Inductive Reactance:

[rattus]

We have buried the dead z(t) horse or at least embalmed him. Now for something important:

Let?s derive the formula for inductive reactance.

Lenz?s Law states that the emf induced in an inductor by a changing current is given by,

1) e = - L*di/dt, where the minus sign indicates an opposition to a change in current. Now let,

2) i = Im*sin(wt) then,

3) di/dt = Im*[-cos(wt)*w], now substituting eqn. 3 into eqn. 1

4) e = wL*Im*[cos(wt)] Then at t = 0,

5) e = Em = wL*Im, and

6) Xl = Em/Im = wL = Vrms/Irms

Inspection of eqns. 2 & 4 shows that v leads i by 90 degrees, but that angle is not carried by Xl; neither is Xl preceded by the operator ?j?. It is a real number with the units of ohms.

One might be tempted to take the ratio of e to i, but since these sinusoids are displaced by 90 degrees, the ratio would range between +/- infinity?useless!.

Note also, that inductive reactance only has meaning when used with sinusoidal waveforms.

 

[cadpoint]

So will it be; how high a ping pong ball will bounce, or will it be with the distance and amptilude between touch points of a bounce with in a dimensioning ratio of no power presented, of and all of one cycle of power?

Reading Len's Law is opposite of our everyday thinking of cause and effect, or even action reaction. Here is a good overview of it in laymans terms for all of you'll playing at the house! physicsforums. lease feel free to seek another defination!

I had to look-up equation number two to understand and conceive my ping pong statement. I frankly got tripped up with with the, "now let"; "now substituting eqn...", Parts of which, I had to have a friend explain it correctly!
My clue was Sin turning to Cos... which I missed with the word play, again noted but missed.

If it's a cause of a mystery wrapped in an imgana, there is no question mark within in your statement and while a math subsatution as you did is allowed only if all the properties and charartistic's are with-in the quantize masses. I frankly missed it, all things considered.

 

[rattus]

By the way:

By the way:

Who can spot the offsetting errors in this derivation?

And, who can derive the formula for Xc?

 

[Hameedulla-Ekhlas]

5) e = Em = wL*Im,


Note also, that inductive reactance only has meaning when used with sinusoidal waveforms.

rattus: one question, According to

"Basic electricity By M. Fogiel, Research & Education Association, Staff of Research Education Association, U S Naval Personnel page 164 "

e = Emsin(wt) or e = Em sin(0)

But I have seen in above that e = Em. Can you give some explaination ?

 

[rattus]

rattus: one question, According to

"Basic electricity By M. Fogiel, Research & Education Association, Staff of Research Education Association, U S Naval Personnel page 164 "

e = Emsin(wt) or e = Em sin(0)

But I have seen in above that e = Em. Can you give some explaination ?

Em*sin(wt) is a periodic function of time.

Em*sin(0) is the sine of the angle 0, zero that is.

In the post I evaluate the expression,

e = Em*cos(0) = Em or I could say,

e(0) = Em*cos(0) = Em

 

[__dan]

Z is frequency dependent

Z is frequency dependent

I'm not sure what you're looking for with this inquiry although I did learn something from the other thread, instantaneous values for I or V do not carry enough information to determine reactive circuit components, inductance, L, or capacitance, C. I do believe you are trying to help the reader understand rather than asking for something specific for yourself.

E = I*R is linear and the value of R is constant as E or I vary in frequency, in the ideal case. R converts power to heat.

Current flow through the capacitor, I = C*dv/dt and voltage drop across the inductor E = L*di/dt.

Trying to substitute Z for R, the value of Z is nonlinear and has to vary with frequency. L or C may be constant but their contribution to V or I is very frequency dependent and varying. In the ideal case, L or C in the circuit convert zero power to heat, they drop voltage or pass current, impedance or conductance, in response to the frequency.

The elements are filters or conductors for various frequencies. The result depends on the circuit applied frequency. The quantitative solution involves a Laplace transform and results for steady state and transient response, in the frequency domain. The course I had in it was a long time ago. Quantitatively I would draw a blank today, but the coursework does condition the mind to the elements behavior in a circuit. Capacitors are conductors, short circuits, at high frequencies and non conductive at steady state DC where dv/dt = 0. As frequency increases, dv/dt becomes larger, and I = C dv/dt increases.

Similarly for inductance, E = L di/dt, the inductor is conductive for low frequency and a short circuit for steady state DC where di/dt = 0. As frequency increases, di/dt becomes larger, and E, voltage drop across the inductor, increases. The inductor can be non conductive at high frequency or for very fast microsecond transients.

In the instantaneous case where the value of I or V does not convey information about frequency or rate of change, if dv/dt or di/dt are undefined, then Z can also or must also be undefined. Z has a value that is neither zero nor infinity but the instantaneous case does not convey sufficient information to determine Z.

 

[Hameedulla-Ekhlas]

Inspection of eqns. 2 & 4 shows that v leads i by 90 degrees, but that angle is not carried by Xl; neither is Xl preceded by the operator “j”. It is a real number with the units of ohms.


Note also, that inductive reactance only has meaning when used with sinusoidal waveforms.

rattus: I am agree with you it is a constant but in this we can not say X without j , it shows a reactance with ohm. But we can not add directly R + X

R = 6, X = 2 ; R + X = 8; we have three method.

Rcos(0) + Xsin(0) ; Z = (R^2 + X^2 )^0.5 or Z = R + jX

The term j comes only when we add reactance with resistance.

The above formula and poof shows that it is completely a reactive power without any real part.

Same thing in instantaneous power has also two parts it is only because we can not directly add these two parts to make one.

Anyway, it is good discussion .

 

[Hameedulla-Ekhlas]

Em*sin(wt) is a periodic function of time.

Em*sin(0) is the sine of the angle 0, zero that is.

In the post I evaluate the expression,

e = Em*cos(0) = Em or I could say,

e(0) = Em*cos(0) = Em

Rattus: As I have read some books it says like this

e is the instantaneous voltage, Em is the maximum voltage, 0 is the angle of the generator armature, and "sine" is one of the trigomoetric function.

The instantaneous voltage, e depends on the sine of the angle. It rises to a maximum positive value as the angle reaches 90 degree. this occurs because the conductor cuts directly across the flux at 90 degree. It falls to zero at 180 degree because the conductor cuts no lines of flux at 180 degree and etc.

Another form of the trigonometric equation for a sine wave of voltage involves the angular velocity refers to the number of degrees(angle) through which a voltage vector rotates per second. generally omega(w) given in radians per second rather than degrees per second.

There are pi radians in half a circle and 2pi radian in the circumference of a complete cricle.

the equation is

e = Emsin(wt)


So, if I take the equation by e=Emcos(0) at 90 degree it will be zero. and at zero it will be maximum.

 

[__dan]

on a different note

on a different note

on a different note

Given the choice of being either correct or having good timing, would you choose being right over having good timing.

This question has only one answer actually. Experience shows being right at the wrong time can be catastrophic while being right or wrong can be inconsequential compared to having good timing.

For economic activity to take place there has to be money in circulation for the output of goods offered for sale to be bought, the transaction. Price is not determined by the good offered, it is determined by the quantity of money in circulation, in sum, by the timing.

One wants X, the placing of a bucket to be filled with money by methods beyond comprehension. The bucket is not necessarily filled by exchanging Y for X although that would be nice. The bucket is filled by placing it when it is raining money. Since money is created by the act of borrowing, the federal deficit spending, it is quantitatively raining money ... somewhere.

It bears repeating because it is true historically through time, not in the instantaneous case. Having the bucket filled is not determined by the quality of the good offered for sale, it is determined by placing it at the time and place it is raining money. ... has value, so says the bucket.

http://www.amazon.com/Peter-Principle-Laurence-J/dp/1568491611

 

[rattus]

So, if I take the equation by e=Emcos(0) at 90 degree it will be zero. and at zero it will be maximum.

Ham, you have a conflict of terms here:

cos(0) means the cosine of angle zero. You can only evaluate it at zero degrees. Its value is 1.

cos(90) means the cosine of angle 90 degrees. Its value is 0.

 

[rattus]

rattus: I am agree with you it is a constant but in this we can not say X without j , it shows a reactance with ohm. But we can not add directly R + X

R = 6, X = 2 ; R + X = 8; we have three method.

Rcos(0) + Xsin(0) ; Z = (R^2 + X^2 )^0.5 or Z = R + jX

The term j comes only when we add reactance with resistance.

The above formula and proof shows that it is completely a reactive power without any real part.

Same thing in instantaneous power has also two parts it is only because we can not directly add these two parts to make one.

Anyway, it is good discussion .

Ham, reactance, as derived, is expressed as a single real number. There are no complex numbers in the derivation.

Now, if we make it part of an impedance we introduce the operator j to form the complex quantity we express as R +/- jX

And, instantaneous powers add algebraically. Average power must be treated as a complex quantity.

 

[rattus]

dan,
Linearity refers to the circuit elements. Frequency affects the value, but not the linearity, and we must assume constant frequency and constant values of R, L, and C in order to perform a linear circuit analysis.

 

[Hameedulla-Ekhlas]

Ham, you have a conflict of terms here:

The instantaneous voltage, e depends on the sine of the angle. It rises to a maximum positive value as the angle reaches 90 degree. this occurs because the conductor cuts directly across the flux at 90 degree. It falls to zero at 180 degree because the conductor cuts no lines of flux at 180 degree and etc.

Another form of the trigonometric equation for a sine wave of voltage involves the angular velocity refers to the number of degrees(angle) through which a voltage vector rotates per second. generally omega(w) given in radians per second rather than degrees per second.

cos(0) means the cosine of angle zero. You can only evaluate it at zero degrees. Its value is 1.

cos(90) means the cosine of angle 90 degrees. Its value is 0.

let me state it again. According to above information and the maximum voltage is 90 degree and e=Emcos(90) = 0. According to cosine 0 is maximum and 90 degree is zer0.
Is not better e=Emsin(0)
I am sure you got what I mean.

 

[Hameedulla-Ekhlas]

Em*sin(wt) is a periodic function of time.

Em*sin(0) is the sine of the angle 0, zero that is.

In the post I evaluate the expression,

e = Em*cos(0) = Em or I could say,

e(0) = Em*cos(0) = Em

Rattus: let me make more clear my question:


e is the instantaneous voltage, Em is the maximum voltage, ∅ is the angle of the generator armature, and "sine" is one of the trigomoetric function.

The instantaneous voltage, e depends on the sine of the angle. It rises to a maximum positive value as the angle reaches 90 degree. this occurs because the conductor cuts directly across the flux at 90 degree. It falls to zero at 180 degree because the conductor cuts no lines of flux at 180 degree and etc.

Another form of the trigonometric equation for a sine wave of voltage involves the angular velocity refers to the number of degrees(angle) through which a voltage vector rotates per second. generally omega(w) given in radians per second rather than degrees per second.

There are pi radians in half a circle and 2pi radian in the circumference of a complete cricle.


now the voltage is maximum at 90 degree and minimum at 0 degree.

if we go with this formula e=Emsin∅ it matches the above statement.

if we go by e=Emcos∅; ∅ = 90 voltage = 0 ; ∅ = 0 voltage = maximum


e=Emcos∅; did not match the above information.

Hope to be clear for you now

 

[rattus]

Hom, perhaps there is a typo in your earlier post.

"cos(0)" appeared, but I think you mean cos(phi). Right?

 

[Smart $]

let me state it again. According to above information and the maximum voltage is 90 degree and e=Emcos(90) = 0. According to cosine 0 is maximum and 90 degree is zer0.
Is not better e=Emsin(0)
I am sure you got what I mean.

Ham,

While using the sine function in instantaneous analysis is not incorrect, one runs into a problem when also using vector analysis or transposing from one analysis type to the other. The latter type analysis has to use the cosine function for the real part of the equation, not to mention using Euhler's Formula to remove the imaginary from the equation. Thus when using the cosine function, max amplitude occurs at 0? instead of 90? when using the sine function.

 

[Hameedulla-Ekhlas]

Ham,

While using the sine function in instantaneous analysis is not incorrect, one runs into a problem when also using vector analysis or transposing from one analysis type to the other. --QUOTE]

Rattus and Smart$:

Thanks for clearing for me both.

 

[Besoeker]

One might be tempted to take the ratio of e to i,

I think one, with just the basic understanding of electrical theory, would not be so tempted.

 

[rattus]

I think one, with just the basic understanding of electrical theory, would not be so tempted.

"Should not be tempted"?

 

[cadpoint]

I think one, with just the basic understanding of electrical theory, would not be so tempted.

"Should not be tempted"?

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 substatution 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!