First-order system simulation in Simulink

Introduction

We use first-order system simulation to understand how simple systems change over time. It helps us see how things like temperature or speed react to changes.

To model how a room heats up when you turn on a heater.

To see how water level changes in a tank when filling or draining.

To understand how a car's speed changes when you press the gas pedal.

To predict how a battery charges over time.

To study how a simple electrical circuit responds to voltage changes.

Syntax

Simulink

1. Open Simulink.
2. Create a new model.
3. Add a 'Transfer Function' block.
4. Set numerator and denominator for first-order system (e.g., numerator = [1], denominator = [T 1]).
5. Add input source (e.g., Step block).
6. Add Scope block to see output.
7. Connect blocks: Step -> Transfer Function -> Scope.
8. Run simulation.

The denominator defines the system's time constant (T) and behavior.

The Step block simulates a sudden change input.

Examples

This sets a first-order system with time constant 5 seconds.

Simulink

Numerator: [1]
Denominator: [5 1]

This system has a gain of 2 and time constant 3 seconds.

Simulink

Numerator: [2]
Denominator: [3 1]

Sample Program

This example simulates a system that slowly reaches a value of 1 after a step input, showing how the system reacts over time with a 10-second time constant.

Simulink

1. Open Simulink and create a new model.
2. Add a Step block (input).
3. Add a Transfer Function block.
4. Set Transfer Function numerator to [1].
5. Set Transfer Function denominator to [10 1] (time constant 10 seconds).
6. Add a Scope block.
7. Connect Step -> Transfer Function -> Scope.
8. Run the simulation for 50 seconds.

// The Scope will show the output rising slowly to 1 over time, showing the first-order response.

Important Notes

The time constant (T) controls how fast the system responds.

Use Scope to visualize the system output over time.

Step input helps see how the system reacts to sudden changes.

Summary

First-order system simulation shows how simple systems change over time.

Use Transfer Function block with numerator and denominator to define the system.

Step input and Scope help test and visualize the system response.

Practice

1. What does a first-order system simulation in Simulink typically show?

easy

A. How the system output changes over time after an input

B. The exact solution of differential equations analytically

C. The internal code of Simulink blocks

D. The hardware specifications of the computer running Simulink

Solution

1. Step 1: Understand the purpose of first-order system simulation

   It models how the output of a simple system changes over time when given an input.

2. Step 2: Identify what simulation shows in Simulink

   Simulink simulates the system response graphically, showing output behavior over time.

3. Final Answer:

   How the system output changes over time after an input -> Option A

4. Quick Check:

   Simulation = output change over time [OK]

Hint: Simulation shows output change over time [OK]

Common Mistakes:

• Confusing simulation with code details
• Thinking simulation shows hardware specs
• Assuming simulation gives exact analytic solutions

2. Which of the following is the correct way to represent a first-order transfer function in Simulink's Transfer Function block?

easy

A. Numerator: [5], Denominator: [1 0]

B. Numerator: [1 5], Denominator: [1]

C. Numerator: [1], Denominator: [1 5]

D. Numerator: [1], Denominator: [5 1]

Solution

1. Step 1: Recall first-order transfer function form

   A first-order system has denominator of form s + a, so denominator coefficients are [1 a].

2. Step 2: Match numerator and denominator arrays

   Numerator is usually constant [1], denominator [1 5] means s + 5, correct for first-order.

3. Final Answer:

   Numerator: [1], Denominator: [1 5] -> Option C

4. Quick Check:

   Denominator = [1 a] for first-order [OK]

Hint: Denominator starts with 1 for s term [OK]

Common Mistakes:

• Swapping numerator and denominator
• Using wrong order of coefficients
• Using zero in denominator leading coefficient

3. Given the Simulink Transfer Function block with Numerator = [2] and Denominator = [1 3], and a step input of magnitude 1, what is the steady-state output value?

medium

A. 2

B. 0.67

C. 0.5

D. 3

Solution

1. Step 1: Calculate steady-state gain from transfer function

   Steady-state gain = numerator constant / denominator constant = 2 / 3 ≈ 0.67.

2. Step 2: Apply step input magnitude

   Step input is 1, so output steady-state = 0.67 * 1 = 0.67.

3. Final Answer:

   0.67 -> Option B

4. Quick Check:

   Gain = 2/3 = 0.67 [OK]

Hint: Steady-state = numerator/denominator constant term [OK]

Common Mistakes:

• Using numerator over denominator reversed
• Ignoring step input magnitude
• Confusing transient with steady-state

4. You created a first-order system in Simulink with Numerator = [1] and Denominator = [1 0], but the simulation shows no response. What is the likely error?

medium

A. Denominator [1 0] represents an integrator, not a first-order system

B. Numerator should be zero for first-order systems

C. Step input magnitude is missing

D. Scope block is not connected

Solution

1. Step 1: Analyze denominator [1 0]

   Denominator [1 0] means s + 0 = s, which is an integrator, not a first-order system with constant term.

2. Step 2: Understand simulation behavior

   Integrator causes output to ramp, but if input is zero or setup wrong, no visible response occurs.

3. Final Answer:

   Denominator [1 0] represents an integrator, not a first-order system -> Option A

4. Quick Check:

   Denominator must have constant term for first-order [OK]

Hint: Denominator must have constant term, not zero [OK]

Common Mistakes:

• Assuming zero constant term is valid
• Ignoring block connections
• Forgetting to set step input

5. You want to simulate a first-order system with time constant 4 seconds and gain 3 in Simulink. Which numerator and denominator coefficients should you use in the Transfer Function block?

hard

A. Numerator: [1], Denominator: [1 0.25]

B. Numerator: [3], Denominator: [4 1]

C. Numerator: [3], Denominator: [1 0.25]

D. Numerator: [3], Denominator: [1 4]

Solution

1. Step 1: Write transfer function formula for first-order system

   Transfer function = gain / (time_constant * s + 1) = 3 / (4s + 1).

2. Step 2: Convert to Simulink coefficients

   Numerator [3], Denominator [1 4] for s + 4.

3. Final Answer:

   Numerator: [3], Denominator: [1 4] -> Option D

4. Quick Check:

   Denominator = [1 tau] for first-order [OK]

Hint: Denominator coefficients = [1 tau] [OK]

Common Mistakes:

• Swapping numerator and denominator
• Using time constant as denominator constant
• Not normalizing denominator leading coefficient to 1