and is driven up into the suspension, the spring compresses and the shock absorber builds pressure in the reservoir. The impact is "filtered" by the suspension, you feel the bump but don't break your back from it.

The tire has an instant when upward travel is max and upward velocity is zero. The tire begins the down stroke. On the upstroke the suspension received energy, received movement (displacement), and stored it. Now on the down stroke there is air under the tire, the tire moves down, powered mostly by the stored energy from the suspension system.

That is one half of the sine wave, up and down over the rail. If the tire fell into a hole, immediately after the upward impact of the rail, that would be the second half of the sinewave as the tire falls into the hole and is driven back out.

Note that at any instant we could look at our recording equipment and know instantaneous F=Kx, force on the spring due to displacement, shock absorber reservoir pressure. However the instantaneous values do not contain essential information. We do not know if the tire is traveling up or down, how fast it is moving, and we do not know the forcing function, the road ahead, is it another rail, a hole, or a turnpike. Isolated instantaneous values only convey the static case. The suspension could be compressed but not moving, or if moving, how fast? We need to know the instantaneous di/dt, dv/dt , rate of change, frequency of the input. Given frequency of the input and the constants for L, C, the output frequency response can be calculated.
 

rattus

Senior Member
No one?

No one?

Who can spot the offsetting errors in this derivation?

And, who can derive the formula for Xc?

No one has noticed that the negative signs in the expressions below are wrong!

e = -L*di/dt

d(sin(wt))/dt = -cos(wt)*w

Or maybe you are just too polite?

No one has come forth to derive the formula for Xc either.
 

Cold Fusion

Senior Member
Location
way north
No one has noticed that the negative signs in the expressions below are wrong!

e = -L*di/dt
...
Some may have noticed you are the only person in the world that uses this convention.

However, they also may have figured you had a 500 post response as to why the model should have this sign convention. I know, it's hard to believe someone would think that.

cf
 

rattus

Senior Member
Some may have noticed you are the only person in the world that uses this convention.

However, they also may have figured you had a 500 post response as to why the model should have this sign convention. I know, it's hard to believe someone would think that.

cf

I doubt it cf, that is the way the law is written. Only when I obtained a negative result for the Xc formula did I realize my mistake. You see, two wrongs did make a right!

As for the long threads, some responders answer questions not asked and also try to use techniques not applicable to the problem at hand. Those issues must be addressed.
 

mivey

Senior Member
I thought it was rather clear to start with. I would bet that you got it on the first pass!
In light of the other thread, this one was clear. The other thread was like trying to bury the idea one spoonful at a time. At any rate, the idea of z(t) is not utter nonsense in every situation, but could be considered nonsense in the steady-state circuit analysis you outlined.

On a side note, here is a fun applet. Change the imaginary portion of Z load from 0 to 1, hit the re-calc button, then hit play to watch the infinities fly by in the impedance graph:
http://www.eecg.utoronto.ca/~bradel/projects/transmissionline/index.html
 

rattus

Senior Member
In light of the other thread, this one was clear. The other thread was like trying to bury the idea one spoonful at a time. At any rate, the idea of z(t) is not utter nonsense in every situation, but could be considered nonsense in the steady-state circuit analysis you outlined.

As I have said, impedance does sometimes change, but I don't know how to handle that extra variable, and I doubt that many if any others do either. Furthermore, I doubt that this changing impedance can be made into a function of time unless it is artificially changed.

Give us an example?
 

mivey

Senior Member
As I have said, impedance does sometimes change, but I don't know how to handle that extra variable, and I doubt that many if any others do either. Furthermore, I doubt that this changing impedance can be made into a function of time unless it is artificially changed.

Give us an example?
For the transmission line, you could say the impedance changes as the waves are reflected and make it a function of impedance changing with time even though it is really a superposition of sinusoidal waves.


For the fault analysis case, we model the impedance within specific time periods differently (sub-transient and transient) so it would be a piece-wise function unless you really wanted to continuously plot the "impedance" as it changes from the initial fault to steady-state (like by using infinitesimally small time%2
 
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rattus

Senior Member
For the transmission line, you could say the impedance changes as the waves are reflected and make it a function of impedance changing with time even though it is really a superposition of sinusoidal waves.

But Zo does not change; Zload does not change. It is more of a phase shift. You could even use a lumped model and perform a linear analysis with discrete elements.

For the fault analysis case, we model the impedance within specific time periods differently (sub-transient and transient) so it would be a piece-wise function unless you really wanted to continuously plot the "impedance" as it changes from the initial fault to steady-state (like by using infinitesimally small times)

Yep, iron does saturate, inductance does change, but you still perform linear analyses with the assumption that R, L, and C are constant. And too, a fault would cause rapid changes in inductance. I would think the change would occur in just a few cycles--not enough time for the reactance to perform linearly. Only at maximum fault current could you perform a linear analysis. Never done it, but these are my ramblings on the matter.

In a word, impedance is defined for the sinusoidal steady state assuming constant values of R, L, and C. It is not defined for any other waveforms. So, from the definition of impedance, the very notion of z(t) is nonsense.

Oh! BTW, have a nice holiday all, or should I say y'all?
 
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rattus

Senior Member
Chapter 8:

Chapter 8:

Now that we have finished Chapter 1, let?s jump ahead to Chapter 8 and use some complex numbers.

CF was stumped in his attempt to make sense of,

1) sin(wt)/sin(wt + phi).

If we let IM mean the imaginary value, and let Im and Vm mean the peak current and voltage of a sinusoid, then:

2) v = Vm*sin(wt) = Vm*IM[exp(jwt)]

3) i = Im*[sin(wt + phi)] = Im*IM[exp(jwt+phi)] = Im*{IM[exp(jwt)]}*[exp(jphi)]

Now divide v by i and you are left with,

4) [Vm/Im]/[exp(jphi)] = [Vm/Im]*exp(-jphi)] = Z @ -phi

which is a constant, not a function of time. The division has removed the time variable and left us with a new expression which is an impedance complete with phase angle.

Please note, this is not the answer to the v/i question.
 

mivey

Senior Member
Let me fix that for you:

In a word, impedance is usually defined for the sinusoidal steady state assuming constant values of R, L, and C. In steady-state circuit analysis, it is not defined for any other waveforms. So, from the usual definition of impedance in circuit analysis, the very notion of z(t) is nonsense.
 
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