The operator j and complex numbers.

As suggested in http://www.vintage-radio.net/forum/s...6&postcount=35 I have started this thread.

My first comment is by using complex numbers inductor, resistor and capacitor circuits can be analysed using good old ohms law.

I think complex in this meaning is it's a complex of two (only two) numbers, not complicated.

[kellys_eye]

I recall the Maplin Magazine (correct me if anyone knows this to be the wrong magazine) also doing such an article some years ago.

It too was well written and not at all 'scary' as many seem to believe j-notation to be :lol:

Turns out to be surprisingly useful knowledge so thanks for posting that link MM.

[Craig Sawyers]

Mathematicians use i and not j. Reason the we in electronics chose j instead is to prevent confusion with instantaneous current i.

But the biggest barrier to comprehension is the term "complex numbers" which implies something complicated. If they were called called Mary and Peter numbers, it would have overcome the frightening term "complex" at a stroke ;-)

[G6Tanuki]

I recall back in the early-1980s being the only one in a team-of-five who had enough of an understanding of this to make free use of the COMPLEX(X,Y) type-definition in the FORTRAN77 programs we used for computational fluid dynamics.

Strange, considering that I was then a botanist(!) and the other people in the team were various 'engineers' by first-degree.

[ms660]

Never fully understood the j thing, in the field of radio what would j be used with/for?

Lawrence.

[TonyDuell]

Another way to think of it is that multiplying by j means a 90 degree phase shift of a sinusoidal signal.

That implies that multiplying by j again is a 180 degree phase shift, which of course is the same as multiplying the signal by -1. So j^2=-1 from that...

As to where it's useful, it's a mathematical notation that (for example) simplifies calculations of impedance in circuits with resistance, capacitance and inductance.

The impedance of a resistor = R (a real number, since current and voltage are in phase for a resistor)

The impedance of an inductor = 2*pi*f*L*j (the j giving the quadrature phase shift between current and voltage in a perfect inductor).

Likewise the impedance of a capacitor = 1/(2*pi*f*C*j).

Now you can use the normal rules for resistors in series and parallel to calculate the impedance of a complicated circuit. You have to use the mathematical way of handling complex numbers, but that's the only 'gotcha'. You can't 'forget' about the phase shift, it naturally comes through the calculation.

[julie_m]

And that 2 * π * f term keeps cropping up so often, it even gets its own symbol, ω (lower case omega).

[ms660]

Originally Posted by TonyDuell

Now you can use the normal rules for resistors in series and parallel to calculate the impedance of a complicated circuit. You have to use the mathematical way of handling complex numbers, but that's the only 'gotcha'. You can't 'forget' about the phase shift, it naturally comes through the calculation.

So what you're saying is that if I multiply the inductive reactance by j (first formula) then the bottom half of the capacitive reactance formula by j (the second formula) then divide the bottom line of the second formula into 1 to complete the second formula calculation I can just add either of the two results of those calculations to say a resistor value that say is in series with the inductance or capacitance?

Lawrence.

[G6Tanuki]

"J" becomes deeply useful when understanding the FM "Quadrature" detector - it highlights the strange underlying "duality" of SSB and FM.

FM is fundamentally SSB with the carrier reinserted 90 degrees out of phase - a detail which escapes many people who've looked at circuits using the intriguing 6BN6 "Gated Beam" valve.

These days "Multiplying by J" in software is a fundamental aspect of any DSP receiver to handle the I and Q components.

[Sidenote: somewhere I have a taken-at-work christmas-party photograph of my late father and his then co-workers from about 1946 in which he is wearing a peaked-cap of the kind then beloved by commissionaires, bus-inspectors etc. Tucked into the peak is a card with the letter "j" drawn on it. He was, of course, "Operator J". Another person wears an ex-German tin helmet with "5" stencilled on it, apparently there was a wartime skit where one of the performers said, in a German accent, "This is Funf speaking".]

[Herald1360]

Originally Posted by ms660

So what you're saying is that if I multiply the inductive reactance by j (first formula) then the bottom half of the capacitive reactance formula by j (the second formula) then divide the bottom line of the second formula into 1 to complete the second formula calculation I can just add either of the two results of those calculations to say a resistor value that say is in series with the inductance or capacitance?

Lawrence.

Sort of. If the capacitor reactance is, say, -j5, the resistor is R and the inductor is j5 then the complex arithmetic result depends on associating the R with the correct reactance it's in series or parallel with, before working out the final value. If they're all in series that's easy, you get -j5 +j5 +R = R which is exactly what you get for a series tuned circuit at resonance with some R in series with the L. For various series/parallel and parallel combinations you have to work through the complex arithmetic for series and parallel circuits to arrive at the end result.

Eg: multiplying two complex numbers 2+3j and 4-5j for instance is done by multiplying out the two brackets (2+3j) and (4-5j) remembering that j*j= -1 you get 8+12j-10j+15 (the +15 is actually -15j^2 where j^2 = -1).

It can get a bit messy to keep track of, but it's the quantity rather than complexity of the calculations that can make it tricky sometimes.

[ms660]

I think it's tricky for me most of the time, maths is not my strong point, anything above simple then I need my calculator.

Lawrence.

[mark_in_manc]

Originally Posted by ms660

So what you're saying is that if I multiply the inductive reactance by j (first formula) then the bottom half of the capacitive reactance formula by j (the second formula) then divide the bottom line of the second formula into 1 to complete the second formula calculation I can just add either of the two results of those calculations to say a resistor value that say is in series with the inductance or capacitance?

Yes.

If you have a fraction with a j on the bottom (like capacitive reactance 1/jwC) then if you multiply by j/j (which is just one, so you're not doing anything weird) you get j/j^2wC which is -j/wC. Stick this in series with an inductor (jwL) and a resistor (R!) and you've got

jwL + R - j/wC

It's handy to collect the j terms, so you get

R + j (wL - 1/wC)

If you want to make it clearer, you can plot this as a vector on an 'argand diagram' - which is just a fancy name for a 2D plot where the y axis is terms in j - imaginary axis - and the x axis is terms without j - real axis.

So for a big 'L' the line points 'up' - impedance leads by 90 deg. For a big C the line points 'down' - impedance lags by 90 deg. And as R gets bigger the line tends to point to the right - impedance is more 'real' (current and voltage are more in-phase).

Loads more good stuff follows - like there is a frequency w_res when positive inductive reactance is equal and opposite to negative capacitive reactance and the contents of the j bracket goes to zero.

w_res.L=1/{w_res.C}

so w_res^2=1/LC, w_res=root(1/LC)

This is resonance, where the impedance goes real and current is only prevented from going infinite by the presence of some R. Nice, isn't it!

[ms660]

No worries, I'm familiar with tuned circuits properties at resonance etc, also familiar with zero, + - 90 phase shift as in vector plot etc, it's just the j thing, I know XL is plus and XC is minus but I've never needed j to tell me that.

Lawrence.

anything above simple then I need my calculator

That is what they are for!

[kalee20]

Originally Posted by Herald1360

Eg: multiplying two complex numbers 2+3j and 4-5j for instance is done by multiplying out the two brackets (2+3j) and (4-5j) remembering that j*j= -1 you get 8+12j-10j+15 (the +15 is actually -15j^2 where j^2 = -1).

Which means, although you don't actually summarise it, that 2+3j times 4-5j is 23+2j.

Complex numbers are really elegant, because you can add them, subtract them, multiply them, divide them, take square roots (or cube roots, or complex roots!), and you never get worse than another complex number, a+bj.

[G8HQP Dave]

Originally Posted by G6Tanuki

FM is fundamentally SSB with the carrier reinserted 90 degrees out of phase

Not quite. Low deviation FM is asymptotically equivalent to DSB with the carrier 90 degrees out of phase. Higher deviations have other sideband elements at various phases.

[ms660]

Originally Posted by merlinmaxwell

That is what they are for!

Now there's a revelation

Lawrence.

[Radio Wrangler]

Let's do that in stages, it'll be clearer, I think. (2+3j) * (4-5J)

expands to 2*3 plus 3j*4 minus 2*5j minus 3j *5j

Which is 6 +12j -10j -15j^2

j^2 is -1

So we get 6 +2j --15

The double negative signs cancel so that fifteen is positive and adds to the six

Voila! 23 + 2j

It looks tedious going the long way round, but we don't do long division either, we just make sure we have batteries in our calculator.

Now, like adding resistors in series, we can add impedances in series

A string of 15 Ohms resistive, with something whose impedance is (25 - 3j) and something whose impedance is (11+8j) is simple. We just add together all the real terms, and we add together all the j terms separately so the result is (15+25+11) + (-3+8)j

so that's 51 + 5j for the total.

Because we've used complex numbers for the complex impedances, our book keeping has kept tally of both the magnitude and phase.

So we can plug complex impedances into all the familiar formulae for handling resistors in series and parallel combinations. We can now analyse quite complicated networks with just a piece of paper and a decent pocket calculator.

The formula for handling a resistive potentiometer is

Vout/Vin = R2/(R1+R2) where R1 is the top resistor and R2 is the bottom resistor.

We can now play with our shiny new tool and stick complex impedances in place of R1 and R2.

So now we can analyse networks in terms of gain and phase when the networks consist of a mess of resistors, inductors and capacitors.

Often a resistive potentiometer is connected around an amplifier to provide feedback. Done with a gainy amplifier, the gain is set as the inverse of the potentiometer's Vout/Vin ratio. OH I'd better say that two fixed resistors scaling down a voltage is a also called a potentiometer... it's just not a variable one and there's nowhere to stick a knob. So now we can put complex impedances in place of the simple resistors and we can wind up with anactive tone control (Baxendall) or an RIAA equalising stage.

Suddenly, the toy shop has opened its doors.

You can analyse an amplifier, accounting for all the resistive and reactive components, accounting for all the device gains. It's a bigger job, but it's all repetition of the same stuff. It takes longer but it doesn't need anything harder.

At the end, you have the gain and phase of your amplifier. Analyse for different frequencies, and plot.

Yes, this is tedious, very very tedious, but isn't that exactly what computers were made for? It's how mankind gets its own back on them for all the trouble they've caused

You might have noticed I've been banging on a lot about analysis. So what about design?

It's one of those cruel twists of fate that universities spend almost all of their time teaching analysis to their victiXXX ...er ...um students. Those students graduate and get shinynew certificates with fancy curlicues and illegible signatures all over them. The certificate gets their bum on an interview seat, and then they're on their own. They learned all about analysis, but now this guy wants to give them a job designing new things... performing synthesis... the opposite of analysis!

The boss wants a circuit to do some particular job and has a load of specs all written in mont blanc ink on a napkin from a very expensive restaurant. Pity the poor graduate engineer. If an angel descended from heaven and handed him a golden circuit diagram, he could analyse the living daylights out of it. But there's no angel. Does he just keep guessing and analysing in the hope of hitting on something that works?

So we need something extra, something which puts us in command of things.

Next instalment we'll find out how to use just a resistor and a capacitor to blow up the universe. Don't tell any politicians or military people, they'd only want it turning into the next stage beyond their beloved nukes. Good job it's impossible.

David

[kalee20]

Originally Posted by G8HQP Dave

Not quite. Low deviation FM is asymptotically equivalent to DSB with the carrier 90 degrees out of phase. Higher deviations have other sideband elements at various phases.

Absolutely! Low deviation FM has minimal side bands except the first pair - so it's like DSB plus the carrier, with the carrier at 90 degrees out from what it would be if it was plain AM.

If you listen to a musician playing a note with just a bit of 'wobble', it's almost impossible to tell if he's using vibrato (frequency modulation) or tremolo (amplitude modulation). Because the spectrum is almost the same between them, and our ears are very insensitive to phase.

[Skywave]

If you're dealing with series-connected elements, use Z = R ±jX; tan φ = X/R, where φ is the angle between Z and R. However, using those relationships for parallel-connected devices produces a lot of tricky algebra. However, using Y = G ± jB , tan φ = B/G instead, where Y = admittance, G = conductance and B = susceptance makes that algebra a lot easier to manipulate, especially as Y = 1/Z.

Al.