# Is there anyway to solve a diode circuit with differential equations?

#### ellsmirip25

Joined Feb 10, 2018
5
Generally speaking, is there an "easier" way to evaluate diode circuits? Normally using laplace transforms makes life a lot easier but due to the non-linear behavior I understand there's no way to use Laplace transforms. However is there anyway to use differential equations or some other method? Some computational method I may not have learned yet?

Thanks

#### WBahn

Joined Mar 31, 2012
26,398
If you think Laplace makes life a lot easier compared to having to solve a set of linear differential equations, just wait until you try to solve set of nonlinear differential equations!

Depending on how you are using the circuit, you can "linearize it" about an operating point -- that's really all small signal analysis is.

You can also use the device model to set up the equations and then solve them numerically -- that's really all a simulator does.

• ellsmirip25

#### Jony130

Joined Feb 17, 2009
5,211

#### jarkky

Joined Jan 10, 2020
20
The results can be the Lambert W -function. But that case the Laplacian and inverse Laplacian seems hard to be solved even by Wolfram Alpha.

Using the spice simulation most likely is the best way if there is no good way to handle these kind
of special functions. One could keep searching Google how to make it.
It looks like mainly the Laplace is suitable for very small circuits and physics problems.

Also extracting circuit power seems little bit hard for the Laplace. Voltages and currents work
maybe the best.
Because there is Kirchoff's voltage and current laws (KVL & KCL), but not Kirchoff's law for power.

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#### crutschow

Joined Mar 14, 2008
25,991
Unless you need a numerical solution, the easiest is probably a Spice simulator.
It can calculate the diode voltage, current, and power dissipation.
I use the free LTspice simulator from Analog Devices.

For example below is the LTspice simulation of the common 1N4148 diode showing its forward voltage drop (blue trace) and power dissipation (yellow trace) versus forward current (horizontal axis). #### MrAl

Joined Jun 17, 2014
7,877
Generally speaking, is there an "easier" way to evaluate diode circuits? Normally using laplace transforms makes life a lot easier but due to the non-linear behavior I understand there's no way to use Laplace transforms. However is there anyway to use differential equations or some other method? Some computational method I may not have learned yet?

Thanks
There are various methods for doing circuits that contain diodes. The first thing you have to do is decide what class or type of diode circuit you are dealing with. Unfortunately there are several different classes but the most common two are where one is with the diode just acting like a nonlinear circuit element like a nonlinear voltage or current source, and the other where the diode behaves like a switch.
So really how you approach the circuit depends highly on how the diode is being used and what accuracy is expected from the analysis. For example, in a log antilog circuit accuracy may be very important so you have to use an equation that is very much like a spice model if not exactly like one, but in a switching circuit you can often think of the diode as just a switch that turns on and off depending on external conditions and the losses of the diode can be either zero or you can make it a voltage source with fixed voltage or you can even make it a resistor with fixed resistance.

In the case of a static nonlinear voltage or current source, you will most likely eventually be solving a set of numerical equations so you need a numerical solver, and sometimes this comes in a form similar to curve fitting. In the case of a nonlinear voltage or current source you will have to write a partial differential equation or just a set of equations similar to that.

In the case of the diode acting like a switch it gets very interesting too because Laplace Transform Theory comes into play. In the most extreme case you need to write a partial differential equation because you have to solve for both voltage (or current) and time. But often you can work out the time solution independently from the voltage solution because there are other things that come into play which become obvious, and that of course is the timing of the diode. What that does is allows us to separate the circuit into a set of modes where each mode comes in the form of a different circuit which is usually just part of the original circuit. You end up with two or more circuits that lend themselves to solution by Laplace Transform theory. This gets even more interesting because there are other methods that come into play also that allow us to determine some important facts about the circuit operation and there is a lot written on this subject in the form of a state vector differential equation.

It would be interesting to look at a switching circuit with just switches first to see how these methods work. An interesting circuit that is not too difficult to solve is where we have a DC voltage source switching on and off with one switch SW1 into a network of a parallel resistor and that resistor in parallel with a second switch SW2 with different timing in series with another resistor in series with a capacitor and we want to know the average DC voltage across the capacitor.

An interesting but somewhat complicated example is a full wave rectifier. The solution is both voltage (and current) and time variable, so there are two independent variables. With only a load resistor though the solution becomes simpler because we can figure out the time part of it without too much difficulty.
Another interesting example that contains just a single diode is the Zeta converter we were just looking at in another thread. Solving for the diode turn on and turn off times is a challenge, but once solved the solution starts to get very simple.

The two switch circuit is not too complicated and interesting to look at though maybe we could look at that.

If you have any particular circuits you are interested in perhaps you can post a few and we can discuss in more detail how you would go about solving them.