Definition of impedance

[Stefan]

Ahhh, I don't understand Impedance!


I read everywhere that the Impedance is defined as the ratio of voltage and current:

Z = u(t)/i(t)

So lets take a capacitor:

If we define u(t) = sin(wt), then i(t)=Cw*cos(wt). Thous (edit: see olsenn's post)

Z = sin(wt)/(C*wcos(wt)) = tan(wt)/Cw.   (edit: see olsenn's post)

Oops... We expected Z=1/(j*w*C).

We get that result if we start with

u(t) = e^(jwt) and then continue from there, ok, but...

e^(jwt) is not sin(wt). sin(wt) is (e^jwt - e^-jwt) / 2j

In fact e^(jwt) it is a complex number, and a phsical voltage is not complex.
So how does this make any sense?

Wikipedia states this is due to superposition.

Superposition would allow us to look at the two exponential terms of sin
independently, which would give the expected result.
However, superposition works for linear functions, and Z=u/i is
a division, which is not linear at all.

So what is wrong here?

My bet is that the definition of impedance, as the ratio of voltage
and current, is inaccurate. Can anyone give me the correct
definition?

thanks,
Stefan

[olsenn]

If we define u(t) = sin(wt), then i(t)=C*cos(wt). Thous

First of all, the derivative of sin(wt) is w*cos(wt), not just cos(wt).

[Stefan]

Got me there. I fixed the post, thanks.

Still, I miss something more fundamental than that.

[c4757p]

In fact e^(jwt) it is a complex number, and a phsical voltage is not complex.
So how does this make any sense?

Herein lies the problem. Spend some time reading about phasors, because that's what exp(j w t) is. It's a complex number which is a scalar stand-in for a sinusoid. exp(j w t) = 1 exp(0j) exp(j w t), whose real part is equivalent to cos(w t) by Euler's formula.

Typically it would just be written as the 1 exp(0j) part, where 1 = amplitude and 0 = phase. (A exp(theta j) is equivalent to A cos(w t + theta))

[Stefan]

Ok, thanks for the hint. I think I got it now.

The correct definition of impedance is

The ratio between the phasors representing voltage and current.

Not the ratio between voltage and current directly.

Its actually a mathematical trick. The Real-part of a phasor is a sinusoidal, creating a bijective mapping between phasors and sinusoidals. Lots of math is simpler to do with phasors. To we first transform our sinusoidals into phasors, do the math there, and then transfrom them back into sinusoidals. Phasor arithmetic tells us which operations on phasors produce equivalent results for their sinusoidals.

So, say I have the current i(t) through a component with impedance Z, I can get the voltage v(t) by first finding the phasor I(t) with Re(I(t))=i(t), then multipying it with Z to get the phasor of the voltage U(t) = ZI(t). I can than get the voltage v(t)=Re(U(t)) easily.

Elegant. I didn't realize that Impedance was defined in a different domain. I bit like using Fourier for convolution and such. Is the name of the fellow known that came up with this?

Come to think of it, this phasor-transformation (or whatever it is called), is it related to the laplace transformation?

[c4757p]

Phasors, Laplace and Fourier are all pretty closely related. The Laplace transform is sort of a generaliztion of phasors to include transient as well as steady-state.

I've got no idea who first used phasors, and Google doesn't seem to know either, but it's tied so tightly to Euler's formula that it's pretty much an obvious extension thereof.

[Stefan]

Somehow my interpretation of 'obvious' tends to differ from that of my teachers. You are no exception here   

I guess when one plays around with Euler's formula (one of maths mankinds nature's true marvels), one eventually realizes that several operations are equivalent to their sinusoidals. So it was probably Euler himself.

[Excavatoree]

I've got no idea who first used phasors, and Google doesn't seem to know either, but it's tied so tightly to Euler's formula that it's pretty much an obvious extension thereof.

I was told by one of my EE professors that it was Charles Proteus Steinmetz.

[c4757p]

Somehow my interpretation of 'obvious' tends to differ from that of my teachers. You are no exception here   

I meant to a mathematician! Like this guy:

Charles Proteus Steinmetz.

Thanks!

[IanB]

OK, so I think you have it now, but the most important thing about impedance is that it is an abstraction that works when you are dealing with regular sinusoidal AC signals. It has no meaning when you are dealing with arbitrary, non-periodic voltages.

The other thing is that those j's that crop up everywhere are imaginary. Voltages and currents in real life are real numbers. Complex numbers do not exist in actual circuits, AC or otherwise.

What is really going on is that in AC circuits with reactive elements, sinusoidal signals tend to get phase shifted so that the current wave is out of phase with the voltage wave. It turns out that any degree of phase shift (e.g. 30 degrees) can be represented by adding together two signals of the same frequency, one at 0 degrees and one at 90 degrees. (If the phase angle needs to be -30 degrees or 120 degrees you can just invert either or both of the waves so that the multiplier is negative.)

So, here the magic happens. In the complex plane the imaginary axis is at 90 degrees to the real axis. So the mixture of two waves (0 degrees and 90 degrees) can be represented by a complex number where the real part is the 0 degree component and the imaginary part is the 90 degree component).

Mathematically, you end up with a complex number like "a + jb" representing a mixture of signals as in "a sin(wt) + b cos(wt)".

The impedance is still the ratio of voltage to current, but now the magnitude of the impedance is the amplitude of the voltage divided by the amplitude of the current, and there is a phase angle involved to show how much the current wave is out of phase with the voltage wave.

[Stefan]

Wikipedia agrees with you on Steinmetz, thanks Excavatoree

Euler probably realized it too, but found it far to obvious to bother writing it down; or maybe he just run out of paper like Fermat 


@IanB
Thanks for summing that up. This might help others that read this thread very much. I was already familiar with Euler's formula and implicitly assumed sinusoidal signals. Its a good thing to have this written down explicitly.

[IanB]

For reference, here is how the phase angle mathematics works without using complex numbers:

[IanB]

Ahhh, I don't understand Impedance!


I read everywhere that the Impedance is defined as the ratio of voltage and current:

Z = u(t)/i(t)

So lets take a capacitor:

If we define u(t) = sin(wt), then i(t)=Cw*cos(wt). Thous (edit: see olsenn's post)

Z = sin(wt)/(C*wcos(wt)) = tan(wt)/Cw.   (edit: see olsenn's post)

Oops... We expected Z=1/(j*w*C).

We get that result if we start with

u(t) = e^(jwt) and then continue from there, ok, but...

e^(jwt) is not sin(wt). sin(wt) is (e^jwt - e^-jwt) / 2j

In fact e^(jwt) it is a complex number, and a phsical voltage is not complex.
So how does this make any sense?

...

My bet is that the definition of impedance, as the ratio of voltage
and current, is inaccurate. Can anyone give me the correct
definition?

Here is a brief answer to the question above. Impedance really is the ratio of voltage to current, when you deal with amplitudes of AC signals:

[Kremmen]

For reference, here is how the phase angle mathematics works without using complex numbers:

Sure, that will work but don't you agree that lugging along a mess of sines and cosines and doing overly complicated manipulations is needless work when there is an elegant representation based on e?.

Bearing in mind that Euler's exponential notation describes rotations around the unit circle in the complex plane, where the basis (axes) is Re/Im, it is a perfect medium for representing frequencies and shifting phases. "Nasty" trig is replaced by simple algebra and you don't need to remember all the trig formulas for each case. Also the calculations themselves are "nicer" with far fewer terms. That is why electrical and electronic engineers favor the e notation without exception. All processing of time-varying signals is heavily dependent on Euler's notation and all commonly used methods and algorithms use it.

[IanB]

Sure, that will work but don't you agree that lugging along a mess of sines and cosines and doing overly complicated manipulations is needless work when there is an elegant representation based on e?.

Absolutely, the complex phasor notation is what you use in actual calculations.

But I always think it helps to have some understanding of what is going on before using a special notation. Especially since there are no complex numbers in real voltages and currents. The basis for the whole j notation is not something that was ever explained well to me, it was just assumed that it was easy to understand (it wasn't).

So my notes above are just intended as an aid to understanding, not an alternative calculation approach.

Even now, I don't think it is entirely obvious or intuitive that manipulations of complex numbers represented as (a,b) or (r,theta) maps onto superpositions or convolutions of sine waves with different magnitudes and phase angles.

[Footnote: mathematically, a complex number a + jb is represented as a pair of numbers (a,b) and a set of rules for manipulating them. Whether you treat your pair of numbers in rectangular coordinates as (a,b) or polar coordinates as (A,w), you are still really carrying along sines and cosines and doing manipulations of them.]

[Kremmen]

Absolutely, the complex phasor notation is what you use in actual calculations.

But I always think it helps to have some understanding of what is going on before using a special notation. Especially since there are no complex numbers in real voltages and currents. The basis for the whole j notation is not something that was ever explained well to me, it was just assumed that it was easy to understand (it wasn't).

I agree that the e notation may be somewhat counterintuitive initially, and more time should perhaps be spent in the initial introduction. I do recall a similar hurdle in my own time, when first wrapping my brain around the concept.

So my notes above are just intended as an aid to understanding, not an alternative calculation approach.

Understanding the correlation between the e notation and trigonometry is a must, of course so in that sense you do need the trig once you come back up for air after a long calculation. At the same time, it is almost as if by design, how well the complex notation and time-varying signals fit together.

Even now, I don't think it is entirely obvious or intuitive that manipulations of complex numbers represented as (a,b) or (r,theta) maps onto superpositions or convolutions of sine waves with different magnitudes and phase angles.

This is actually one of the central fundaments of digital signal processing. I take an example - it is a bit involved but bear with me.

The whole idea of the Fourier transform in its various forms starts from the assumption of a superposition of sine waves. The Discrete Fourier Transform exmplifies this beautifully: you start with a signal of N samples in length. You can think of the signal as a vector in a N-dimensional (Hilbert) space C^N, extending from the origin to some direction in the hyperspace, defined by the coefficients n (n=0,1,...n) of the signal sample. Since the sample was collected in time, the space where the vector lies is defined in the time domain by its basis (i.e. the vectors that tune the N-space). To visualize, think of the familiar 3-space and the cartesian coordinates x/y/z. Those coordinates are tuned by the unit vectors i,j,k that together form the basis of the coordinate system. This basis is orthonormal since the vectors are all at mutual 90 degree angles and of equal norm , i.e. equally long (of length 1). You can see this easily in your head when you think about it. So all coordinates we give in x/y/z are expansions in this space and the coefficients are multiples of the unit vectors. Thus point 3,5,-2 = 3*i,5*j,-2*k). Mathematically, the basis vectors don't need to be orthonormal, or indeed even orthogonal. They can be at any mutual angles as long as no two vectors are collinear so that coordinates in this space are a linearly independent combination of the basis vectors.

So we have this sample vector in the n-space in time domain. The time domain signal x can be expressed as the sum of x(n)*e^k where each sample is a Dirac delta function d(n-k). Next we expand the vector out of the time domain, into another basis. This being the Fourier transform the new basis is of course the frequency domain. Expansion just means that we project the signal vector into each individual tuning vector in the new basis, to obtain the signal coefficient for that coordinate vector. In practice we calculate the vector inner product between the signal and each of the basis vectors.
Enter e.
The orthogonal basis of the frequency domain is defined in a Hilbert space of dimensionality C^N as a set of vectors w_k(n) = e^j*2*pi/N*n*k; (n,k=0,1,...,N-1). This basis can be shown to be orthogonal but i am not going to go there.
The change of basis expresses the signal not as a linear combination of delta functions, but as a linear combination of sinusoids. If you inspect the basis of the frequency domain more carefully, you can see that it is a set of sinusoids of varying frequency. In fact from 0 to N/2. The Fourier transform "maps" the signal against all of these sinusoids and extracts the "amount" of match between the particular sinusoid and the signal (i.e. projects the signal on this particular basis vector).

OK, sorry about this lengthy yarn. But the point is that a) there are plenty of manipulations and superpositions of sinusoids and b) that doing all of the vector operations in sin/cos notation would be about as nice as a hot poker in the ass. Not to mention that there are critical optimizations for real life performance, that are only obvious in e notation, really.

[Footnote: mathematically, a complex number a + jb is represented as a pair of numbers (a,b) and a set of rules for manipulating them. Whether you treat your pair of numbers in rectangular coordinates as (a,b) or polar coordinates as (A,w), you are still really carrying along sines and cosines and doing manipulations of them.]

To be sure. But it makes all the difference in the world if you know how to pick the optimal representation for each case. For technical calculations, especially DSP, the e notation can't be beaten.