How capacitor blocks dc current?

[Ratch]

sophiecentaur,

Umm. I'm not sure what current flowing "physically" means.

You will never observe me using the phrase "current flow". That literally means "charge flow flow", which is redundant and ridiculous. Charge does not flow twice, so you are correct to wonder about it. I always refer to current as "existing" or "passing through" or having a "direction", but never flowing. Charge can flow, however.

Let's ignore the electrons bit because that just clouds the issue.

Electrons cannot be ignored. They are the primary charge carriers in metals. When electrons move, current exists.

A current flows in one end and out of the other end of a resistor or a capacitor. Kirchoff's laws work perfectly well in most circuits. How are the two cases different?

Kirchoff's Current Law (KCL) is not violated in either case. Here's why. KCL simply says that all charges have to be conserved. They cannot travel down a wire and just disappear. There is no problem with charges passing through a resistor, but it is a little more subtle when charges encounter a capacitor. A capacitor is a energy storage element, and it stores charges on one side of the dielectric insulator and supplies electrons from the opposite of the dielectric, as I said several times before. This separation and accumulation of electrons causes a back voltage to form which diminishes the current in the branch containing the capacitor. It takes energy to accumulate and deplete the electrons, and this energy is stored in an electric field within the dielectric. Nevertheless, every electron is accounted for according to KCL. There is also a transitory current in the circuit branch containing the capacitor.

Is it really worth labouring the point that 'charging a capacitor' is not the same thing as 'charging' a comb by rubbing it?

It is good to know how a capacitor works, even if it is different than generating high voltage by rubbing a comb.

I think we agree that 'yer actual DC' does not exist, because that would involve infinite time for it to be established. So, if DC is a pragmatic term for 'constant value for long enough', then the reactance (let's just call it Impedance, in fact) to DC is just as meaningful as at any frequency of AC. At our newly defined version of DC frequency (<0.0001Hz, say), the impedance is (to all intents and purposes) infinite.

A capacitor is going to follow its energizing voltage. Unless there is no resistance in the circuit, it will have a time delay that a resistor does not have. This time delay is caused by having to imbalance or even out the charge between its plates. A capacitor energized by a step voltage of constant amplitude will have an impedance whose magnitude is infinite, but with an orthoginal orientation. That makes it different than just an open circuit. Mathematically, it is described as -j∞.

But I still don't see why you guys don't want to use Maths (or at least refer to it) to describe what goes on. The exponential charge / discharge of a CR network describes exactly what goes on and that can be re-stated in terms of frequencies and Impedance. The results of experiment agree so well with that simple theory and it isn't difficult to approach the 'ideal case' in practice. That's why we can design filters and other circuits to work in such a predictable way.

We are using a minimum of mathematics, because for nonsinusoidal circuits, differential equations (DE) are necessary to calculate and understand the response. Not everyone is up to speed on DE.

Discussing "what's really happening" is not really getting one any closer to an understanding unless you really want to get into QM and how materials behave.

I think we are closer to the micro level than the quantum level. I do think it is necessary to understand what really goes on rather than using hydraulic analogies and other fool's aids to describe what is not really happening.

Ratch

 

[sophiecentaur]

I do think it is necessary to understand what really goes on rather than using hydraulic analogies and other fool's aids to describe what is not really happening.

Ratch

I am pleased that we are at one in that respect!

A capacitor is going to follow its energizing voltage. Unless there is no resistance in the circuit, it will have a time delay that a resistor does not have. This time delay is caused by having to imbalance or even out the charge between its plates. A capacitor energized by a step voltage of constant amplitude will have an impedance whose magnitude is infinite, but with an orthoginal orientation. That makes it different than just an open circuit. Mathematically, it is described as -j∞.

Not sure what this all means. It is a bit wooly and you appear to be wanting to make a distinction that is quite artificial. You are mixing your domains here, in any case. 'Impedance' and 'orthogonal' are frequency domain terms and 'step function' is a time domain term.

 

[Ratch]

sophiecentaur,

'Impedance' and 'orthogonal' are frequency domain terms

They shouldn't be. You can't plot a frequency spectrum from a impedance, and orthogonal is simply a definition which means "at right angle to".

Ratch

 

[sophiecentaur]

sophiecentaur,



They shouldn't be. You can't plot a frequency spectrum from a impedance, and orthogonal is simply a definition which means "at right angle to".

Ratch

Time and frequency domains are interchangeable with the appropriate transforms but the appropriate operations in each domain are different. Impedance refers to the ratio of Volts to Current for a given frequency. It's a 'frequency domain term'. It is used to calculate the effect of a circuit component on the frequency spectrum of a signal. Can you show an example where Impedance comes into the description of an operation in the time domain (without some sort of transform being involved)? If you want to work in the time domain then you need to specify impulse responses and not impedances.

Is there a "right angles' (I think you actually mean 90 degree phase, in this context) in the time domain? In fact, of course, the term Orthogonal has many other connotations than just "at right angles".

I guess this flurry of attempted pedantry is due to a perceived slight concerning "current flow". In fact, if you read back, you'll see that I was picking up on someone else's post and not yours so you can calm down.

 

[Ratch]

sophiecentaur,

Impedance refers to the ratio of Volts to Current for a given frequency. It's a 'frequency domain term'.

All right, I will concede that impedance is frequency dependent if storage elements are involved.

Is there a "right angles' (I think you actually mean 90 degree phase, in this context) in the time domain? In fact, of course, the term Orthogonal has many other connotations than just "at right angles".

Yes, I was. As in a duplex (complex) number, where x + jy means x units on a reference scale and b units on a scale 90° CCW from it.

I guess this flurry of attempted pedantry is due to a perceived slight concerning "current flow". In fact, if you read back, you'll see that I was picking up on someone else's post and not yours so you can calm down.

You guessed wrong on that one. I was not even thinking of "current flow" when I answered your post.

Ratch

 

[sophiecentaur]

sophiecentaur,



All right, I will concede that impedance is frequency dependent if storage elements are involved.



Yes, I was. As in a duplex (complex) number, where x + jy means x units on a reference scale and b units on a scale 90° CCW from it.



You guessed wrong on that one. I was not even thinking of "current flow" when I answered your post.

Ratch

I don't understand that at all. Surely you can describe the reactance of a pure resistor as beingzero just the same as the reactance of a 'capacitor' can be -753Ω. You seem to have a problem with 'zeros' in this business - as, also, with DC means f=0. What's so special about "storage elements"?

 

[Ratch]

sophiecentaur,

I don't understand that at all.

What don't you understand?

Surely you can describe the reactance of a pure resistor as being zero just the same as the reactance of a 'capacitor' can be -753Ω.

Certainly the reactance of an ideal resistor is zero ohms, and it also has a real part that is frequency independent. The reactance of the ideal capacitor you to which you refer would be -j753 ohms, and have no real frequency independent component.

You seem to have a problem with 'zeros' in this business - as, also, with DC means f=0.

That seems to be a simple enough concept. What make to think that?

What's so special about "storage elements"?

Their ability to receive energy from the circuit, and return the same energy back to the circuit at a different times, makes their voltage and currents different than nonstorage elements like resistors which dissipate energy away from the circuit. This is best observed in sinusoidal waveforms where phase differences occur between current and voltage of coils and caps, but no phase differences are present in resistive components.

Ratch

 

[sophiecentaur]

The term "storage element" is a bit too non-specific, to my mind. After all, you can make a simple chemical battery with virtually no reactive impedance (certainly no inherent reactance). Why not just use the term 'Reactive'?
And, incidentally (I expected you to pick me up on this one so I looked in wiki and other places, to make sure I got it right) X stands for reactance and is measured in Ohms. Impedance is a complex quantity and is R+jX. So the reactance of a Capacitor is -1/ωC, with no j. I was correct to quote a reactance of -753Ω. If it were not this way, we would write
Z = R + X
Pedantry can turn round and bite you.

I suggested that you may have a problem with zeros because, on two occasions, you have treated a zero value (Reactance or Frequency) as somehow different from finite values.

 

[Ratch]

sophiecentaur,

The term "storage element" is a bit too non-specific, to my mind. After all, you can make a simple chemical battery with virtually no reactive impedance (certainly no inherent reactance).

The term "storage element" does indeed describe a coil and capacitor, but it is not a comprehensive definition. A battery or DC supply acts differently because they do not take energy from the circuit and return it later. "Reactive" also describes coils and capacitors, but does not describe the storage aspects of those elements. The bottom line is that one cannot describe everything about an object in one word.

the reactance of a Capacitor is -1/ωC, with no j. I was correct to quote a reactance of -753Ω.

Wiki is wrong abou that, as they are wrong about a lot of things. Reactance of a coil or capacitor is a complex quantity, and it needs its "j". Look at the link below, and notice that they get it right. If you don't put in a "j", then how do you distinguish a reactance from a resistance mathematically?

Pedantry can turn round and bite you.

Not this time.

I suggested that you may have a problem with zeros because, on two occasions, you have treated a zero value (Reactance or Frequency) as somehow different from finite values.

Infinities and infinitesimals have to be treated with caution. Perhaps you can provide an example in a previous post where I have been wrong about that subject. I hate to be wrong.

Ratch

http://www.st-andrews.ac.uk/~www_pa/Scots_Guide/info/signals/complex/react.html

 

[SunnyBoyNY]

The term "storage element" does indeed describe a coil and capacitor, but it is not a comprehensive definition. A battery or DC supply acts differently because they do not take energy from the circuit and return it later. "Reactive" also describes coils and capacitors, but does not describe the storage aspects of those elements. The bottom line is that one cannot describe everything about an object in one word.

The term storage element is circuit analysis is bound to inductive and capacitive elements, because those are the ones causing dynamic behavior. Reactive impedance are equal to the eigenvalues of a system (poles).

A battery could be simply modeled as a huge capacitor such that the associated eigenvalue would be so small that all other pertinent transients would be long finished before the battery energy changes ever-so-slightly. Since poles in typical RLC circuits are rather fast, we could reduce the order of the system by modeling a battery by a constant voltage source.

 

[Ratch]

SunnyBoyNY,

A battery could be simply modeled as a huge capacitor...

A huge capacitor will resonate with a huge coil; a battery will not. As I mentioned before, a battery does not take energy from the circuit and return it back in equal amounts at a different time as a capacitor would. Therefore, resonance is not possible with a battery, and the circuit will not respond the same way to huge capacitor as it would to a battery.

Ratch

 

[SunnyBoyNY]

SunnyBoyNY,



A huge capacitor will resonate with a huge coil; a battery will not. As I mentioned before, a battery does not take energy from the circuit and return it back in equal amounts at a different time as a capacitor would. Therefore, resonance is not possible with a battery, and the circuit will not respond the same way to huge capacitor as it would to a battery.

Ratch

A non-ideal battery will not resonate with a coil because a good-battery model consists of a number of non-linear elements. A large capacitor is one of them. In an LC circuit energy flow reverses when one element is completely discharged and the other is fully charged. That impossible to do with a battery that is almost completely depleted at a voltage not far from its maximum.

The approximation of a battery as a large capacitor would be valid for analysis where the run time is much shorter than the resonant frequency of the battery model and the smallest galvanically connected inductance coil in the circuit.

How else would you model a charge-dependent voltage source? Say ones wants to model a battery powering a dc/dc converter that has an LC output filter and a resistive load. The battery voltage (=capacitor voltage with an initial condition) will drop over time because there is no other initial energy in the system. The converter will go through a number of cycles that will not be affected by the additional capacitive element at all.

 

[Ratch]

SunnyboyNY,

A non-ideal battery will not resonate with a coil because a good-battery model consists of a number of non-linear elements.[

How else would you model a charge-dependent voltage source? Say ones wants to model a battery powering a dc/dc converter that has an LC output filter and a resistive load. ...

This thread is concerned with linear circuits, not nonlinear ones. That means linear elements. Also, you are drifting off topic by asking for nonlinear models that work conditionally for specific situations.

For the reasons I gave before, a battery is not a universal linear substitute for a capacitor.

Ratch

 

[SunnyBoyNY]

For the reasons I gave before, a battery is not a universal linear substitute for a capacitor.
Ratch

If we are limited to linear circuits then a battery is simply modeled as a voltage source combined with small-value series resistor and inductor. Provided low-importance of such parasitic components we can reduce the model to a simple voltage source.

Anyway, is this much ado about nothing?

 

[sophiecentaur]

sophiecentaur,

The term "storage element" does indeed describe a coil and capacitor, but it is not a comprehensive definition. A battery or DC supply acts differently because they do not take energy from the circuit and return it later. "Reactive" also describes coils and capacitors, but does not describe the storage aspects of those elements. The bottom line is that one cannot describe everything about an object in one word.

Wiki is wrong about that, as they are wrong about a lot of things. Reactance of a coil or capacitor is a complex quantity, and it needs its "j". Look at the link below, and notice that they get it right. If you don't put in a "j", then how do you distinguish a reactance from a resistance mathematically?

Not this time.
Infinities and infinitesimals have to be treated with caution. Perhaps you can provide an example in a previous post where I have been wrong about that subject. I hate to be wrong.

Ratch

http://www.st-andrews.ac.uk/~www_pa/Scots_Guide/info/signals/complex/react.html

The battery in my car does just that and so does the one in my iPod. You can't assign it an equivalent capacity either because it doesn't have an exponential time characteristic when discharging or charging through a resistor.

It's not just wiki. It is everywhere. X stands for reactance. The Impedance of a reactance X is jX. Check it out.

I agree in principle but I don't think that there is much of a problem in the 'limits' in these cases, though.

 

[Pythagorean]

Basically, capacitors and inductors both have transients involved with them; the system does not immediately reach equilibrium in response to a perturbation. In an environment where there's always perturbations (like AC) the circuit never gets a chance to reach equilibrium.

 

[SunnyBoyNY]

It's not just wiki. It is everywhere. X stands for reactance. The Impedance of a reactance X is jX. Check it out.

Ratch, Sophiecentaur,

By looking at the formula, the voltage/current relationship is determined by impedance; reactance is just the imaginary part of impedance.

However, the reactance term ought to be negative for one element and positive for the other element.

E.g., reactance can be -500 ohm (cap) or 500 ohm (inductor) at a resonant frequency. These two values add up in case the two elements are in series. Only this way the total impedance can be zero at the resonant frequency.

 

[Ratch]

SunnyBoyNY,

If we are limited to linear circuits then a battery is simply modeled as a voltage source combined with small-value series resistor and inductor. Provided low-importance of such parasitic components we can reduce the model to a simple voltage source.

I believe the question was whether we can substitute a capacitor for a battery. I don't think so except under certain circumstances.

Anyway, is this much ado about nothing?

Perhaps. It is a personal decision.

By looking at the formula, the voltage/current relationship is determined by impedance; reactance is just the imaginary part of impedance.

Correct.

However, the reactance term ought to be negative for one element and positive for the other element.

E.g., reactance can be -500 ohm (cap) or 500 ohm (inductor) at a resonant frequency. These two values add up in case the two elements are in series. Only this way the total impedance can be zero at the resonant frequency.

And, the reactance terms should show their "j"s, so that one knows whether it is a reactance or instead a positive/negative resistance.

sophiecentaur,

The battery in my car does just that and so does the one in my iPod. You can't assign it an equivalent capacity either because it doesn't have an exponential time characteristic when discharging or charging through a resistor.

What does the battery in your car and ipod do? Are we in agreement that a capacitor cannot be a universal substitute for a battery?

It's not just wiki. It is everywhere. X stands for reactance. The Impedance of a reactance X is jX. Check it out.

Not everywhere, elsewhere has got it wrong too. If you don't give it a sign and a "j", then the description of the reactance is incomplete.

Ratch

 

[SunnyBoyNY]

Not everywhere, elsewhere has got it wrong too. If you don't give it a sign and a "j", then the description of the reactance is incomplete.

Ratch

This pretty much boils down to this question:

Is reactance jX or X? According to Agarwal & Lang (MIT - Foundations of Analog and Digital Electronic Circuits, 1st Edition):

Let us examine first the power delivered to some arbitrary impedance Z =
R + jX by a sinusoidal source, as depicted in Figure 13.51. The quantity X is
usually referred to as the reactance of the circuit.

Based on the sentence above I assume that reactance is just X and is completely determined by its magnitude and sign. The fact that reactance is the imaginary part of impedance is taken of by the "j" before the "X".

 

[Studiot]

It's not just wiki. It is everywhere. X stands for reactance. The Impedance of a reactance X is jX. Check it out.

Not everywhere, elsewhere has got it wrong too. If you don't give it a sign and a "j", then the description of the reactance is incomplete.

I've been following this thread with interest and some amusement.

Mostly I think Ratch has made some very good points and defended them well, however I think it's time to put the record straight on this one.

Complex (circuit) analysis is only one way to analyse circuits, allbeit a very convenient one.

If you use complex quantities you have to go the whole hog and employ complex voltage current and admittance as well. Older engineers will always talk of 'complex impedance' to emphasise the point.

Impedance, admittance and reactance are not inherently complex quantities.

They can be specified using a real modulus and a real phase angle and the calculations performed by trigonometry. Not is single i, j or k will be seen.

Alternatively complex notation can be used, but again here there is a choice as we can either use the R+jX format or use the e^jα.e^jωt format. The latter comes into its own in calculus based operations since exponentials make differentiation and integration particularly easy.

So it is possible to specify a circuit, say 3 ohms and 7.96 millihenrys with 60 (real) volts at 50 cycles/second applied.

The resistance is 3 ohms, the reactance is 2.5 ohms, the impedance is 3.91 ohms the phase angle is 39°48' and the current is 15.4 amps.

These are all real (measurable) quantities. Not a single imaginary one is needed, or directly measurable.

I will leave it up to those who like complex notation to put these figures into that form.

go well

 

[Ratch]

SunnyBoyNY,

Based on the sentence above I assume that reactance is just X and is completely determined by its magnitude and sign. The fact that reactance is the imaginary part of impedance is taken of by the "j" before the "X".

I don't have any trouble agreeing with the above.

Studiot,

Impedance, admittance and reactance are not inherently complex quantities.

I can show you a good book that says that immittance is not a phasor quantity, but is a complex quantity. It has to be. What else can you get when you divide a sinusoidal voltage by a sinusoidal current?

They can be specified using a real modulus and a real phase angle and the calculations performed by trigonometry. Not is single i, j or k will be seen.

Exponential representation can be easily converted into rectangular or polar form by Euler's Theorem. Anytime the angle θ is not zero, there is a reactance involved, and an implied orthogonal component to the term.

Ratch

 

[Pythagorean]

I think the point with reactance is that you could just as well describe your position on the surface of Earth as a complex number, defining some origin as 0 and doing the x y vector as x + iy. It's an arbitrary notation for a 2D system. You could just as well use a vector. As long as the two dimensions are orthogonal (which they are in space, too), using complex notation instead of a vector works.

 

[Ratch]

Pythagorean,

I think the point with reactance is that you could just as well describe your position on the surface of Earth as a complex number, defining some origin as 0 and doing the x y vector as x + iy. It's an arbitrary notation for a 2D system.

I don't see any relationship between reactance and a position on the round surface of the Earth. What point with reactance did you have in mind?

You could just as well use a vector. As long as the two dimensions are orthogonal (which they are in space, too), using complex notation instead of a vector works.

With a vector describing a position, at least you would not be restricted to an orthogonal coordinate system. So using one vector, would it point through the Earth from, say Greenwich, England, to a position in, say Australia?

I think that would be a solution in search of a problem.

Ratch

 

[SunnyBoyNY]

I don't have any trouble agreeing with the above.

Ratch,

this also means that reactance is negative for capacitors and positive for inductors (e.g. -500 Ohms for a capacitor as suggested by sophiecentaur).

 

[Pythagorean]

space itself is orthogonal in all three directions. Lots of sets of variables are (it just means they're independent of each other).

All I mean, is in circuit theory, you just use complex notation to describe two states (current and potential) in one number, a complex number. You could just as easily use a vector and assume orthogonality in your vector, which is exactly how we are able to get from cartesian to polar coordinates and back again.

 

[sophiecentaur]

SunnyBoyNY,

Not everywhere, elsewhere has got it wrong too. If you don't give it a sign and a "j", then the description of the reactance is incomplete.

Ratch

Can you give us a (good) reference where jX is called reactance, then?
Of course, a sign is necessary - hence my arbitrary value of negative reactance for a capacitor.

 

[Ratch]

SunnyBoyNY,

this also means that reactance is negative for capacitors and positive for inductors (e.g. -500 Ohms for a capacitor as suggested by sophiecentaur).

As long as '"j" gets tacked on at some point, I am happy.

Ratch

 

[Ratch]

Pythagorean,

space itself is orthogonal in all three directions. Lots of sets of variables are (it just means they're independent of each other).

I thought you were referring to the "surface" of the Earth, not space.

All I mean, is in circuit theory, you just use complex notation to describe two states (current and potential) in one number, a complex number. You could just as easily use a vector and assume orthogonality in your vector, which is exactly how we are able to get from cartesian to polar coordinates and back again.

What does a "state" mean? What does a transform from cartesian to polar and vice versa have to do with anything? Give a simple made up example of a "state" being expressed with one number.

Ratch

 

[Pythagorean]

Pythagorean,
I thought you were referring to the "surface" of the Earth, not space.

You can describe the surface of the Earth in spatial coordinates much like you describe the state of the elements of a circuit in "I-V space". Pick any coordinate system, xyz or phi/theta/r. All eigenbases are still orthogonal.

What does a "state" mean? What does a transform from cartesian to polar and vice versa have to do with anything? Give a simple made up example of a "state" being expressed with one number.

Ratch

Cartesian to polar coordinate transform is only valid in orthogonal systems. It applies to circuits as well as space

A state is the minimum set of variables needed to describe your system. For instance, a particle has position and velocity in it's state vector, [x,p]. A complex number is a single number, but can also represent two numbers if you think of it as a vector instead. In fact, j (or i) is just an orthogonal transform operator.

 

[Ratch]

sophiecentaur,

Can you give us a (good) reference where jX is called reactance, then?
Of course, a sign is necessary - hence my arbitrary value of negative reactance for a capacitor.

All my reference material uses the notation Xc or Xl. But I think we are in agreement that there are two kinds of reactance, and these have to designated as reactance as opposed to resistance, and the polarity has to be shown.

Ratch

 

[Ratch]

Pythagorean,

You can describe the surface of the Earth in spatial coordinates much like you describe the state of the elements of a circuit in "I-V space". Pick any coordinate system, xyz or phi/theta/r. All eigenbases are still orthogonal.

Cartesian to polar coordinate transform is only valid in orthogonal systems. It applies to circuits as well as space

A state is the minimum set of variables needed to describe your system. For instance, a particle has position and velocity in it's state vector, [x,p]. A complex number is a single number, but can also represent two numbers if you think of it as a vector instead. In fact, j (or i) is just an orthogonal transform operator.

How is this going to help anyone do circuit analysis, especially figuring out whether a capacitor blocks DC current?

Ratch

 

[Pythagorean]

Pythagorean,How is this going to help anyone do circuit analysis, especially figuring out whether a capacitor blocks DC current?

Ratch

I'm not sure why you suddenly bring up pedagogy. My posting is in response to this:

Reactance of a coil or capacitor is a complex quantity, and it needs its "j".

Perhaps this is the pedagogical benefit, though; that if you use vectors instead of complex values, you won't get confused into thinking real measurable values are complex. This is probably only relevant to a science/math student though. If you're purpose is only electrical engineering, it's unlikely you'll see complex numbers in any other context, so it's probably never an issue.

 

[Ratch]

Pythagorean,

I'm not sure why you suddenly bring up pedagogy. My posting is in response to this:


Reactance of a coil or capacitor is a complex quantity, and it needs its "j".

I did not think an Earth coordinate system had any relevance to circuit analysis.

Perhaps this is the pedagogical benefit, though; that if you use vectors instead of complex values, you won't get confused into thinking real measurable values are complex. This is probably only relevant to a science/math student though. If you're purpose is only electrical engineering, it's unlikely you'll see complex numbers in any other context, so it's probably never an issue.

I believe the energy stored in the electric field of a capacitor or the magnetic field of an inductor is every bit as "real" as the energy dissipated in a resistor. Yet that reactive energy is designated as "complex" in a sinusoidal circuit.

Ratch

 

[SunnyBoyNY]

Perhaps this is the pedagogical benefit, though; that if you use vectors instead of complex values, you won't get confused into thinking real measurable values are complex.

To my understanding the two approaches are (and ought to be) equivalent in the end.

In the big picture we could:

1) Use the usual circuit model, set up differential equations, do a nightmarish trigonometry (or transform to vectors and then do complex algebra to get the result.
2) Immediately switch to the impedance-based circuit model and do complex algebra to get the final result.

Bts. measurable values can be and are complex with respect one to another. E.g. power factor measurement. In that case we of course need multiple measurements.

This is probably only relevant to a science/math student though. If you're purpose is only electrical engineering, it's unlikely you'll see complex numbers in any other context, so it's probably never an issue.

Complex numbers are used throughout the EE field. Non-linear systems analysis, adaptive filtering, etc. While your vector-based example is interesting, it has little in common with the actual problem.

 

[Studiot]

this also means that reactance is negative for capacitors and positive for inductors (e.g. -500 Ohms for a capacitor as suggested by sophiecentaur).

As long as '"j" gets tacked on at some point, I am happy.

Reactance of a coil or capacitor is a complex quantity

You seem to have some difficulty with terminology.

We distinguish three distinct types of numbers (there are others but they are irrelevant in this context)

Real numbers,
imaginary numbers which can be expressed as the product of a real number and j
complex numbers which canbe expressed as the sum of a real number and and an imaginary number.

Each have their own special properties.

In particular only real numbers can be positive or negative.
So one of the above quotes is false.

A reactance is a real number, usually given the sign X.

This may be combined with j into an imaginary number and added to or subtracted from a real resistance to achieve a complex impedance.

Impedance, admittance and reactance are not inherently complex quantities.

I can show you a good book that says that immittance is not a phasor quantity, but is a complex quantity. It has to be. What else can you get when you divide a sinusoidal voltage by a sinusoidal current?

You get exactly what I wrote in post #90 a real modulus and a real phase angle.

They can be specified using a real modulus and a real phase angle and the calculations performed by trigonometry. Not is single i, j or k will be seen.

Exponential representation can be easily converted into rectangular or polar form by Euler's Theorem. Anytime the angle θ is not zero, there is a reactance involved, and an implied orthogonal component to the term.

So what?
I can convert or transform 3 into 6 by doubling.
Again so what?

Of course all valid methods must produce the same answer from the same data.

Your posts seem to imply that there only method available is that of complex analysis, which is simply not the case.