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Linearity and our pencil and paper models for Electronics.

Independent Sources

Independent Sources

Our goal now is to come up with a way to create simple models of electronic circuits. From the discussion last time “Design, Models, and Simulations“, we understand our model will be very simplified and therefore inaccurate. However, it will be good enough for our first shot before going to the trouble of inputting it into the computer and getting an accurate simulation.  That in turn will be much easier than breadboarding and testing. However, the main reason we will do it this way instead of going directly to the computer is to get a very good understanding of the effect of various components in our circuit.

One of the most important assumptions and compromises we will make in our paper models of circuits is we will assume linearity.  Linear means that the mathematics will be simple addition and multiplication.  This will allow us to do things like: Signal out = (Signal In X Gain) – An offset.  This is exactly the same form as the equation for a line:  Y=mX+b.

Sometimes, we will have to play some tricks with our assumptions.  A common one is to make the circuit “piecewise linear”.   I did this when I talked about a diode in “Magic Crystals and Magic Potions… no just a real diode“.  In that post we created a model where the current was zero when the diode was biased below 0.65 Volts and the current followed Ohms law above that voltage.  This is two lines to roughly describe a very non-linear circuit.  Computer simulations will replace that with an exponential equation.

The same thing will be true when we get to Bipolar Junction Transistors.   We will use three lines to roughly model the transistor.  These will represent three zones of operation of the transistor; cut off, active, and saturation.   If we are designing an amplifier we will want to be operating in the active zone.  If we are designing something for digital operation, for example a relay driver, we may very well be operating in the cut off and saturation zones and transitioning  through the active zone.  We will get there when we get there.

The linear assumption helps us in other ways also.   When we have multiple signals or sources coming into our circuit we can look at each one separately and then sum the total outputs together.   This is especially useful once we start working with AC.  Often we will have AC riding on top of a DC signal and we can look at each part by itself and then combine the two parts.   This cannot be done if a non-linear device is assumed, because some parts of the signal will be amplified more than other parts.   This will really be a weakness with our pencil and paper model and why we will want to go on to the computer simulation.

Our first two ideal devices we will use in our models are independent sources.   There are two types of these.  A Voltage Source, and A Current Source.  The voltage source will have some symbols inside of it describing the polarity if it is a DC source.  It could be an AC source, or even a special waveform source such as a square wave or pulse.   The current source will have an arrow inside of it indicating the direction of current flow in conventional current.  (Sorry, but this is basically engineering now so everything will be in conventional terms; hole current, or the opposite direction than electron current.)   These devices are shown in the first picture.   We have used the ideal voltage source before in many posts.

Dependent Sources

Dependent Sources

Now we have two new “devices”. These are called dependent sources. Again these can be either current or voltage sources and again an arrow will be used to denote a current source. These not only have an output, but there is an input to each of these. The input can either be the current through a device or the voltage across a device. The big difference in our symbols is the independent sources are denoted by a circle, and the independent sources are denoted by a diamond shape.

A circuit witha Voltage Source and an Resistor

A circuit with a Voltage Source and an Resistor




Our final device for right now is the reistor. Later we will get back into AC and add the capacitor and inductor but we have lots that we can and have to do with our transistors just working with DC for right now.

If you are just now getting into this blog this is probably a good time to go back and review all the electrical theory to this point.  The best way is to use the category Theory -Electricity on the menus at the top of the blog and go through the pages of blog postElectrical Power and Answers to the Meter Scalings and find a good starting point. I will not review Ohms Law and Power since those have been covered before.  Some key posts are:

As always, it is my sincere goal to provide some knowledge to allow you to create the things you want to build.

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4 comments to Linearity and our pencil and paper models for Electronics.

  • […] Linearity and our pencil and paper models for Electronics. […]

  • First calculator I saw of any value was the HP-35. Fortunately for me, my employer of the day was RCA and they offered to provide them to any engineer for $200, via salary deductions of $25 pe paycheck. Took about 5 seconds to respond to that offer!

    We were doing RF propagation calculations for proposed radio sites in Alaska. Had to also do great circle calculations to satisfy FCC. Using log tables and slide rules was not a fun process. About 1 hr by hand to do a site; about 5 min with the HP-35. I still have a couple of the follow-ons to them. Still have the slide ruules too, but now they are mounted to the wall along with my Japanese and Chinese Abacuses.


  • Memories! Wow! Piece-wise linear models! Haven’t heard that expression since Electrical Engineering 101 Fall of 1957. But no computer simulations then.
    Instead bought my Pickett aluminum slide rule which I still have in the desk. Cost the same as a Raspberry Pi model A does now.


    • Gary

      I can remember those slipsticks but by the time I got around to finishing my degree calculators were cheap and common. I think I have only heard the word piecewise linearization in school also. However, they don’t call it that, but just about every controller I ever used had some function that consisted of a table that essentially fit a series of lines to a curve. For example, on DC drives if the field of the motor was being weakened to speed up the motor and reduce torque, the field curve was determined and put into one of those tables. Now all that is done in some self-tuning mode on newer drives and probably that curve is hidden in the program somewhere.
      In a somewhat joking mode I have been told there are two types of people into technology; Nerds and Geeks. Nerds are like me and really want to know the fundamentals of what is going on. Geeks are fascinated by the latest and greatest technology and are known for being gamers. I think I am fast becoming a relic.


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