---
title: "Continuous system simulation; analog computer example"
subject: "Simulation and Modeling"
chapter: 3
key: "semester_8/simulation_and_modeling:continuous-simulation-analog-example"
source: https://www.bctengineer.com/q/semester_8/simulation_and_modeling/continuous-simulation-analog-example
site: BCT Engineering
---

# Continuous system simulation; analog computer example

Simulation and Modeling · Chapter 3: Continuous system simulation

- Asked in: 2072 Magh, 4+6 marks
- Video: [Analogue Method in system Simulation](https://www.youtube.com/watch?v=tcPHAq0GBD4) — YANTRAM-DTH (Design and Technology Hub)
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_8/simulation_and_modeling/continuous-simulation-analog-example
- Whole chapter: https://www.bctengineer.com/md/semester_8/simulation_and_modeling/chapter-3

## Question

What is continuous system simulation? Explain analog computer with a practical example.

## Short answer

Continuous system simulation studies systems whose state varies smoothly with time, solved by analog or digital computers. An analog computer uses voltage to represent variables and units like integrators and potentiometers. For decay dy/dt=-ky, a single integrator with feedback via pot k produces output y0 e^{-kt} from initial condition -y0.

## Answer

#### Continuous system simulation
A **continuous system** is one whose state variables change continuously with time, such as the position of a vehicle, the temperature of a furnace or the concentration of a chemical. **Continuous system simulation** is the study of such systems by a model, usually a set of differential equations, solved by a computer. There are two ways:
- **Analog computer:** the equation is wired as an electrical circuit.
- **Digital computer:** the equation is solved by numerical integration (Euler, Runge-Kutta) in small time steps, using languages such as CSMP or tools such as Simulink.

Examples: aircraft flight, chemical reactors, population growth, electrical circuits.

#### Analog computer
An **analog computer** is a computing device that represents each variable by a continuously varying physical quantity (usually voltage) and solves equations by connecting electronic units that behave like the mathematical operations. The **analog method** solves a differential equation by building a circuit whose voltages obey the same equation (an analogue of the system) and recording the output.

Main units:

| Unit | Function |
|---|---|
| Summer | Adds input voltages, output is the negative sum |
| Integrator | Integrates the sum of its inputs with respect to time, with sign inversion |
| Coefficient potentiometer | Multiplies a voltage by a constant between 0 and 1 |
| Inverter | Changes the sign of a voltage |
| Multiplier / function generator | Product of two variables, nonlinear functions |
| Recorder / scope | Displays the output against time |

#### Practical example: radioactive decay or capacitor discharge
Consider `dy/dt = -k y`, with `y(0) = y0`. This may be radioactive decay, or discharge of a capacitor through a resistor with `k = 1/RC`.

An integrator gives the negative of the integral of its input. If its output is `-y`, its input must be `dy/dt = -k y`. This input is obtained by feeding the output `-y` through potentiometer k.

```text
        +--------[ pot k ]<-------+
        |                          |
        +--------> [ Integrator ]--+--> -y
                   (initial value -y0)
```

Steps:
1. Set the integrator's initial condition to `-y0`.
2. Connect the integrator output `-y` to its input through pot k.
3. Start the run; the output follows `-y0 e^(-kt)`, so the recorder (with an inverter if needed) shows `y = y0 e^(-kt)`.

If k = 0.5 per second and y0 = 10 V, the voltage falls to 3.68 V in 2 s (`10 e^(-1)`). Changing the pot k changes the decay speed immediately.

For second-order systems such as the car suspension, `M x'' + D x' + K x = F`, two integrators in series are used.

#### Advantages and disadvantages
**Advantages of the analog computer**
- Operates in parallel, so the solution is fast, even real time.
- Continuous solution; no step size or numerical round-off error.
- Interactive: parameters can be changed by a knob and the result seen immediately.
- Shows the physical meaning of the equation; good for control studies.

**Disadvantages**
- Accuracy is limited (about 0.1 to 1 percent) by component tolerance, noise and drift.
- Amplitude scaling and time scaling are needed, and are tedious.
- The size of the problem is limited by the number of units available.
- Hard to implement nonlinearities, logical decisions and data storage.
- Re-patching for a new problem takes time; poor repeatability and documentation.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is continuous system simulation?
   A. Studying a system whose state changes continuously with time using a model, usually differential equations
   B. Studying a system that changes only at discrete event times using a queue model
   C. Generating random numbers to sample a system at fixed intervals
   D. Building a physical prototype and testing it in real time
   Answer: A. Continuous systems have state variables changing continuously, and are modelled by differential equations solved on a computer. (Key point: Continuous simulation)

2. How does an analog computer solve a differential equation?
   A. It converts variables to binary numbers and integrates in small time steps
   B. It represents variables by voltages and wires units into a circuit obeying the same equation
   C. It looks up stored solutions in tables
   D. It uses logic gates to make decisions at each step
   Answer: B. Each variable is a continuously varying voltage and the circuit is an analogue of the system. (Key point: Analog computer principle)

3. For dy/dt = -k y, the integrator output is -y. What is fed to its input?
   A. +y directly with no potentiometer
   B. k times the integral of y from an inverter
   C. dy/dt = -k y, obtained by passing -y through potentiometer k
   D. y0 only, as a constant voltage
   Answer: C. An integrator gives the negative integral of its input, so for output -y the input must be dy/dt = -ky, made by feeding -y through pot k. (Key point: Integrator sign)

4. In the decay example, what is set as the integrator initial condition and what does the output follow?
   A. Initial value +y0; output rises as y0 e^(kt)
   B. Initial value 0; output stays at zero
   C. Initial value -y0; output falls linearly to zero
   D. Initial value -y0; output follows -y0 e^(-kt), giving y = y0 e^(-kt) after inversion
   Answer: D. The integrator starts at -y0 and decays exponentially; an inverter gives y = y0 e^(-kt). (Key point: Initial condition and result)

5. With k = 0.5 per second and y0 = 10 V, what is the voltage after 2 s?
   A. About 3.68 V
   B. About 5.00 V
   C. About 6.07 V
   D. About 1.35 V
   Answer: A. y = 10 e^(-0.5x2) = 10 e^(-1) = 3.68 V. (Key point: Numerical check)

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