Simulating an RC Network with an Electronic Analog Computer

published 2025-08-22

By Christopher Howard

Yes, irony abounds. Continuing on...

We model an RC network like so:

 ╭── R ──╮
 │       │
 +       │
 S       C
 -       │
 │       │
 ╰───────╯

Per Kirchhoff's voltage law:

V_S(t) + V_R(t) + V_C(t) = 0

Differentiate:

d/dt{V_S(t)} + d/dt{V_R(t)} + d/dt{V_C(t)} = 0

Since, for a resistor, V = IR, we have

d/dt{V_R(t)} = R d/dt{I(t)}

Per Wikipedia, the voltage-current relation for capacitors is

I(t) = C d/dt{V(t)} or

d/dt{V_c(t)} = I(t) / C

We can substitute these into our original equation. To keep things simple, let's declare the following:

F = V_s(t)

Y = I(t)

Using engineering dot notation, we now have:

Ḟ + RẎ + Y/C = 0

Put that in normal form:

Ẏ = (- Y - CḞ) / RC

Here is a corresponding analog computer diagram. To simplify, I combine RC into one coefficient potentiometer, {M}, but you could have two coefficients {R}{C} there instead. Strictly speaking, a diagram modeling the above equation would only give us the current. So I added an extra integrator on the right to give the voltage of the capacitor. I.e.

                        t  
V_c(t) = V_c(to) + (1/C)∫I(τ)dτ
                        t0

Here is the diagram:

╭─────╴{1/M}───────────╮
│                      │
│ ╭╮      ╷            │  ╭╮
╰─┤│\ -Y  │\ Y         │  ││\
  ││ ╶────│ ╶───┬──────┼──┤│ ╶─{1/C}╶─(V_c(t)
  ││/     │/    │      │  ││/
  ╰╯      ╵     │      │  ╰╯
                │ ╷    │
                ╰─┤\   │ -Y-CḞ
                  │ ╶──╯
 F)╶─╴C1╶─┬──{C}──┤/
          │       ╵
          R1
          │
         GND

The C1-R1 RC network — ha ha! — gives us a differentiator to get Ḟ from F. F is our arbitrary, variable supply voltage V_S(t), which can come from a signal generator or whatever you like. I just arbitrarily picked a 470nF polyester cap and a 100kΩ resistor.

Implementing this as in the diagram above, I got drift in the output of the integrator that provides V_c(t). To fix that, I put an 8 MΩ resistor, the largest I had on hand, into the feedback loop for the integrator, parallel to the capacitor. I.e., across output and summing junction.

Here is the oscilloscope output. The results seems plausible:

oscilloscope output showing supply voltage, current, and capacitor voltage in the simulation

Copyright

CC BY-SA 4.0 Deed