# Impedance defined?

1. Jul 31, 2014

### EEngineer91

Hi,

I was reading Charles Steinmetz "Theory and Calculation of Alternating Current Phenomenon" and on page 89 (can be found online easily) Steinmetz defines impedance as: Z = R - jX, but see it commonly defined as R + jX. I read on wikipedia there are two impedance equations for capacitive impedance and inductive (R - jX, R + jX), but Steinmetz doesn't mention either or suggest this is a special case of impedance, just that it applies for alternating waves.

So, did Steinmetz make a mistake in his definition? How do we know which impedance he is talking about?

I am confused because I doubt Steinmetz defined it incorrectly, yet don't know where the sign difference comes from? I get the wikipedia explaination of phase difference, but Steinmetz says its the impedance, I don't think he meant just capacitive or inductive.

EDIT: Looking further into the book, he only uses Z = R - jX throughout the whole book, seems to be the general definition, regardless of inductive or capacitive

Thoughts? Thanks.

Last edited: Jul 31, 2014
2. Jul 31, 2014

### olivermsun

It's just a convention. If you use -jX then your X will have flipped sign relative to the X that you would have using the +jX convention.

3. Jul 31, 2014

### EEngineer91

If it's just a convention, then why didn't Steinmetz define it as R + jX .... one implies lagging by 90 degrees, one implies leading by 90 degrees...I see that you get same amplitudes and phases calculations

4. Jul 31, 2014

### AlephZero

According to Wikipedia, Steinmetz invented the use of complex numbers in circuit analysis. Considering the book was published more than 100 years ago, it's not surprising that his notation conventions are not exactly the same as what is used now.

(I haven't read any of the book beyond the title page and the publication date).

5. Jul 31, 2014

### AlephZero

The interpretation of "lagging" or "leading" depends how you define Z. If you interpret the notation used in book written in 1897 in terms of conventions used in 2014, you can expect to be confused.

6. Jul 31, 2014

### EEngineer91

....still doesn't explain why he chose - instead of +....if it is the same, then why not define it like he did admittance, with a + sign? We define Z by its rectangular components, intensity and phase from the positive horizontal axis.... there is a huge difference between R + jX and R - jX graphically, although the phase difference is the same

7. Jul 31, 2014

### olivermsun

Well, if admittance is 1/Z = 1/(R - jX), then what happens when you rewrite it with the complex part in the numerator?

And again but more explicitly: R + jX1 and R - jX2 work out to be exactly the same if X1 = -X2.

8. Jul 31, 2014

### EEngineer91

It gives the same values, yes, but why originally choose -? Why not use +?

9. Jul 31, 2014

### olivermsun

I dunno. Maybe he liked $+j\omega$ to be clockwise.

10. Aug 1, 2014

### AlephZero

There is a basic choice to be made in using complex numbers here.

Option 1 is to describe time varying quantities as $p \cos \omega t + q \sin \omega t$. The "obvious" names for $p$ and $q$ are the "(in) phase" and "quadrature" parts of the quantity.

Option 2 (which is now more or less universal) is to use the real part of $Ae^{j \omega t}$, where the real and imaginary parts of $A$ correspond to "phase" and "quadrature".

The sign of the quadrature term is different for the two options.

11. Aug 1, 2014

### EEngineer91

Again, one implies lagging by 90 degrees, one is leading by 90 degrees....im sure this is two different situations...

12. Aug 1, 2014

### olivermsun

And again, leading and lagging phase depends on the actual value of the impedance, not on the sign convention. Once you know your sign convention then you follow that convention when you compute the complex impedance for, e.g., a capacitor. The current is always going to lead the voltage across a capacitor no matter how you write it.

Also, as I alluded to above, if you like your admittance to look like Y = G + jB, with a positive sign, then Z = 1/Y = C * (G - jB) kind of implies a negative sign.

13. Aug 1, 2014

### AlephZero

I was right - the OP's question is about history not engineering.

From the Preface to the 5th Edition, 1916 (see https://archive.org/stream/ed5theorycalcula00steiuoft#page/n12/mode/1up)

(followed by a whine that his original method published in 1897 was "better".)

14. Aug 1, 2014

### psparky

Here's my two cents

Impedance is defined as R +Jx

x is the reactance which is either defined as ωL (inductive) or a vector pointing straight up.

Or the reactance is defined as -ωC (capacitive) or a vector pointing straight down.

Works for me.

15. Aug 1, 2014

### olivermsun

Nice find!

He goes on to explain in Ch. VII (p. 49–52) how the crank and polar diagramming conventions (essentially, clockwise vs. anti clockwise phase progression) can be reconciled by a corresponding sign change everywhere.