Impedance part 1

[spacelabstudio]

In case anyone is interested:

http://ampnerds.spacelabstudio.com/impedance/part1.html

[HotBluePlates]

"A good operational definition of resistance ..."

... is the opposition to the flow of current, which is uniform at all frequencies.

But yeah, the ratio way of thinking of it is handy with respect to ohm's law.


"One last pedantic point. Thévenin's theorem, and therefore his equivalent circuit, really only apply to circuits composed entirely of linear components."

What you really say with the following explanation is the Thevenin resistance is only constant for linear circuits. For non-linear circuits, it may be non-constant.

If we used an infinitely-small-signal approximation, it would be the same as differentiation (in calculus) and would be the same as how we practially find the a.c. plate resistance of a triode gain stage; i.e., you find the tangent to the grid voltage curve at the point of operation, or the slope of that curve with an infinitely small-signal.

But as you say in the slide-show, that is a pedantic point. It is a helpful thing to realize, though. It also makes me not feel like I wasted time in a calculus class (yes, you really can use math, if you choose to!).


For the Maximum power case, I know you really wanted to display the math. But for those whose eyes glaze over when looking at equations, you can help prove it to them with a graph showing how voltage and current change with the ratio of output-to-input impedance. A second line on the graph showing voltage * current will help them see that power is truly at a maximum when output impedance = input impedance.


Nice work on the presentation!

[spacelabstudio]

For the Maximum power case, I know you really wanted to display the math. But for those whose eyes glaze over when looking at equations, you can help prove it to them with a graph showing how voltage and current change with the ratio of output-to-input impedance. A second line on the graph showing voltage * current will help them see that power is truly at a maximum when output impedance = input impedance.

That's a good idea, thanks!  My brain tends to work in the abstract, but most people are way more visual.  That sounds like a good way to illustrate the point in a non-mathy way.

Chris

[silverfox]

I went through it and got gained some knowledge. As for the calculus- I was just looking at one of those books in a thrift store today... What are the odds...

I'll read part two.

Thanks,

Fox.

[HotBluePlates]

Ready for what calculus really tells you (at least "differential" calculus)?

If you have a linear function (or a linear electronic device like a resistor or a theoretical perfect tube), the rate of change of some variable is constant. It exhibits the same behavior all the time.

Calculus becomes interesting when it tells you about something whose behavior is non-constant. Like a triode's internal plate resistance. When something is non-linear, there is some characteristic about it which changes depending on some other condition.

If you could describe a tube's various curves each with a mathematical formula, you could apply calculus to each and see how non-linear one tube is compared to another. The secret is, you can skip the equation, and understanding what a graph implies, see the same information and make comparisons more intuitively.

I'm starting to realize math classes would have been a lot better/easier if the teaches didn't have to follow a curriculum with a set timeline, and spent more time teaching how the math tells you useful information, rather than focusing on the mechanics of how to manipulate an equation.

Oh well...

[spacelabstudio]

Ready for what calculus really tells you (at least "differential" calculus)?

If you have a linear function (or a linear electronic device like a resistor or a theoretical perfect tube), the rate of change of some variable is constant. It exhibits the same behavior all the time.

Calculus becomes interesting when it tells you about something whose behavior is non-constant. Like a triode's internal plate resistance. When something is non-linear, there is some characteristic about it which changes depending on some other condition.

If you could describe a tube's various curves each with a mathematical formula, you could apply calculus to each and see how non-linear one tube is compared to another. The secret is, you can skip the equation, and understanding what a graph implies, see the same information and make comparisons more intuitively.

I'm starting to realize math classes would have been a lot better/easier if the teaches didn't have to follow a curriculum with a set timeline, and spent more time teaching how the math tells you useful information, rather than focusing on the mechanics of how to manipulate an equation.

Oh well...

Yep.  I gave this presentation last night and definitely got glassy looks whenever there was an equation on the screen, no matter how basic.  Between that and your input, I'm seeing that part two is going to need to have a lot more charts.  It may be the nuts and bolts math stuff should just be a separate page linked to from the main presentation for people who are into that kind of thing.

Chris

[silverfox]

Ready for what calculus really tells you (at least "differential" calculus)?

.. Like a triode's internal plate resistance.

Let's see- Temperature in K, electron cloud density, electrical potential, constant for internal resistance of plate material, capacitance, frequency and ??

with all due respect: I'm ready, tell me. I understand basic algebra however a limited mathematical explanation will serve best. Word pictures and analogies work good for me.

Silverfox.

[HotBluePlates]

Ready for what calculus really tells you (at least "differential" calculus)?

.. Like a triode's internal plate resistance.

Let's see- Temperature in K, electron cloud density, electrical potential, constant for internal resistance of plate material, capacitance, frequency and ??

No... look at a set of plate curves for a triode.

If the tube were linear, each of those curves would be straight lines, parallel with each other and equally spaced.

What you see instead are lines that may be roughly straight over one portion, but there is curvature to the individual grid voltage lines (one type of non-linearity; varying change of current with change of plate voltage).

If you replace the whole family of curves with straight lines whose angle matches the angle formed by the longest, straightest portion of the grid line, the enter family is fan-shaped rather than being parallel (2nd instance of non-linearity). And the lines are hardly equally-spaced along their lengths or among individual lines (3rd instance of non-linearity).

These things rear their heads in various ways, but basically you can determine graphically that there will be distortion in a triode due to changing internal plate resistance (which cause the positive and negative portions of a signal to be amplified differently), and that the gain and distortion of a tube will change with different operating points.

[silverfox]

..."Calculus becomes interesting when it tells you about something whose behavior is non-constant. Like a triode's internal plate resistance. When something is non-linear, there is some characteristic about it which changes depending on some other condition."

I understand we are dealing with a non linear system. Why is it non linear? I think it has to do with the density of the electron cloud (interacting with volts and frequency and other previously listed variables), and the properties of the materials. These interactions create a non linear system. How? Don't know except for speculation.

Is this the heart, (nexus) of the tone within the valve?

Fox.

[PRR]

> Why is it non linear?

A useful amplifying device biased-up at a useful current has useful gain.

Biased at ZERO current they all have ZERO gain.

If you play around, you'll see the gain doesn't just vanish at a certain current, it fades-away.

In general, gain increases with current, down to zero and usually up as far as we dare (melt-down ruins gain).

Just as gain falls to zero, gain falls when device current changes. If we bias a 12AX7 at 1.0mA, then swing it to 0.5mA and 1.5mA, the low-current half-wave is smaller and the high-current half-wave is larger.

BJTs, the gain is exactly proportional to current over many decades. (And gain is generally so high that we "must" throw some away, usually with linear resistors.)

Field-Effect devices (including tubes) at "normal" currents, the change of gain is less than change of current. Perhaps square-root (math!!) of current. So we can often listen to small FET/VT amps without additional linearization.

________________________________

> "Thévenin's ...only apply to circuits composed entirely of linear components."

Thévenin is always true.

But with non-linear parts, there's no direct solution. Or none you can reasonably do by hand on non-trivial circuits.

If you are given the nonlinear part's properties and assume a solution, you can work backward and show that the assumed solution isn't correct (unless you got lucky). SPICE is all about Thévenin. It can't forward-solve a nonlinear circuit, so it assumes (say) that three series elements have equal currents and split the supply voltage equally. So 10K 1K Diode off 9V, it assumes 9V 6V 3V 0V and computes the currents. 0.3mA, 3mA, and 99,999,999+ Amps. Uh, wrong. It adjusts its assumptions, less on the diode. After a few hundred rounds of guessing-game, it gets within 8-place agreement, which is close-enough for practical work. But Thévenin guides every step.

Richard Kuehnel's 5F6a book solves nonlinear circuits with some heavy matrix math. Apparently this trick was known for decades but remained a "trick" until cheap computing.

[tubeswell]

I'm starting to realize math classes would have been a lot better/easier if the teaches didn't have to follow a curriculum with a set timeline, and spent more time teaching how the math tells you useful information, rather than focusing on the mechanics of how to manipulate an equation.

Oh well...

Me too. Most of the school teachers I remember had a 'don't ask me I just work here' agenda.

[silverfox]

..."Field-Effect devices (including tubes)"..

Okay, that explains it better for me. A tube is a field effect device; Hence the need for Calculus. Didn't know that.

The electron cloud is influenced by the characteristics of the materials near the cloud and the potential and frequency components superimposed.

A variable impedance device is created?

A variable resistance device is created. No! Only within a portion of the graph.

The variable impedance modulates the power supply thus amplifying the input signal?

This sounds like weather... Hmmm. Ever heard of HAARP? What if portions of the atmosphere were treated as though it were a gas discharge tube or a capacitor?

This still pertains to impedance and I suspect by the time part two of the course appears this discussion will be most beneficial to me.

Silverfox.

[thermion]

Hmmm. Ever heard of HAARP?

No, what does that do?

[PRR]

> A tube is a field effect device; Hence the need for Calculus.

Calculus is useful (for BJTs also) but NOT essential.

[HotBluePlates]

> A tube is a field effect device; Hence the need for Calculus.

Calculus is useful (for BJTs also) but NOT essential.

Right. It's not about the process of the math, but what the math tells you.

..."Calculus becomes interesting when it tells you about something whose behavior is non-constant. Like a triode's internal plate resistance. When something is non-linear, there is some characteristic about it which changes depending on some other condition."

I understand we are dealing with a non linear system. Why is it non linear?

I'm talking about 2 dogs barking, and saying one's bark is a higher pitch than the other. You just asked, "Why do dogs bark?"

That's a whole other study of how tubes are made and the geometry/chemistry involved in each, as well as how different tube types differ.

..."Field-Effect devices (including tubes)"..

A variable impedance device is created?

Yes, you could call a tube a variable impedance.

*If* it were linear, the impedance would always vary uniformly with input signal.

But it doesn't, so it's non-linear. Some operating points for a given tube are more non-linear than others. Some tube types are more non-linear than others.

[silverfox]

Thanks one and all. I shall process the information submitted.

Fox.

[spacelabstudio]

> "Thévenin's ...only apply to circuits composed entirely of linear components."

Thévenin is always true.

Ok, but the equivalent "resistor" is non-linear, and therefore breaks the definition of resistor.  (R=V/I)  Right? 

I understand we are dealing with a non linear system. Why is it non linear? I think it has to do with the density of the electron cloud (interacting with volts and frequency and other previously listed variables), and the properties of the materials. These interactions create a non linear system. How? Don't know except for speculation.

Well, yeah, there's the physics of it, but easier than that, you can look at the grid curves on a datasheet and see that they're curves instead of straight lines.  We tend to talk about linearity of a component with regards to how its V and I relate to each other.  In a resistor, that relationship is obviously a linear function.  They don't publish curves for resistors because they'd all just be straight lines.  Three (or more) terminal devices get tricky, too, because then the current through the device may be dependent on more than one voltage, or there may be more than one current (in the case of a BJT).  And none of those VI relationships really work out to be straight line linear.  If linearity is your goal, you just operate the device within a range of voltages where the behavior is "close enough".

After viewing your presentation, Impedance Part I might be slightly mislabeled.  I believe the title should be introduction to impedance, or the foundation to understanding impedance. 

Ok, well it is the first part. 

Its been a hundred years since I had calculus, and I look forward to your take on impedance.  (Yes, my mind was like a glazed doughnut, the really important info was missing from the middle).

I won't be using any calculus on that one.  I'll even try to keep the algebra to the minimum and rely on graphical aids.  Making people do the math seems to obscure the concept.  Once they have the concept and they want to use it design something, they'll need to learn the math (or just be really good at breadboarding), but loading the math up front seems to be a bad idea, from what I've seen so far.

Do you have this presentation someplace that I could sit down with a hardcopy and work the details.  As any good chef, says, now lets take a little red wine and deglaze the pot. (That's where the flavor really is).  You can now call me a mixer of metaphors. 

Not really.  The presentation is just an HTML file, though, so I'll take some time soon and try to massage it into something that can be printed out in a little while.  Hopefully won't be too hard.

[HotBluePlates]

> "Thévenin's ...only apply to circuits composed entirely of linear components."

Thévenin is always true.

Ok, but the equivalent "resistor" is non-linear, and therefore breaks the definition of resistor.  (R=V/I)  Right? 

Because pure resistance is linear. But tubes are non-linear. However, their internal plate resistance is still V/I; it's just that V and I do not change at the same rate.

Look at a triode data sheet (12AX7 will do). See the plate characteristics? V on the x-axis, I on the y-axis. The grid voltage lines are V/I, but they're curves (linear, or constant resistance, would be a straight line at some angle depending on the amount of resistance). The fundamental theory is they follow is Child's Law, or about an exponent of 3/2. That varies with the tube and operating point; non-linearity in a couple ways.

That's why PRR said Thevenin is always true, but may not be simple to calculate. It's also why tubes have characteristic curves on the data sheet, because it's easier to generate and figure the math in graphical form than in equation form.

And, because for a non-linear device V/I is not a uniform value for all V and I (that is, the ratio of the two changes because the rate of change for each is different), you cannot use algebra to find the value of V/I. You must use ΔV/ΔI, and if the deltas get infinitely small (for a more accurate calculation) you are finding the derivative of the curve.

That's where calculus comes in, though as I said before, we don't have a handy equation for the gridlines. Each is also different (if I was a math genius, I might say the gridlines form a 3-D surface that is like a section of a cone, but I'm not that smart and I failed Calc II). Again, it's easier to perform the calculus operation of finding the derivative of the gridlines, or rate of change of the slope, through graphical means.

I understand we are dealing with a non linear system. Why is it non linear?

Well, yeah, there's the physics of it, but easier than that, you can look at the grid curves on a datasheet and see that they're curves instead of straight lines.

More than that. Pretend the gridlines are all straight. Is the tube now linear?

If the gridlines each share a common origin, but have different angles (so that they "fan out" at the extremities) then each line is a different resistance. Prove that to yourself by plotting 100Ω, 1kΩ, 10kΩ, 100kΩ on graph paper with V for x-axis and I for y-axis.

Since we know that we plot a loadline on these "curves" to show how the tube and the load divide voltage among themselves with an input signal, then if each gridline is a different resistance the tube is still non-linear. It will show a different gain on the positive input signal swing than on the negative input signal swing, because the internal resistance has changed.

So for an ideal linear tube, the gridlines must be parallel straight lines, which implies a uniform internal resistance, which is equal at every grid voltage.

Making people do the math seems to obscure the concept.  Once they have the concept and they want to use it design something, they'll need to learn the math (or just be really good at breadboarding), but loading the math up front seems to be a bad idea, from what I've seen so far.

Math is a compact way to describe a complex concept that make take pages or chapters of a book. Ideally, it shows complex inter-relationships in an easy to understand fashion.

I once had a great math teacher (my Calc I teacher, as opposed to my Calc II teacher who was the worst I've ever seen anywhere). The guy was a grad student, teaching a class to help fill his degree requirements. Because he really understood the stuff he was teaching, he could break down scary looking equations into simple ideas, and help us see the overall concept it was communicating.

Don't make people do math equations, help them see graphically what the math represents. Unless you have to do proofs for a class requirement, there's probably little need in showing them how to derive each equation. The hard work is in figuring out how to make the concept simple.

[silverfox]

drgonzonm said: "Spacelab, please provide the calculus,"

First I coincidentally looked at a Calculus book the day this came out and now I went and bought one for 50 cents.

Calculus Early Transcendentals, James Stuart 5e. It came with a solutions book.

I started reading it this afternoon while in the back of my head I'm wondering why at this stage of life should I bother. Anyway- I'd be interested in seeing some of the math associated with the Topic.

Perhaps the introductory information in this book relates to tube curves in that, there was a graph with a bunch of little rectangles under the curve and I could see how perhaps that related to tube properties-characteristics. Each rectangle describing a point of the curve in whatever parameters the rectangle maps back to.

Like I said I just started so I don't know if that's even close.

Silverfox.

[jazbo8]

That's where calculus comes in, though as I said before, we don't have a handy equation for the gridlines. Each is also different (if I was a math genius, I might say the gridlines form a 3-D surface that is like a section of a cone, but I'm not that smart and I failed Calc II). Again, it's easier to perform the calculus operation of finding the derivative of the gridlines, or rate of change of the slope, through graphical means.

Grid lines are really 3D surface like you said. Here is one I like that shows -Vgk as well as +Vgk.



Jaz

[burt_toast]

whew! very nice work, even just from a presentation standpoint.

if kid's minds are sponges, then mine is a 51 yr old dry, crusty one. will try to absorb as much as i'm able from this (and following ones).

maybe i should join a pottery forum instead (or making balloon animals, or finger puppets).

man, you guys are sharp!

scott