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I am not from DSP background and also not sure if I am posting this in right community. I have basic understanding of resistors, capacitors and filters. My requirement is as follows-

The supply voltage is fixed and I would like to simulate the output of the filter using Python. With a fixed input voltage, I would like to know what is the output voltage at R3. The current is sampled at a frequency of 100 milliseconds. The input is a DC voltage, which can change.

I am confused where should I start from. Should I use scipy library? Since I want a time domain output, what is the best approach? Is there some specific reading material for such problem? Which kind of filter would be this?

Please excuse for a basic question.

enter image description here

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    $\begingroup$ Are you specifically looking for a solution using python libraries or is it ok to use pure mathematical code (based on hand solution)? Also what are the initial conditions of the capacitors ? Also will your solution include capacitor transients or only the DC final values ? What about the dynamic load ? As It may change the solution drastically... $\endgroup$ – Fat32 Mar 14 '16 at 13:36
  • $\begingroup$ I would like to assume resistive load. Rather we can assume it to be a current sink which draws current which can change over time. Also we can assume zero/constant initial conditions. I would like to use python libraries, because my application is based in python. However if we describe using mathematical equations, I can port those equations to python. $\endgroup$ – knowledge_seeker Mar 14 '16 at 20:19
  • $\begingroup$ Please better describe that dynamic load. A current sink? what is its terminal I-V characterisation? Without that you won't be able to solve the circuit hence find the current that is being sinked. Also please clarify if you are seeking for a simple DC solution or all the transient? And needless to remind any further circuit theory can depart your question into hardware store as there would be little place left to apply any DSP theory to solve it. $\endgroup$ – Fat32 Mar 14 '16 at 22:48
  • $\begingroup$ What is the name of the imaged system what kind of electronic component is it, what information can you give us about the image contents. $\endgroup$ – com.prehensible Mar 15 '16 at 12:36
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Obtaining the system equations

Using Kirchhoff's voltage law: $$ \begin{align} V(t) &= V_{R_1}(t) + V_{R_2}(t) + V_{R_3}(t) + V_I(t) \tag{1}\\ V_{R_1}(t) &= V_{C_1}(t) \tag{2} \\ V_{R_2}(t) &= V_{C_2}(t) \tag{3} \end{align} $$ And similarly using Kirchhof's current law: \begin{align} I_{R_3}(t) &= I_{R_1}(t) + I_{C_1}(t) \tag{4}\\ &= I_{R_2}(t) + I_{C_2}(t) \tag{5} \end{align}

Now using the Voltage-Current relationships of the resistances and capacitors (assuming ideal components):

\begin{align} V_{R_1}(t) &= R_1 I_{R_1}(t) \tag{6}\\ V_{R_2}(t) &= R_2 I_{R_2}(t) \tag{7}\\ V_{R_3}(t) &= R_3 I_{R_3}(t) \tag{8}\\ I_{C_1}(t) &= C_1 \frac{d\,V_{C_1}(t)}{dt} \tag{9}\\ I_{C_2}(t) &= C_2 \frac{d\,V_{C_2}(t)}{dt} \tag{10} \end{align}

Based on your comments, a current sink with a resistive internal resistance can be assumed. For analysis, it is common to replace such a load with its Norton equivalent circuit (see Norton theorem) which consists of putting the internal resistance (say $R_4$) accross an ideal current sink $I(t)$. \begin{align} V_I(t) &= R_4 I_{R_4}(t) \tag{11} \\ &= R_4 (I_{R_3}(t) - I_I(t)) \tag{12} \end{align}

Numerical solution

To discretize the problem to discrete time step $t_n = n \Delta t = n/f_s$, we can make approximations for $(9)$ and $(10)$: \begin{align*} I_{C_1}[n] &\approx C_1 \frac{V_{C_1}[n+1] - V_{C_1}[n]}{\Delta t} \\ &\approx C_1 f_s \left(V_{R_1}[n+1] - V_{R_1}[n]\right) \\ I_{C_2}[n] &\approx C_2 \frac{V_{C_2}[n+1] - V_{C_2}[n]}{\Delta t} \\ &\approx C_1 f_s \left(V_{R_1}[n+1] - V_{R_1}[n]\right) \end{align*} This gives us the recurrence equations: \begin{align*} V_{R_1}[n+1] &\approx V_{R_1}[n] + \frac{1}{C_1 f_s} I_{C_1}[n] \\ &\approx V_{R_1}[n] + \frac{1}{C_1 f_s} \left(I_{R_3}[n] - I_{R_1}\right) \\ &\approx V_{R_1}[n] + \frac{1}{C_1 f_s} \left(\frac{V_{R_3}[n]}{R_3} - \frac{V_{R_1}[n]}{R_1}\right) \\ V_{R_2}[n+1] &\approx V_{R_2}[n] + \frac{1}{C_2 f_s} I_{C_2}[n] \\ &\approx V_{R_1}[n] + \frac{1}{C_2 f_s} \left(I_{R_3}[n] - I_{R_2}\right) \\ &\approx V_{R_1}[n] + \frac{1}{C_2 f_s} \left(\frac{V_{R_3}[n]}{R_3} - \frac{V_{R_2}[n]}{R_2}\right) \end{align*} And $V_{R_3}[n]$ begin obtained by substituting $(8)$ and $(12)$ into $(1)$: \begin{align*} V_{R_3}[n] &= \frac{V[n] - V_{R_1}[n] - V_{R_2}[n] + R_4 I[n]}{1+R_4/R_3} \end{align*}

which can readily be implemented in python (with $V(t)$ and $I_I(t)$ being constant for illustration, but the extension to arbitrary function is trivial) with:

# numerical solution
T    = 1.0/fs
Tmax = 20
t = np.arange(0,Tmax,T)

vr3 = np.zeros(len(t))
vr1 = 0
vr2 = 0
for i in np.arange(len(t)-1):
    vr3[i] = (V-vr1-vr2+R4*I)/(1+R4/R3)
    i3  = vr3[i]/R3
    ic1 = i3 - vr1/R1
    ic2 = i3 - vr2/R2
    vr1 = vr1 + ic1/(C1*fs)
    vr2 = vr2 + ic2/(C2*fs)
vr3[-1] = (V-vr1-vr2+R4*I)/(1+R4/R3)

Steady-state solution

As a validation of the above numerical solution, we can check whether it converges to a steady state which can be computed analytically fairly easily (assuming a steady state can be reached, which typically requires $V(t)$ and $I_I(t)$ to be near constant). To do this we note that in the steady state derivative terms go to zero. As a result, $I_{C_1}(t) = I_{C_2}(t) = 0$ and consequently, $I_{R_1}(t)=I_{R_2}(t)=I_{R_3}(t)$. Thus $(1)$ can be written as: \begin{align*} V(t) &= R_1 I_{R_1}(t) + R_2 I_{R_1}(t) + R_3 I_{R_1}(t) + R_4\left(I_{R_1}(t) - I_I(t)\right) \\ &= \left(R_1+R_2+R_3+R_4\right) I_{R_1}(t) - R_4 I_I(t) \end{align*} From which we can solve for $V_{R_3}(t)$: \begin{align*} V_{R_3}(t) &= \frac{R_3}{R_1+R_2+R_3+R_4} \left(V(t) + R_4 I_I(t)\right) \end{align*}

For illustration, the graph below shows an example for the following parameters:

# parameters
R1 = 2200.0
C1 = 0.001
R2 = 1000.0
C2 = 0.0047
R3 = 5600.0
R4 = 100000.0
V  = 5
I  = 1e-5
fs = 10

Example solution with above paramters

As a final note, for voltage source $V(t)$ and current sink $I_I(t)$ which are constant or are from a set of fairly simple functions, the solution could also be obtained analytically with the use of Laplace transforms.

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  • $\begingroup$ this is something I really needed. thanks a lot for your solution. $\endgroup$ – knowledge_seeker Mar 16 '16 at 8:36

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