# RC Circuit - Mathematical Model

The exponential equation found earlier (in the page examining the circuit's step response) can be viewed as a solution, and tells us much about the behaviour of the circuit - it tells us what the output is for a step input and allows us to see the effect of changing component values. However, a fuller description can be obtained by making a complete mathematical model. We can return to the basic current and voltage equations,

 dVc Vin - Vc = RC (1) dt

and by rearranging (1) we get:

 1 dVc = ( Vin - Vc ) dt CR

So:
 1 Vc = Vin - Vc dt RC

 1 Vin - Vc Vc = dt C R

This relationship can be modelled (using VisSim you may download the VisSim file) as below (Figure 3); this is a direct implementation of the formula (as, I hope, you can see).

• Readers without VisSim can download VisSim Viewer 6 - a free utility that allows VisSim models to be viewed and run (File size: 3.15 MB).

Figure 3

Reassuringly, the behaviour of Vc in the simulation is the same as that in Figure 2.

What is useful about the model is the appearance in it of such things as voltages and currents (Vin, Vc, and ic) as well as component values. Every aspect of the circuit can be mathematically modelled. And we are not restricted to step inputs - any waveform can be used (provided the modeller is able to synthesise it).

### A Word about Models and Simulations

The mathematics of the RC circuit (or any other system) can be simulated by various means. In the above example; the input and output voltages are present and can be plotted out, as can the voltage across the resistor. This feature allows the full behaviour of the circuit to be studied; whereas, a 'solution' looks at only one aspect.

A few years ago, an electronic analogue computer would be used for this sort of thing. For anyone not old enough to have used such a machine I will explain. In its simplest form an analogue computer would consist of analogue integrators, summers (which add two more voltages together) and gain stages. Such building blocks would be made using operational amplifiers. The computer would be programmed by connecting the various blocks together and adjusting time constants and gains by choosing suitable component values. This range of blocks is sufficient to model the sort of differential equations that I have been discussing. Indeed the topology of analogue computers is still used to realise analogue active filters.

A major drawback of the analogue computer was its accuracy; the better ones achieved accuracies of a few percent. When digital computers became widely available, simulations using languages such as FORTRAN and BASIC became possible and later, software (such as VisSim) specifically designed to provide a 'user friendly' interface was developed.

So what we see in the simulation above is functionally identical to the RC network. All the simulated voltages and currents in the model have their counterparts that can be measured in the real network.

A small word of warning - although it may look a bit like one, the pictorial representation of the simulation is not a circuit diagram and although the 'signals' (i.e. the voltages and currents) have counterparts in the real world, the blocks (summers and integrators do not). So do not think that a perfect integrator is lurking somewhere in the network - it isn't (after all, the 'real' network is just a resistor and capacitor). At best the RC circuit is a poor imitation of an integrator (as we shall see later) but that is little more than coincidence. Integrators appear over and over again in simulations even though the systems they're used to model do not contain anything resembling one.

Last updated: 29 May 2008;   © Lawrence Mayes, 2006-2008