Hi,

I think i agree with that, but not sure what you mean by "in general you have to allow for two inductances and two capacitances".

I think the simplified impedance formula is just made for a series RLC.

Doing the parallel case the long way, we get:
N=j*(w*L)/(w*C)
D=1/(w*C)-w*L

where D is the denominator and N the numerator, and in this form it is clearly in the form of:
xL*xC/(xC-xL)

when we define xC=1/(w*C) and xL=w*L.

So that agrees with your conclusion, but we dont actually have to test with the inequality do we?

Are you saying you want to have four reactances in total like xL1, xL2, xC1, xC2, in order to qualify for a simplified formula? That is interesting yes, but i am not sure they take it that far normally. We could look at it though in more detail.

 

Hi,

I think i agree with that, but not sure what you mean by "in general you have to allow for two inductances and two capacitances".

I think the simplified impedance formula is just made for a series RLC.

Doing the parallel case the long way, we get:
N=j*(w*L)/(w*C)
D=1/(w*C)-w*L

where D is the denominator and N the numerator, and in this form it is clearly in the form of:
xL*xC/(xC-xL)

when we define xC=1/(w*C) and xL=w*L.

So that agrees with your conclusion, but we dont actually have to test with the inequality do we?

Sure we do. What if xL > xC? Then you have a negative value and you've defined all reactances to be positive. So a negative reactance is undefined. Therefore you have to perform tests and use a variant of the formula that will result in a positive value and that also tells you whether that positive number is capacitive or reactive.

It is simply not good enough to plug into a formula and come up with a result that says the combined reactance is 12.34 Ω. Is that capacitive or inductive? You don't know. So you have to have a decision tree that considers all possible cases and, for each case, tells you whether the result is capacitive or inductive and uses a formula that results in a positive value.

 

then to find the effective reactance of two reactances in parallel:

if (X1 and X2 are both inductive)
---- X = (X1·X2)/(X1 + X2) (inductive)
else if (X1 and X2 are both capacitive)
---- X = (X1·X2)/(X1 + X2) (capacitive)
else if (X1 is inductive and X2 is capacitive)
---- if (X1 > X2)
---- ---- X = (X1·X2)/(X1 - X2) (capacitive)
---- else
---- ---- X = (X1·X2)/(X2 - X1) (inductive)
else if (X1 is capacitive and X2 is inductive)
---- if (X1 > X2)
---- ---- X = (X1·X2)/(X1 - X2) (inductive)
---- else
---- ---- X = (X1·X2)/(X2 - X1) (capacitive)
else
---- X = infinity

Obvious error corrected in red.

 

Obvious error corrected in red.

Thanks! I had almost commented in that post that the procedure was so complicated that there was a good chance that I had a mistake in it that I hadn't caught.

 

Sure we do. What if xL > xC? Then you have a negative value and you've defined all reactances to be positive. So a negative reactance is undefined. Therefore you have to perform tests and use a variant of the formula that will result in a positive value and that also tells you whether that positive number is capacitive or reactive.

It is simply not good enough so plug into a formula and come up with a result that says the combined reactance is 12.34 Ω. Is that capacitive or inductive? You don't know. So you have to have a decision tree that considers all possible cases and, for each case, tells you whether the result is capacitive or inductive and uses a formula that results in a positive value.

Hello again,

Ok, so you want to define all reactances as positive, always. Then yes, you'll have to do a comparison. If you at least allow one to be negative though, then you dont have to compare anything because the logic follows from the sign of the result.

But i thought we were talking about two different things:
1. A formula
2. Defining either all positive reactances or allowing the capacitive to be negative.

To define a formula, we can do it either way right?
We can either:
Make the formula include the right sign, or make the formula general and then allow the capacitive reactance xC to be negative.

I dont see too much difference here because for one example we can always do this:
X=abs(parallel(x,x))
then L or C type reactance is defined by:
S=sgn(parallel(x,x))
where S is the type, and S=1 means inductive and S=-1 means capacitive, and S undefined means X=infinity.

So we sort of remove the requirement for the restriction on the sign of the reactance by simply calling the result an intermediate result, then finding the type and the value.

Is it that you are wanting to create a general purpose formula for three or four reactances?

 

Hello again,

Ok, so you want to define all reactances as positive, always. Then yes, you'll have to do a comparison. If you at least allow one to be negative though, then you dont have to compare anything because the logic follows from the sign of the result.

THAT'S WHAT I'VE BEEN SAYING!!!!

You can't have it both ways (unless you want to invoke more magical mystery minus signs!).

If you want to define both capacitive and inductive reactances to be positive, they negative reactances are undefined.

But i thought we were talking about two different things:
1. A formula
2. Defining either all positive reactances or allowing the capacitive to be negative.

Formulas need to be consistent with the definitions of the variables that are plugged into them.

To define a formula, we can do it either way right?
We can either:
Make the formula include the right sign, or make the formula general and then allow the capacitive reactance xC to be negative.

But how can you allow capacitive reactance to be negative if you've defined capacitive reactance to be positive? You can't have it both ways!!!!

I dont see too much difference here because for one example we can always do this:
X=abs(parallel(x,x))
then L or C type reactance is defined by:
S=sgn(parallel(x,x))
where S is the type, and S=1 means inductive and S=-1 means capacitive, and S undefined means X=infinity.

But if you have defined xL=wL and xC=1/(wC), then both of the x's that are put into parallel(x,x) are positive values and the result is then positive. Unless, of course, you start throwing magical mystery minus signs at everything in sight in order to force things to work out the way you want them to.

 

THAT'S WHAT I'VE BEEN SAYING!!!!

You can't have it both ways (unless you want to invoke more magical mystery minus signs!).

If you want to define both capacitive and inductive reactances to be positive, then negative reactances are undefined.



Formulas need to be consistent with the definitions of the variables that are plugged into them.



But how can you allow capacitive reactance to be negative if you've defined capacitive reactance to be positive? You can't have it both ways!!!!



But if you have defined xL=wL and xC=1/(wC), then both of the x's that are put into parallel(x,x) are positive values and the result is then positive. Unless, of course, you start throwing magical mystery minus signs at everything in sight in order to force things to work out the way you want them to.

Hi,

He he, ok then.

But my main point was more or less in reference to the general statement you made, which is true but needs a contextual background:
"If you want to define both capacitive and inductive reactances to be positive, then negative reactances are undefined."

It sounds very clear and concise, but if you investigate, it's not as clear as we would like. The reason is because we can decide to make the reactance negative in the FORMULA, and then keep the rest positive, and come out with a signed result, then convert that into a comparative result. Or, we could keep the formula positive, then use a negative capacitive reactance. The only thing we cant do is make them both positive or both negative.
In your formula, we can make them both positive because we do the compare directly.

So that's what i was getting at really, that's all. It works one way or another

 

If you define reactance to be positive for both inductive and capacitive reactance and don't have a problem with doing so, they is it too much to ask that you be consistent with that choice? If reactance is positive, then what does a negative reactance mean? If negative reactance isn't well-defined as to what it means, then how is using formulas that produce negative reactances a good idea? What you are doing is, just like the people using electron flow current, using formula that are inherently inconsistent with your definitions and then magical mystery minus signs to patch things up. I know that you are okay with that. I'm not. So, as I've said before, go ahead and use your magical mystery minus signs all you want. Just don't expect me to ever say that it is proper.

 

If you define reactance to be positive for both inductive and capacitive reactance and don't have a problem with doing so, they is it too much to ask that you be consistent with that choice? If reactance is positive, then what does a negative reactance mean? If negative reactance isn't well-defined as to what it means, then how is using formulas that produce negative reactances a good idea? What you are doing is, just like the people using electron flow current, using formula that are inherently inconsistent with your definitions and then magical mystery minus signs to patch things up. I know that you are okay with that. I'm not. So, as I've said before, go ahead and use your magical mystery minus signs all you want. Just don't expect me to ever say that it is proper.

Hi again,

Well please dont get me wrong, i like what we are calling "your way" of doing it too, or rather, the other way that you mention. I am just not sure where you were headed with that. Certainly nothing wrong with it, unless of course that it doesnt carry over to more complex calculations, but then we could just specify it for a limited number of components.

In a program i dont think it would matter because neither one of us would ever see the comparison being done or the sign. We'd probably have the magnitude and phase angle spit out.

But i might be interested to see where you were going with this, If it was for a multiple element circuit we'd have to look at all the comparisons we'd have to do. I just like using a sign sometimes when it carries the meaning through the entire problem, and especially if it ends up being very universal. For example, and i know this isnt exactly the same thing, but in the complex math form we might end up with a sign for the imaginary part, which we have to keep in order to proceed with the calculations if we add anything else later. If we drop the sign, we have to remember that it gets subtracted.

On that site "hyperphysics" they say there is no formula for parallel impedances and they jump right into complex math. Not sure what's up with that.

 

Hi again,

Well please dont get me wrong, i like what we are calling "your way" of doing it too, or rather, the other way that you mention. I am just not sure where you were headed with that. Certainly nothing wrong with it, unless of course that it doesnt carry over to more complex calculations, but then we could just specify it for a limited number of components.

In a program i dont think it would matter because neither one of us would ever see the comparison being done or the sign. We'd probably have the magnitude and phase angle spit out.

But i might be interested to see where you were going with this, If it was for a multiple element circuit we'd have to look at all the comparisons we'd have to do. I just like using a sign sometimes when it carries the meaning through the entire problem, and especially if it ends up being very universal. For example, and i know this isnt exactly the same thing, but in the complex math form we might end up with a sign for the imaginary part, which we have to keep in order to proceed with the calculations if we add anything else later. If we drop the sign, we have to remember that it gets subtracted.

On that site "hyperphysics" they say there is no formula for parallel impedances and they jump right into complex math. Not sure what's up with that.

Not sure where I'm headed? How about what I stated back in Post #13: "When you have X_C being positive and using (X_L - X_C), you set yourself up for problems and limit yourself to only being able to work problems in small chunks and for actual values instead of parametrically." Just look at how complicated it is to add two unknown reactances in series, let along in parallel. The complexity literally explodes exponentially, so imagine what a nightmare expressing a generic delta-wye conversion or Wheatstone bridge.

Pherhaps what's up with Hyperphysics saying that there is no formula for parallel impedances (could you post a link to where they state that, I couldn't find it) is because they use positive capacitive reactance and thus things are to complicated to reduce to a consistent formula. But, even so, their site is inconsistent. They state that capacitive reactance is positive:

http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html

Yet when they introduce complex impedance they say that it is Z = R + jX

http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html

So while they talk about negative and positive phase angles, their own definition of capacitive reactance doesn't allow for negative phase angles to ever occur since X is positive for both capacitive and inductive reactance. Unless, of course, they had magical mystery minus signs running around -- which is exactly what happens.

 

Hello again,

I cant find it again on Hyperphysics either. They state that "it is considerably more difficult" and that is because they dig right into complex numbers. I must have seen it on another site, but i checked Wikipedia and it's not there either. Therefore i set out to see if i could find a formula, but every formula doesnt seem to help much although it does come out to the same formula given some tricks we have to use, such as in the case of one L and one C in parallel and that in series with one R. If we keep the L and C part separate and include the R as the last step, then the formula comes out the same, but it may not always be apparent what to do here. It may be that we calculate all the L's and C's first, then later include the pure R's, then keep any extra terms due to that included R as part of the real part and keep the original L and C part as the imaginary part. I havent gone over this too well yet though, but it does seem to work with the L and C and then R in series. Trouble is, it's not that much simpler anyway, if at all. The only simplification is that we get away with not having to use complex numbers, which isnt that great because it's still fairly complicated. If it came out a lot simpler i would say yeah, let's do it, but it's still too complicated a procedure even without using complex numbers.

What we could do is look at a few more examples. I did the LC in series with R, and i did four impedances in parallel, and they work out similar using a breadbox formula, but it is still so complicated that i had to keep asking myself, "Is this really going to help even if it does work?"

 

Not sure where I'm headed? How about what I stated back in Post #13: "When you have X_C being positive and using (X_L - X_C), you set yourself up for problems and limit yourself to only being able to work problems in small chunks and for actual values instead of parametrically." Just look at how complicated it is to add two unknown reactances in series, let along in parallel. The complexity literally explodes exponentially, so imagine what a nightmare expressing a generic delta-wye conversion or Wheatstone bridge.

Pherhaps what's up with Hyperphysics saying that there is no formula for parallel impedances (could you post a link to where they state that, I couldn't find it) is because they use positive capacitive reactance and thus things are to complicated to reduce to a consistent formula. But, even so, their site is inconsistent. They state that capacitive reactance is positive:

http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html

Yet when they introduce complex impedance they say that it is Z = R + jX

http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html

So while they talk about negative and positive phase angles, their own definition of capacitive reactance doesn't allow for negative phase angles to ever occur since X is positive for both capacitive and inductive reactance. Unless, of course, they had magical mystery minus signs running around -- which is exactly what happens.

Those links only get you to the top of their pages. However, if you traverse the tree until you reach the major topic "Capacitive Reactance", the very first sentence after that says "The frequency dependent impedance of a capacitor is called capacitive reactance."

How do you like that? "...impedance...is called...reactance"

 

Those links only get you to the top of their pages. However, if you traverse the tree until you reach the major topic "Capacitive Reactance", the very first sentence after that says "The frequency dependent impedance of a capacitor is called capacitive reactance."

How do you like that? "...impedance...is called...reactance"

I actually don't have too much of a problem with that. I might phrase it differently and say something like, "the impedance of a capacitor is purely reactive, meaning that it has no resistive component."

 

Hello again,

I cant find it again on Hyperphysics either. They state that "it is considerably more difficult" and that is because they dig right into complex numbers. I must have seen it on another site, but i checked Wikipedia and it's not there either. Therefore i set out to see if i could find a formula, but every formula doesnt seem to help much although it does come out to the same formula given some tricks we have to use, such as in the case of one L and one C in parallel and that in series with one R. If we keep the L and C part separate and include the R as the last step, then the formula comes out the same, but it may not always be apparent what to do here. It may be that we calculate all the L's and C's first, then later include the pure R's, then keep any extra terms due to that included R as part of the real part and keep the original L and C part as the imaginary part. I havent gone over this too well yet though, but it does seem to work with the L and C and then R in series. Trouble is, it's not that much simpler anyway, if at all. The only simplification is that we get away with not having to use complex numbers, which isnt that great because it's still fairly complicated. If it came out a lot simpler i would say yeah, let's do it, but it's still too complicated a procedure even without using complex numbers.

What we could do is look at a few more examples. I did the LC in series with R, and i did four impedances in parallel, and they work out similar using a breadbox formula, but it is still so complicated that i had to keep asking myself, "Is this really going to help even if it does work?"

If we have only positive reactances, it is always going to get so complicated so fast that, in any practical sense, we are stuck with working with actual values (which means that we can't work out general formulas for any but the very simplest circuits) and cranking the numbers on those in little bits so that we always know whether our calculated reactance value is capacitive or inductive as we move on.

But if we define capacitive reactance to be negative, then we can combine things very simply -- not quite as simply as when we use complex impedances, to be sure, but we have a lot more flexibility and expressive power.

 

Hi,

Yeah either way it gets hairy fast. Complex numbers is the best way in the more general case. Using a number cruncher there is less chance of human error that way too.

 

Hello again,

Hey, i found a simple transformation for the series to parallel impedance calculation that sort of makes it easier, or at least simpler to construct for the three elements R, L, and C.

First, the series circuit RLC:
|Z|=sqrt(R^2+X^2)=sqrt(R^2+(xL-xC)^2)

and written out this is:
|Z|=sqrt(R^2+(w*L-1/(w*C))^2)

That's the series circuit again. Now for the parallel, we just use the dual of EVERYTHING, and when i say everything i mean everything

I'll do this in small steps, starting with the solution for the series circuit, so the intermediate steps wont be valid, just the first and last step here:

Series:
sqrt(R^2+(w*L-1/(w*C))^2)

Start by taking the dual of L and C, which here means we swap L and C, and get:
sqrt(R^2+(w*C-1/(w*L))^2)

Now we replace R with it's dual, and get:
sqrt(1/R^2+(w*C-1/(w*L))^2)

Now we take the dual of that entire expression, and get:
1/sqrt(1/R^2+(w*C-1/(w*L))^2)

and that last step is the |Z| magnitude of the impedance for a parallel RLC circuit.

Pretty cute

 

AC impedences are represented by complex numbers.
A pure resistive complex number has just a positive real part, whose value corresponds to the resistance.
A pure capacitive complex number has just a negative imaginary part and the coefficient is the capacitive reactance.
A pure inductive features a positive imaginary part and the value is the inductive reactance.
Series - parallel equivalent impedance can be calculated with the same method used in DC circuits, but you have to operate with complex numbers.
http://andrea1972.altervista.org/files/AC-circuits.pdf

 

AC impedences are represented by complex numbers.
A pure resistive complex number has just a positive real part, whose value corresponds to the resistance.
A pure capacitive complex number has just a negative imaginary part and the coefficient is the capacitive reactance.
A pure inductive features a positive imaginary part and the value is the inductive reactance.
Series - parallel equivalent impedance can be calculated with the same method used in DC circuits, but you have to operate with complex numbers.
http://andrea1972.altervista.org/files/AC-circuits.pdf

What you're missing is that the context of the discussion is about working with impedances and reactances WITHOUT using complex numbers. Sadly, many texts that work at this level insist on treating both capacitive and inductive reactance as positive quantities, meaning that reactance is a combination of a value and a label (e.g., 3.4 Ω capacitive) and so formulas abound with terms such as (Xl-Xc). The end result is that people are unnecessarily limited in the complexity of generic circuits they can deal with because the decision tree to deal with the labels becomes unmanageable very quickly.

 

Since OP is long gone and is not coming back any time soon. Can I post my solution?

 

Since OP is long gone and is not coming back any time soon. Can I post my solution?

I don't see why not.