# A Challenging Question about Resistor's equivalent

Discussion in 'Homework Help' started by mo2015mo, Oct 4, 2013.

1. ### mo2015mo Thread Starter Member

May 9, 2013
157
1
Hi my friends ,,

I have a challenging question for you
As we know the most of elements are assembling by resistance's equivalent
For example inductor impedance, capacitor impedance...
ZL = j XL , Zc= -j Xc ...

But what about the Resistor's equivalent??
Has it a equivalent or no ??

2. ### MrChips Moderator

Oct 2, 2009
17,281
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Resistor's equivalent is ZR = R

3. ### mo2015mo Thread Starter Member

May 9, 2013
157
1
We know that ,, but i don't mean Impedance
What the R's equivalent by ....
thinking deeply

4. ### WBahn Moderator

Mar 31, 2012
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Impedance is NOT equivalent to resistance. Impedance is a relationship between the voltage and current variables in the complex frequency domain that has the same functional form as the relationship between voltage and current in the time domain.

The voltage and current variables in the complex frequency domain are NOT the same thing as the voltage and current variables in the time domain. You can transform back and forth between them, but they are not the same.

The voltage and current variables in the complex frequency domain therefore map to the corresponding voltage and current variables in the time domain. Similarly, an inductance L in the time domain maps to a corresponding impedance jωL in the complex frequency domain while a capacitance C maps to a corresponding impedance 1/(jωC) in the complex frequency domain.

So your question really should be: What impedance does a resistor in the time domain map to in the complex frequency domain?

5. ### LvW Well-Known Member

Jun 13, 2013
685
102
May be I am getting old - however, I don`t understand this "challenging" question.
If somebody speaks about "equivalence" he should - at least - mention which two areas/regions/domains he wants to compare regarding an equivalent behavior.

6. ### bountyhunter Well-Known Member

Sep 7, 2009
2,498
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That's because it's nonsense.

7. ### WBahn Moderator

Mar 31, 2012
23,148
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Actually, there's not much indication of any thinking at all.

By the very pattern you set up

ZL = j XL; ZC = -jXC; ZR = R

Though note that it really should be

ZL = j XL; ZC = jXC; ZR = R

XL > 0
XC < 0

8. ### studiot AAC Fanatic!

Nov 9, 2007
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Ordering relations in the complex plane?

9. ### The Electrician AAC Fanatic!

Oct 9, 2007
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It seems to me that he intends XL and XC to be pure real, ZL and ZC to be complex (imaginary).

10. ### WBahn Moderator

Mar 31, 2012
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It's not that, it's that impedance is defined to be

Z = R + jX

where R is the resistance and is defined as the real part of the impedance and X is the reactance and is defined as the imaginary part of the impedance.

The problem that I have with defining both capacitive reactance and inductive reactance as positive quantities is that now we have to have a ton of new equations to deal with the possibilities.

If we have a capacitor: Z = R - jXC

If we have an inductor: Z = R + jXL

If we have both, then we have to really jump through hoops because we have committed to identifying capacitve vs reactive not on the sign of the reactance, but on the operator use to combine resistance and reactance. So:

Xtot = |XL -XC| and
Ztot = R + jXtot if XL > XC but
Ztot = R - jXtot if XC < XL

Following this through without dropping a sign somewhere becomes a nightmare for anything but the simplest circuits. But if you simply led XC be negative (following directly from the fact that the impedance of a capacitor is 1/(jωC) = j*[-1/(ωC)]) and XL be positive and then simply having the impedance be the sum of the resistance and the imaginary unit times the reactance, the signs take care of themselves just beautifully.

imagi

11. ### WBahn Moderator

Mar 31, 2012
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XL and XC are pure real. I've never seen reactance defined as anything other. But there is an inconsistency in that some define them is such a way that XL and XC are both positive quantities, namely that XL=(ωL) and XC=1/(ωC). I think this is an extremely inferior way of defining them and, instead, the proper definitions are XL=(ωL) and XC=-1/(ωC).

12. ### studiot AAC Fanatic!

Nov 9, 2007
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Don't the problems arise because some authors are trying to avoid doing proper vector arithmetic or complex arithmetic, for simplicity?

13. ### WBahn Moderator

Mar 31, 2012
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I think that is a part of the problem. Mixed in with authors that are trying to avoid complex arithmetic (complex is in complex numbers, not complex as in not-simple) because their audiences lack the necessary math background (so a lot of high school and trade school level courses), you have the authors that are just parroting what they were taught and who have never learned themselves how to do things with a reasonable level of mathematical rigor.

It's a problem that's not going away, but that doesn't mean we shouldn't continue the struggle.

14. ### studiot AAC Fanatic!

Nov 9, 2007
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I think it is a case of
Mathematicians look away now, as I said in Lightfire's Integral thread.

I am off to the seaside for the day but I will have more to say on this later.

15. ### studiot AAC Fanatic!

Nov 9, 2007
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I am still not happy with the proposition that XL>0 and XC<0.

If we take any number whatsover, say a, then all numbers greater a are greater than any number less than a.

Thus the above proposition is suggesting that the reactance of any inductor, however small, is greater than the reactance of any capacitor. So a 1 picohenry inductor has greater reactance than a 100 farad capacitor.

Surely this is not what we want to suggest?

What would a computer program, that operated on your definition of reactance, to determine the largest reactance in a list output?
Of course you would have to use the modulus. But the modulus of something is not the thing itself.

However we look at things we have to specify two numbers in AC circuitry for the circuit variables. Even for pure capacitances or pure inductances the current and voltage are out of phase so this is true. You cannot use X without the j or equivalent.

Last edited: Oct 6, 2013
16. ### DerStrom8 Well-Known Member

Feb 20, 2011
2,424
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First of all, Resistance is the real part of an impedance, the reactance is the imaginary part.

So what's the confusion? I don't see a challenging question here. Equivalent resistance is just R = R. Or R = R/2 + R/2. Or R = R/3 + R/3 + R/3, and so on. R = R1 + R2. It's as simple as that.

I'd like the OP to come back and make the question clearer, because I don't see any problem in this at all....

Matt

17. ### studiot AAC Fanatic!

Nov 9, 2007
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Might I enquire, ever so sweetly, is an impedance the truth, the whole truth and nothing but the truth?

DerStrom8 likes this.
18. ### WBahn Moderator

Mar 31, 2012
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So does that mean that someone should prefer to have -1A of current passing through their heart instead of +10mA, since -1A is less than +10mA?

Surely that's not what we want to suggest.

Or perhaps it is reasonable to say that there are quantities for which the magnitude tells us one piece of information and the sign tells us a different piece of information.

In fact, many signed values carry two pieces of information -- one thing that is conveyed by the magnitude and another that is conveyed by the sign. A financial transaction is often recorded as a signed quantity in which the sign merely indicates the direction of the transaction relative to a specified reference.

Sometimes the ordering of two values is based on the magnitude order sometimes it is based on the arithmetic order, depending on what information is conveyed by the values.

Having X be a signed quantity provides the two pieces of information -- the magnitude gives you the magnitude and the sign gives you the phase.

But let's see the implications of going your route, that both capacitive and inductive reactance are positive and we choose the operator in the impedance equation based on the type of the reactance.

I have five black boxes that have unspecified impedances in them. On each box is a label showing the values of the effective series components, namely the resistance and the reactance (the reactance being a number greater than zero and a corresponding label indicating whether the type of reactance is inductive or capacitive).

Call these boxes Z1 through Z5 and the corresponding resistances R1 through R5, the corresponding reactances X1 through X5 (and, if you wish, the corresponding types T1 through T5). The boxes are wired so that Z1 and Z2 are in series forming the combination Z6, while Z3 and Z4 are similarly wired in series forming combination Z7. These two combinations, Z6 and Z7, are in parallel with each other forming Z8 and the result is then in series with Z5 to form Ztot.

What are the resistances and impedances of Z6, Z7, Z8, and Ztot?

Right from the beginning it's a nightmare.

We have six possibilities for Z6:

Code ( (Unknown Language)):
1.
2. IF: T1 and T2 are both capacitive
3.    Z6 = (R1+R2) + j(X1+X2)
4. IF: T1 is capacitive and T2 is inductive
5.    IF: X1 > X2
6.       Z6 = (R1+R2) - j(X1-X2)
7.   ELSE:
8.       Z6 = (R1+R2) + j(X2-X1)
9. IF: T1 is inductive and T2 is capacitive
10.    IF: X1 > X2
11.       Z6 = (R1+R2) + j(X1-X2)
12.   ELSE:
13.       Z6 = (R1+R2) - j(X2-X1)
14. If T1 and T2 are both inductive:
15.   Z6 = (R1+R2) - j(X1+X2)
16.
We then have six possibilities for Z7. We similarly have six possibilities for Z8 in terms of Z6 and Z7, which means that for each of these six possibilities we have 36 cases that have to be allowed for, giving us 216. Now, some of them will be mutually exclusive, so they will drop out when all is said and done, but identifying which ones do is it's own nightmare. Then, for Ztot, we have six possibilities in terms of Z5 and Z8, so we end up with 1296 separate cases depending on which reactances are what type and which ones are bigger than others.

You might consider ordering things so that you choose your case based on the ordering of the reactances (there are 120 possible such orderings) and the types (there are 32 possibilities there), but that gives you 3840 possibilities that have to be enumerated. Many of them are degenerate and so there will be many combinations that map to the same equation.

Now, why would people subject themselves to this nightmare? The answer, of course, is that they don't. Instead, they wait until they know the specific numbers and then combine each pair using the algorithm above (and that one is simple enough that humans can readily internalize it, just as we do the algorithm for adding and subtracting two signed numbers, which is similarly convoluted when you sit down and think about it). But the result is that if you change one parameter of one component in one box, they have to start almost from scratch (of course, they can keep track of the computations that aren't effected by the change, but they still end up redoing a large fraction of the work in most cases).

Instead, if you let the reactance carry the type information, you end up with the following:

$
Z_6 \, = \, R_6+jX_6 \, = \, (R_1+R_2) + j(X_1+X_2)
\
Z_7 \, = \, R_7+jX_7 \, = \, (R_3+R_4) + j(X_3+X_4)
\
Z_8 \, = \, Z_6||Z_7 \, = \, \frac{(R_6+jX_6)(R_7+jX_7)}{(R_6+R_7)+j(X_6+X_7)}
Z_8 \, = \, \frac{[(R_1+R_2)(R_3+R_4)-(X_1+X_2)(X_3+X_4)]+j[(R_1+R_2)(X_3+X_4)+(R_3+R_4)(X_1+X_2)]}{(R_1+R_2+R_3+R_4) + j(X_1+X_2+X_3+X_4)}
\
\
\ \text{or}
\
Z_8 \, = \, R_8+jX_8
\
\ \text{where}
\
R_8 \, = \, \frac{(R_1+R_2)(R_3+R_4)(R_1+R_2+R_3+R_4) \, + \, (R_1+R_2)(X_3+X_4)^2 \, + \, (R_3+R_4)(X_1+X_2)^2}{(R_1+R_2+R_3+R_4)^2 + (X_1+X_2+X_3+X_4)^2}
\
X_8 \, = \, \frac{(R_1+R_2)^2(X_3+X_4) \, + \, (R_3+R_4)^2(X_1+X_2) \, + \, (X_1+X_2)(X_3+X_4)(X_1+X_2+X_3+X_4)}{(R_1+R_2+R_3+R_4)^2 + (X_1+X_2+X_3+X_4)^2}
\
\
\ \text{and finally}
\
Z_{tot} \, = \, Z_5 + Z_8 \, = \, R_{tot}+jX_{tot}
\
\ \text{where}
\
R_{tot} \, = \, R_5 \, + \, \frac{(R_1+R_2)(R_3+R_4)(R_1+R_2+R_3+R_4) \, + \, (R_1+R_2)(X_3+X_4)^2 \, + \, (R_3+R_4)(X_1+X_2)^2}{(R_1+R_2+R_3+R_4)^2 + (X_1+X_2+X_3+X_4)^2}
\
X_{tot} \, = \, X_5 \, + \, \frac{(R_1+R_2)^2(X_3+X_4) \, + \, (R_3+R_4)^2(X_1+X_2) \, + \, (X_1+X_2)(X_3+X_4)(X_1+X_2+X_3+X_4)}{(R_1+R_2+R_3+R_4)^2 + (X_1+X_2+X_3+X_4)^2}
$

Unlike the prior case where we have all kinds of nexted "if this is capacitive and that is inductive and this is bigger than that then do this" steps, we have a single result that allows us to do something really valuable -- sanity check it. We know that the overall resistance of each impedance HAS to be positive (since we are talking about passive components here). Well, are R8 a Rtot positive (or at least non-negative)? It's obvious that both will be.

NOTE: I actually caught a mistake I made in multiplying by the complex conjugate of the demoninator -- which I did mentally against my better judgement -- and missed the 'conjugate' part of it at one point.

Now, that's not to say that these relationships are exactly "simple" or anyone's rational description of a pleasant dream, but they are very, very simple in comparison to what we were faced with the other way.

19. ### studiot AAC Fanatic!

Nov 9, 2007
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523
Yes but we only have 'signs' because of the phase angle. This refers to a time disfference in reality, not a directional difference. That is the current and voltage are still in the same direction.

sin90 = 1 is an equality, not an identity.

The rest of your post (wipes brow!) will take some studying on.

Last edited: Oct 6, 2013
20. ### WBahn Moderator

Mar 31, 2012
23,148
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But as soon as we start talking about reactance to begin with, we have moved into a transformed space in which "direction" has its own interpretation within that space.