Tag: Plotting

The following non-linear circuit is an uncontrolled three-phase full-wave 6-pulse rectifier. The inputs are $v_{an}(t) = V_m sin (omega t)$, $v_{bn}(t) = V_m sin (omega t-120°)$ and $v_{cn}(t) = V_m sin (omega t+120°)$. $R$ and $L$ are the load’s resistance and inductance, $omega$ and $T$ are voltage sources’ angular frequency and period, and $V_m$ are the sources’ amplitude; the diodes are modeled as ideals. The unknown to solve for is $i(t)$, the load’s current (not shown in the diagram). The circuit’s differential equation is

$i'(t) + dfrac{R}{L} i(t) = dfrac{sqrt{3} V_m}{L} sin{(omega t + theta)} tag*{}$

subject to the initial condition

$i(t_0) = I_0 tag*{}$

I consider it unnecessary to explain the functioning of the circuit, and don’t want to confuse you with circuit theory. Basically, due to the non-linearity, there’s not a single expression that describes the solution of the circuit for all $ t ge 0 $. However, we can obtain a piecewise solution: for each cycle or period $T$, we express $i(t)$ in seven segments/pieces. Each piece has the same general form, which is:

$i(t) = dfrac{sqrt{3} V_m omega L}{R^2 + (omega L)^2} (- cos{(omega t + theta)} + e^{frac{R}{L}(t_0-t)} cos{(omega t_0 + theta)}) + dfrac{sqrt{3} V_m R}{R^2 + (omega L)^2} (sin{(omega t + theta)} – e^{frac{R}{L}(t_0-t)} sin{(omega t_0 + theta)}) + I_0 e^{frac{R}{L}(t_0-t)} tag*{}$

I developed the following algorithm that does what I need:

Given the circuit’s parameters R, L, w, T = 2 Pi/w,
Vm, the initial condition IC, and n (where n is a natural number used to end the loop). In the following, whenever I say “expr“, sustitute it with (Sqrt(3) Vm w L)/(R^2 + (w L)^2) (- Cos(w t + θ) + E^(R/L (t0 - t)) Cos(w t0 + θ)) + (Sqrt(3) Vm R)/(R^2 + (w L)^2) (Sin(w t + θ) - E^(R/L (t0 - t)) Sin(w t0 + θ)) + I0 E^(R/L (t0 - t))

1. Set k = 0.
2. For k T < t ≤ k T + T/12:
   1. Set θ = 90 Degree. If k == 0, set t0 = 0. If k ≥ 1, set t0 = k T.
   2. If k == 0, set I0 = IC. If k ≥ 1, set I0 = i(t0).
   3. Add/append expr (using the new values of t0, I0 and θ) as a new piece of i(t) for the interval k T < t ≤ k T + T/12.
3. For k T + T/12 < t ≤ k T + T/4:
   1. Set t0 = k T + T/12 and θ = 30 Degree.
   2. Set I0 = i(t0).
   3. Add expr (using the new values of t0, I0 and θ) as a new piece of i(t) for the interval k T + T/12 < t ≤ k T + T/4.
4. For k T + T/4 < t ≤ k T + 5 T/12:
   1. Set t0 = k T + T/4 and θ = -30 Degree.
   2. Set I0 = i(t0).
   3. Add expr (using the new values of t0, I0 and θ) as a new piece of i(t) for the interval k T + T/4 < t ≤ k T + 5 T/12.
5. For k T + 5 T/12 < t ≤ k T + 7 T/12:
   1. Set t0 = k T + 5 T/12 and θ = -90 Degree.
   2. Set I0 = i(t0).
   3. Add expr (using the new values of t0, I0 and θ) as a new piece of i(t) for the interval k T + 5 T/12 < t ≤ k T + 7 T/12.
6. For k T + 7 T/12 < t ≤ k T + 3 T/4:
   1. Set t0 = k T + 7 T/12 and θ = -150 Degree.
   2. Set I0 = i(t0).
   3. Add expr (using the new values of t0, I0 and θ) as a new piece of i(t) for the interval k T + 7 T/12 < t ≤ k T + 3 T/4.
7. For k T + 3 T/4 < t ≤ k T + 11 T/12:
   1. Set t0 = k T + 3 T/4 and θ = 150 Degree.
   2. Set I0 = i(t0).
   3. Add expr (using the new values of t0, I0 and θ) as a new piece of i(t) for the interval k T + 3 T/4 < t ≤ k T + 11 T/12.
8. For k T + 11 T/12 < t ≤ k T + T:
   1. Set t0 = k T + 11 T/12 and θ = 90 Degree.
   2. Set I0 = i(t0).
   3. Add expr (using the new values of t0, I0 and θ) as a new piece of i(t) for the interval k T + 11 T/12 < t ≤ k T + T.
9. Increase k by one, that is, evaluate k++ or set k = k + 1.
10. If k < n, return to step 2 and repeat. If k == n, stop the loop/algorithm.

So, I want to implement the previous algorithm in Mathematica, in order to define i(t) as a piecewise function and plot it, where I can assign the values of R, L, w, T = 2 Pi/w, Vm, IC and n before plotting. Obviously the Piecewise() function will be needed, and perhaps Do() or While(), and Module() might be helpful, but I don’t know how to use them to define a piecewise function.

I hope you understand what I’m trying to do, otherwise please ask in the comments.