Background

Setup of basic equation describing a simple inductor circuit - how does an inductor work, what is an inductor used for, how can we model the inductor

Inductor coil in electrical circuit: solving the differential equation for current

Setting up the use of Simpson's Rule to implement Euler's Method and integrate Right Hand side, with Delta x instead of Delta t.

Electrical circuit

Solving a differential equation

Different right-hand sides, different time steps

Content

Introduction

Today we will be using a spreadsheet program to implement Euler’s Method and solve a differential equation. Euler’s Method ties together the concepts of numerical integration and differential equations. We will implement Euler’s Method to solve for the current in the following circuit:

This circuit is described in Example 4 of Section 9.5 of the Stewart Calculus textbook.

Euler's Method

We can use Euler’s Method to solve a differential equation of the general form:

$\frac{dy}{dt} = F(t,y)$

We start by chopping up the time domain into a set of steps of size $\Delta$ . We will evaluate the solution, and the right‐hand side of the equation, at each timestep.

Start by turning the derivative on the left‐hand side into a finite difference quotient:

$\frac{dy}{dt} \approx \frac{\Delta y}{\Delta t}$

Next, using the fact that $\Delta y = y(t+\Delta t)-y(t)$ , we can write a relationship between the solution at a given time step $$ t_i $$ , denoted $$ y_i = y(t_i) $$ , and the solution at the next timestep $t_{i+1} = t_i + \Delta t$ , denoted $y_{i+1}$ :

$y_{i+1} = y_i + \Delta t \left( F(t,y) \right)$

This is the equation you will be implementing in a spreadsheet.

The sequence of timesteps begins at some initial time $$ t_0 $$ corresponding to an initial state of the system $$ y_0 $$ . This initial condition yields an estimate of the solution after the first timestep:

$y_1 = y_0 + \Delta t \left( F(t_0,y_0) \right)$

This yields an estimate of the solution at the next timesteps,

$y_2 = y_1 + \Delta t \left( F(t_1,y_1) \right)$

$y_3 = y_2 + \Delta t \left( F(t_2,y_2) \right)$

and so on.

The Circuit Differential Equation

The General Differential Equation

On the right is a circuit containing four components: a battery (which is a source of electric current), a resistor R, an inductor L, and a switch to turn the circuit on and off.

We are looking for the current as a function of time. However, current is not conserved – we have to start with something that’s conserved. In this circuit, voltage is conserved. The voltage supplied by the battery will match the voltage drops across the components.

The current - that's what we're solving for - is denoted $$ I(t) $$ .

The voltage supplied by the battery is denoted $$ E(t) $$ and is a constant function:

$$ E(t) = 60 V $$

Across the resistor, the voltage dorps by7 an amount proportional to the resistance and the current:

$\Delta V_{resistor} = R I$

Across the inductor, the voltage drop depends on the rate of change of the current:

$\Delta V_{inductor} = L \dfrac{dI}{dt}$

Balancing the supplied voltage with the voltage drops gives us our differential equation:

$L \dfrac{dI}{dt} + RI = E(t)$

The Particular Differential Equation

Let us consider Stewart's Example 4 from Section 9.5, where he uses a circuit with an inductor of 4 Henries, a resistor of 12 Ohms, and a battery with a voltage of 60 Volts, corresponding to L = 4, R = 12, and E(t) = 60. Now we can solve for the current as a function of time.

For the implementation of Euler's Method to solve this circuit, Stewart selected a time step of 0.1 seconds, and a total time of 3 seconds. This makes the particular differential equation:

$\dfrac{dI}{dt} + 3 I = 15$

Worksheet Questions

Question 1: Find the particular differential equation corresponding to a circuit with the following components:

Inductor of 10 Henries, L = 10
Resistor of 20 Ohms, R = 20
Voltage of 120 V, E = 120

Question 2: Implement the solution to this differential equation in an Euler's Method spreadsheet program. When you are finished, you should have a column of values that give you the current as a function of time. What is the limiting value of current, $I_{\infty} = \lim_{t \rightarrow \infty} I(t)$ ?

Question 3: Create a chart with values of current as a function of time. Add proper labels (with units) to your axes, and a title to your chart explaining what is shown.

Question 4: What affect does resistance have on the value of $I_{\infty}$ ? Implement Euler's Method for the three resistances shown below. Replicate your chart of current versus time for each resistor.

Part 4a: Resistor of 5 Ohms

Part 4b: Resistor of 50 Ohms

Part 4c: Resistor of 200 Ohms