I would like to make a discrete time simulation of a 3 phase inductive load but I am struggling with the problem. I have this Y configuration in mind.

enter image description here

I'd like to simulate each current and Vn, given that V1,V2,V3 are not necessarly sinusoidal. Actually, I'm interrested the control of a 3-phase inverter and would like play with the timing of the gate driver pulses. I believe I need a discrete simulation if I want to apply non-linear inputs.

What I did

I started with a simple LR circuit.

enter image description here

I found

$$V_L = V_{in}-I[n]R$$ $$d_I= \frac{V_Ldt}{L}$$ $$I[n+1] = I[n]+dI$$

I tried that with a python script and a simulink discrete simulation and that behave well.

Now if I want to apply that thinking to my 3-phase load, I find these equations (let's jsut talk about branch #1)

$$V_{L_1} = V_1-I_1[n]R_1-V_n$$ $$d_{I_1}= \frac{V_{L_1}dt}{L_1}$$ $$I_1[n+1] = I_1[n]+dI_1$$

With this 3 phase load, I have a new variable: Vn and this is where I get stuck.

With sinusoidal, 120 degrees phase shift and perfectly balanced branch, I know Vn would be 0, but these are assumptions that I don't want to do.

I thought of superposition theorem. To do so, I would need to find an equivalent impedance for R2, R3, L2, L3, which I don't know how to achieve. Since I don't want to assume my inputs are sinusoidal, or even linear at all, I can't just say my inductors are reactive impedances and then simplify.

What are my options here?

I don't think Clarke transform would help here as my goal is to play with the PWM of each of the 3 branches. Do I need to resign myself into making a linear model of my signals? Such as decomposing square pulses into sums of sinusoids and then using superposition?

Thank you!

EDIT - Developped answer

Following UweD answer, I achieved t get a working discrete model of a Y connected 3-phase load. Here's what I got in the end.

Starting with UweD recommandation, we set:

$$ V_1-V_N=R_1I_1+\frac{dI_1}{dt}L_1 $$ $$ V_2-V_N=R_2I_2+\frac{dI_2}{dt}L_2 $$ $$ V_3-V_N=R_3I_3+\frac{dI_3}{dt}L_3 $$ $$ I_1+I_2+I_3 = 0 $$

We can then say:

$$ V_N = V_3+I_1R_3+I_2R_3+ \frac{dI_1}{dt}L_3+\frac{dI_2}{dt}L_3$$

Using first equation, we find dI2/dt

$$ V_1-(V_3+I_1R_3+I_2R_3+ \frac{dI_1}{dt}L_3+\frac{dI_2}{dt}L_3)=R_1I_1+\frac{dI_1}{dt}L_1 $$ $$\frac{dI_1}{dt} = \frac{1}{L_1+L_3}(V_1-V_3-I_1R_3-I_2R_3-R_1I_1-\frac{dI_2}{dt}L_3) $$

$$ \frac{dI_2}{dt}=\frac{1}{L_3}(V_1-V_3-I_1R_3-I_2R_3-I_1R_1-\frac{dI_1}{dt}(L_1+L_3))$$

Using second equation, we find dI2/dt

$$ V_2-V_3+I_1R_3+I_2R_3+ \frac{dI_1}{dt}L_3+\frac{dI_2}{dt}L_3=R_2I_2+\frac{dI_2}{dt}L_2 $$ $$\frac{dI_2}{dt} = \frac{1}{L_1+L_3}(V_2-V_3-I_1R_3-I_2R_3-R_2I_2-\frac{dI_1}{dt}L_3) $$

Then we equate the 2 expression of dI2/dt to find an expression for dI1/dt

$$ \frac{dI_1}{dt}=\frac{ \frac{-1}{L_2+L_3}(V_3-V_2+I_1R_3+I_2R_3+I_2R_2) + \frac{1}{L_3}(V_3-V_1+I_1R_3+I_2R_3+I_1R_1)}{\frac{L_3}{L_2+L_3}-\frac{L_1+L_3}{L_3}}$$

We now have an expression for dI1/dt, 2 expression for dI2_dt. The following discrete Simulink model works well and gives the same result as a SPICE simulator.

Final model

  • \$\begingroup\$ Why don't you use a SPICE simulator for this? You can easily simulate PWM and other arbitrary waveforms. \$\endgroup\$ – Elliot Alderson Jan 6 at 18:51
  • \$\begingroup\$ I guess that eould be a good tool to solve the problem, but I am trying to teach myself solid basics before going forward. What if I want to make a spice model myself, what mathematics should I use? \$\endgroup\$ – Pier-Yves Lessard Jan 6 at 18:54
  • \$\begingroup\$ Also, I'd like the flexibility to use a tool such as Simulink that have the capability of code generation \$\endgroup\$ – Pier-Yves Lessard Jan 6 at 18:57
  • \$\begingroup\$ Creating and solving your own SPICE models is a big task. UC Berkeley created the original SPICE and published academic papers about the algorithms. You are not really asking about electronics design anymore, so I suggest you take this to a site for scientific computing. \$\endgroup\$ – Elliot Alderson Jan 6 at 18:58
  • \$\begingroup\$ I don't plan on making SPICE model... I want to understand the mathematic. \$\endgroup\$ – Pier-Yves Lessard Jan 6 at 18:59

These are your general four equations:

\$V_1 - V_n = R_1 \cdot i_1 + L_1 \cdot \frac{di_1}{dt}\$

\$V_2 - V_n = R_2 \cdot i_2 + L_2 \cdot \frac{di_2}{dt}\$

\$V_3 - V_n = R_3 \cdot i_3 + L_3 \cdot \frac{di_3}{dt}\$

\$i_1 + i_2 + i_3 = 0\$

Now e.g. with

\$i_3 = -i_1 - i_2\$

you can take the third equation, replace \$i_3\$, and express \$V_n\$ as

\$V_n = V_3 + R_3 \cdot i_1 + L_3 \cdot \frac{di_1}{dt} + R_3 \cdot i_2 + L_3 \cdot \frac{di_2}{dt}\$

Replacing \$V_n\$ in our first two equations gives you two equations with the unknowns \$i_1\$ and \$i_2\$ which you can discretisize now.

If you want to go into more complex systems I fully agree with Elliot Alderson that a circuit simulator would be helpful. If you want to develop your own one, I strongy recommend to use the Nodal analysis methode.

  • \$\begingroup\$ That helps. I've been able to move forward with your suggestion. I still don't get good results in simulation. I will mark this as the answer once I figure where I went wrong. Thank you! \$\endgroup\$ – Pier-Yves Lessard Jan 7 at 13:39
  • \$\begingroup\$ I've achieved to make it work. I have edited my answer to include my final answer. Thank you very much \$\endgroup\$ – Pier-Yves Lessard Jan 10 at 2:28

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.