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Thevenin and Norton equivalent sources.

Thevenin equivalent source.

Thevenin equivalent source.

Now that we have talked a little about ideal sources it is time to get real.  Yeah, I really do mean that.  We are going to talk a little about real sources and modelling those.  The first one I am going to talk about is the Thevenin model shown in the first picture.  We have talked about this in the past but I did not use the name Thevenin.   In the post “Technical Datasheets and Real Power Sources“, I introduced the idea that a real power source, a battery, can be considered an ideal voltage source with an internal resistance.

In that post I was primarily concerned with introducing the idea that datasheets and technical data is available for just about any device.  Today we are going to be a little more rigorous and give the model a name and talk about a procedure to determine both the value of the ideal voltage source as well as the resistor value.  The real source could be a complicated circuit with lots and lots of components and it could have more than one internal source.  Our goal is to simplify it to one voltage source and one series resistor, called a Thevenin equivalent.

Finding the ideal voltage value is simple. All we have to do is leave the output terminals open and measure the voltage with a meter with a high input impedance.  The output terminals are represented by the black dots in the diagram.  The term “a meter with high input impedance” means the meter will draw almost zero current.  At the end of this post I will provide links to two previous posts where I talked about this.

Imagine when we measure the voltage at the terminals with no load attached, we get 10 V.  Since we are assuming we are drawing an negligible current through our meter, there would be negligible voltage drop across the internal resistor R.   V ideal = 10 V.

Now imagine we connect a resistor across the terminals and we decrease the value of this resistor until we get 5V on our meter also connected to terminals.  At that point our resistor connected to the terminals is equal to the internal resistance.  Lets say that happens in this case with the resistor is at 1 Ohm.   We now know R is also 1 Ohm.

WARNING:  USE YOUR HEAD… YOU PROBABLY DO NOT WANT TO DROP THE TERMINAL VOLTAGE TO 50%.  IF YOU DID THAT TO YOUR CAR BATTERY FOR EXAMPLE, YOU WOULD BE DRAWING LOTS OF CURRENT AND PROBABLY CAUSING THE BATTERY TO BOIL.   — THIS IS JUST A TEACHING EXAMPLE.

Now that we have those two values, we will see what happens if we use other values for the external resistor.

R External R Total Terminal Voltage Terminal Current
open infinity 10 V 0 A
9 Ohms 10 Ohms 9 V 1 A
1 Ohm 2 Ohms 5 V 5 A
0 (Shorted) 1 Ohm 0 V 10 A

In real life I would use a couple of “reasonable” loads and check both the current and voltage at the terminals and determine the internal resistance. In the case of an automobile battery, that might be the headlights, and maybe the headlights plus the heater blower and the brake lights. These are normal loads for the automobile.

Norton Equivalent Source

Norton Equivalent Source

Like everything else in electricity there is another way of looking at things.  In this case there is another equivalent source called the Norton Equivalent Source.  Where the Thevenin model uses a voltage source, the Norton model uses a current source and in the Norton model the internal resistor is in parallel to the source instead of in series like the Thevenin model.

How do we determine the values?  To determine the ideal I value, we short the terminals and measure the current pumped by the source.  In real life…Don’t do this. or if you must do it, let me stand very far away and watch… I ain’t doing it!

On our paper example we get 10A when we measure the current through a short.  Since a short = 0 ohms, all the current the source can push would go through the short.  This means the I of our current source is 10A.  Since we have a short on the terminals the terminal voltage would be OV.

Now lets say we install an external resistor and gradually increase the value until we get exactly one-half the full available current, or 5A in this case.   This would mean 50% of the available current is going through the internal resistor and 50% would go through the external resistor and both resistors are the same value.   Our external resistor was 10 Ohms so R also equals 10 Ohms.   Now that we have those two values we will try several others.

R External R Total Terminal Voltage Terminal Current
open 1 Ohm 10 V 0 A
9 Ohms 0.9 Ohms 9 V 1 A
1 Ohm 0.5 Ohms 5 V 5 A
0 (Shorted) 0 Ohm 0 V 10 A

We have created two completely different models of the source, but the effects of loads on the terminals is exactly the same! Once we have one of the equivalent circuits it is easy to convert to the other one.

  • The R value is the same for both circuits.
  • I in the Norton = V in the Thevenin divided by R
  • V in the Thevenin = I in the Norton multiplied by R

Of the two circuits I find myself using the Thevenin more often when I am thinking about the effect of load on a power source.   That is mostly because the pictures in my head usually think of voltage as the pusher of electricity and current is the follower or reaction.   However, we will very soon be looking at devices where the model of the device is a dependent current source and Norton will be more useful.
There are a couple more things to learn to do with circuit analysis and those will make the Thevenin and Norton equivalents very useful. However, I will save those to a later date when we need them to get something done and that will not be very far off.

One final word again.  If you are testing a source, do not lower the resistance value to the point where you are overloading the source.  It will almost always destroy the source and may very well destroy you at the same time.  Probably nothing will happen if you short out an AA battery, but if you short out an automobile battery you may very well be eating boiling sulphuric acid.  (I saw it happen once, it is not pretty.)   If you short out the mains in your house, you will get burned and probably burn your house down.  Again this was a pencil and paper exercise only for gaining knowledge.

Gary

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Thevenin and Norton equivalent sources.” by Create-and-Make.com is licensed under a Creative Commons Attribution-ShareAlike 3.0 Unported License.

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