Confusion about Thevenin's theorem for linear circuits with non-linear elements

My question is about **linear circuits** which has non-linear elements.

When I read about Thevenin's theorem, all explanations say that it is basically about representing a linear circuit as a voltage source and an output impedance. And right after they give many examples about how to thevenize a circuit and obtain a Thevenin voltage and Thevenin resistance(output impedance).

In all of their examples the circuits are composed of the combinations of resistors, capacitors, inductors, current and voltage sources. I haven't seen a single example where they use a non-linear element like a transistor or diode in those examples.

"Linear Power Supply are called linear because semiconductor devices are operated in the linear mode unlike switch mode power supply where semiconductor devices are operated in the switch mode i.e. either on or off."

So imagine a power supply SMPS or linear supply. Is the output impedance of a power supply thevenin resistance? If so does that mean that a power supply is a linear circuit/system? And even if it is linear as whole system/circuit, it includes many non-linear elements. Does thevenizing techniques apply to a h-parameter model of non-linear elements for example?

Imagine we have the circuit diagram of an SMPS or linear power supply. Can we thevenize this circuit such that we can employ Thevenin's tricks and find the Thevenin voltage and resistance(output impedance)?

As your quotation says, linear power supplies are called linear because of the transistor's (BJT's) operating region, not because it's a linear circuit.

To find a Thevenin equivalent for a nonlinear circuit, you can do the same things you do for any transistor amplifier:

1. Make the AC components of the input conditions small.
2. Use a nonlinear, large-signal model to find the characteristics of the circuit at the DC bias point.
3. Make a linearizing approximation around the bias point (a so-called "small-signal model").
4. Plug your small AC signal into the small-signal model to get the AC output.

The typical small-signal model for BJTs is the hybrid-pi model. So starting with a linear regulator: simulate this circuit – Schematic created using CircuitLab

You would end up with this: simulate this circuit

That's a linear circuit, so it has a Thevenin equivalent. This model only applies when changes in the voltages and currents are small, though. That's the price you pay for linearity.

You can't do this directly with an SMPS -- the switching can't be linearized. For a constant duty cycle, you can use a transformer to model the switching action. In the comments, Tom mentions a source for linearizing the control loop.

• Your v_pi voltage polarity is reversed it seems. – Mike Oct 22 '17 at 20:38
• The procedure to linearize a switching power supply circuit is in "Modern DC-to-DC Switchmode Power Converter Circuits" by Severns and Bloom. This approach can be used for stability analysis of a switching power supply feedback loop. The loop bandwidth is much lower than the switching frequency. – Tom Anderson Oct 23 '17 at 4:36

Technically, Thevenin's original theorem only included DC and resistors. It is true, however, that it also works for AC and impedance networks.

It is not possible to represent non-linear circuits as Thevenin equivalents if their behaviour is non-linear throughout their operating range. A power supply can be characterised as a Thevenin circuit only within certain limits. Once the response becomes nonlinear, the thevenin model will no longer hold.

• Can we thevenize a linear SMPS power supply such that we can employ Thevenin's tricks and find the Thevenin voltage and resistance(output impedance)? – HelpMee May 10 '17 at 21:12
• As John D said, yes, within limits. You can basically model anything as a Thevenin equivalent, as long as it behaves as a Thevenin equivalent. This sounds a bit dumb, but it's basically the only way to say it once you're talking about non-linear elements – BeB00 May 10 '17 at 21:18

Usually when we make models or equivalent circuits for systems that contain non-linear elements, we linearize the model around a certain operating point. So it works if the non-linear element acts like a linear element if the operating point doesn't change too much.

If the operating point changes, we probably have to change the parameters of the linear model. So you may have several models to cover different circumstances.

So the output impedance of a power supply will be one value for a DC load and something very different if the load disturbance is outside the control bandwidth of the control loop.

For DC and low frequency load currents you could model a supply as a voltage source in series with a source resistance. The model will not be valid for fast transient loads or high frequency load currents.

For an example of linearizing a non-linear device you can look at the Hybrid Pi model of a BJT: Hybrid Pi Model

• Can we thevenize a linear SMPS power supply such that we can employ Thevenin's tricks and find the Thevenin voltage and resistance(output impedance)? – HelpMee May 10 '17 at 21:11
• A SMPS is not typically referred to as linear, but as I said above over limited operating conditions you can model a SMPS as a voltage source in series with a resistor, in the same form as a Thevenin equivalent. – John D May 10 '17 at 21:14
• Forget about using SMPS in abnormal operating condition. Imagine it is working under its rated conditions. So you mean under rated conditions like using a max rated 5V 1A SMPS with load drawing between 100mA upto 300mA; can we thevenize this circuit? Do we need to convert the transistors diodes ect to models to apply thevenin methods? – HelpMee May 10 '17 at 21:18
• I don't know, but i would guess probably not, as thevenin really only deals with AC or DC, and a square wave is neither. – BeB00 May 10 '17 at 21:20
• I wasn't talking about abnormal conditions, load transients are a very normal condition for power supplies. If the load transient is relatively fast the simple "Thevenin" model of the supply will not be accurate. If the load is DC or very low frequency which is also possible it can be a very good model. There are control models that use linearization (PWM switch model, state-space averaging) that can predict output impedance, but then you're getting away from a simple Thevenin model. – John D May 10 '17 at 21:22