Transfer function block in Simulink - Practice Problems & Coding Challenges

Challenge - 5 Problems

🧠 Conceptual

intermediate

Understanding the Transfer Function Block Parameters

In Simulink, the Transfer Function block requires numerator and denominator coefficients. What does the denominator represent in the block?

A The initial conditions of the system states.

B The time delay of the system response.

C The coefficients of the polynomial in the denominator of the transfer function's Laplace variable s.

D The gain factor applied after the transfer function output.

💡 Hint

Think about how transfer functions are written in Laplace form as ratios of polynomials.

The denominator coefficients define the polynomial in s that appears in the denominator of the transfer function, which determines system poles and stability.

❓ Predict Output

intermediate

Output of a Transfer Function Block Simulation

Given a Transfer Function block with numerator [2] and denominator [1, 3, 2], and a step input applied at time 0, what is the expected steady-state output value?

Simulink

Numerator = [2];
Denominator = [1, 3, 2];
Input = step function (unit step at t=0);
Simulate system response;

A 2.0

B 1.0

C 0.5

D 0.0

💡 Hint

Use the Final Value Theorem: steady-state output = limit as s→0 of s * G(s) * 1/s.

The transfer function is G(s) = 2 / (s^2 + 3s + 2). At s=0, G(0) = 2/2 = 1. The step input has Laplace 1/s, so steady-state output is G(0)*1 = 1. The option B (0.5) is incorrect. So correct answer is A.

❓ data_output

advanced

Simulink Transfer Function Block Step Response Data

You simulate a Transfer Function block with numerator [1] and denominator [1, 2, 1] using a step input. Which of the following arrays best represents the output data at times [0, 1, 2, 3] seconds?

A [0, 0.6, 0.85, 0.95]

B [0, 0.5, 0.75, 0.9]

C [0, 0.2, 0.5, 0.7]

D [0, 0.264, 0.594, 0.801]

💡 Hint

The transfer function corresponds to a second order system with repeated poles. Use the known step response formula for (s+1)^2 denominator.

The step response of 1/(s+1)^2 is y(t) = 1 - (1 + t) * exp(-t). Evaluating at t=1,2,3 gives approximately 0.264, 0.594, 0.801 respectively.

🔧 Debug

advanced

Identifying Error in Transfer Function Block Setup

Which option describes the error when setting numerator = [1, 2] and denominator = [1, 0] in a Transfer Function block?

A The denominator polynomial has a zero constant term, causing a division by zero at s=0.

B The numerator coefficients must be integers only.

C The denominator polynomial is not monic (leading coefficient not 1).

D The numerator and denominator vectors have different lengths, which is not allowed.

💡 Hint

Check the denominator polynomial for zero constant term and its effect on system behavior.

A zero constant term in the denominator means the system has a pole at s=0, which can cause instability or division by zero in steady-state calculations.

🚀 Application

expert

Designing a Transfer Function Block for Desired Step Response

You want a Transfer Function block in Simulink that produces a step response with a time constant of 2 seconds and a steady-state gain of 3. Which numerator and denominator coefficients should you use?

A Numerator: [3], Denominator: [2, 1]

B Numerator: [1], Denominator: [3, 2]

C Numerator: [6], Denominator: [2, 1]

D Numerator: [3], Denominator: [1, 2]

💡 Hint

Recall that a first order transfer function is G(s) = K / (τs + 1), where τ is time constant and K is gain.

To get time constant τ=2 and gain K=3, denominator is [2,1] (2s + 1) and numerator is [3].

Practice

1. What does the Transfer Function block in Simulink represent?

easy

A. A mathematical model using numerator and denominator polynomials

B. A block that generates random signals

C. A tool for drawing system diagrams

D. A block that stores data values

Solution

1. Step 1: Understand the Transfer Function block purpose and compare options

   The Transfer Function block models system behavior mathematically using polynomials. Only a mathematical model using numerator and denominator polynomials correctly describes the block.

2. Final Answer:

   A mathematical model using numerator and denominator polynomials -> Option A

3. Quick Check:

   Transfer Function = numerator/denominator polynomials [OK]

Hint: Remember: Transfer function = output/input ratio [OK]

Common Mistakes:

• Confusing with signal generator blocks
• Thinking it stores data instead of modeling
• Assuming it draws diagrams

2. Which of the following is the correct way to enter a transfer function with numerator [1 3] and denominator [1 5 6] in the Simulink Transfer Function block?

easy

A. Numerator: [1 3], Denominator: [1 5 6]

B. Numerator: 1/3, Denominator: 1/5/6

C. Numerator: (1,3), Denominator: (1,5,6)

D. Numerator: {1 3}, Denominator: {1 5 6}

Solution

1. Step 1: Recall Simulink syntax for polynomials and match options

   Simulink requires vectors in square brackets for numerator and denominator coefficients. Only Numerator: [1 3], Denominator: [1 5 6] uses square brackets and space-separated coefficients correctly.

2. Final Answer:

   Numerator: [1 3], Denominator: [1 5 6] -> Option A

3. Quick Check:

   Use square brackets for polynomials [OK]

Hint: Use square brackets for coefficient vectors [OK]

Common Mistakes:

• Using slashes or commas instead of spaces
• Using parentheses or braces instead of brackets
• Entering fractions instead of vectors

3. Given a Transfer Function block with numerator [2] and denominator [1 4 4], what is the expected system type?

medium

A. First-order system

B. Third-order system

C. Second-order system

D. Zero-order system

Solution

1. Step 1: Determine system order from denominator degree and match to options

   The denominator polynomial [1 4 4] has 3 coefficients, so degree is 2 (order = degree of denominator). A second-order system corresponds to denominator degree 2.

2. Final Answer:

   Second-order system -> Option C

3. Quick Check:

   Order = degree of denominator polynomial = 2 [OK]

Hint: System order = degree of denominator polynomial [OK]

Common Mistakes:

• Counting number of coefficients instead of degree
• Confusing numerator degree with system order
• Assuming order from numerator length

4. You entered numerator [1 0] and denominator [1 -2 1] in a Transfer Function block, but the simulation shows unexpected oscillations. What is the likely cause?

medium

A. Numerator should have more coefficients

B. Simulation time is too short

C. Transfer Function block does not support negative coefficients

D. Incorrect sign in denominator coefficients causing instability

Solution

1. Step 1: Analyze denominator polynomial for stability and identify cause

   The denominator [1 -2 1] corresponds to (s-1)^2, which has poles at s=1 (right half-plane), causing instability. Incorrect sign in denominator coefficients leads to this.

2. Final Answer:

   Incorrect sign in denominator coefficients causing instability -> Option D

3. Quick Check:

   Poles in right half-plane cause oscillations [OK]

Hint: Check denominator signs for stability [OK]

Common Mistakes:

• Assuming numerator length causes oscillations
• Believing negative coefficients are unsupported
• Ignoring pole locations

5. You want to model a system with transfer function \( \frac{5s + 10}{s^2 + 3s + 2} \) in Simulink. How should you enter the numerator and denominator coefficients in the Transfer Function block?

hard

A. Numerator: [10 5], Denominator: [1 3 2]

B. Numerator: [5 10], Denominator: [1 3 2]

C. Numerator: [5 10], Denominator: [1 2 3]

D. Numerator: [5 10], Denominator: [3 2 1]

Solution

1. Step 1: Identify numerator and denominator coefficients in descending powers of s

   Numerator is 5s + 10, so [5 10]; denominator s^2 + 3s + 2, so [1 3 2].

2. Final Answer:

   Numerator: [5 10], Denominator: [1 3 2] -> Option B

3. Quick Check:

   Coefficients match polynomial terms in order [OK]

Hint: List coefficients from highest to lowest power [OK]

Common Mistakes:

• Reversing coefficient order
• Mixing up numerator and denominator
• Using incorrect coefficient values