Transfer function block in Simulink - Step-by-Step Execution

Concept Flow - Transfer function block

Input Signal

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Transfer Function Block

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Apply Transfer Function: Output = (Numerator/Denominator) * Input

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Output Signal

The input signal passes through the transfer function block, which applies the mathematical transfer function defined by numerator and denominator polynomials to produce the output signal.

Execution Sample

Simulink

Input = 1.0
Numerator = [2]
Denominator = [1, 3]
Output = tfblock(Input, Numerator, Denominator)

This simulates a transfer function block with numerator 2 and denominator s+3, processing an input of 1.0 to produce the output.

Execution Table

Step	Input	Numerator	Denominator	Calculation	Output
1	1.0	[2]	[1, 3]	Calculate Output = 2 / (s + 3) * Input (steady state s=0)	2 / 3 * 1.0 = 0.6667
2	1.0	[2]	[1, 3]	Output signal produced	0.6667
Exit	-	-	-	Transfer function applied, output ready	-

💡 Transfer function applied to input, output computed as 0.6667 at steady state (s=0).

Variable Tracker

Variable	Start	After Step 1	After Step 2	Final
Input	1.0	1.0	1.0	1.0
Numerator	[2]	[2]	[2]	[2]
Denominator	[1, 3]	[1, 3]	[1, 3]	[1, 3]
Output	undefined	0.6667	0.6667	0.6667

Key Moments - 2 Insights

Why do we use s=0 in the calculation step for the output?

Because the transfer function is evaluated at steady state (s=0) to find the output value for a constant input, as shown in execution_table step 1.

What does the numerator and denominator represent in the transfer function block?

They represent the coefficients of the polynomials in the Laplace variable s, defining the system's behavior, as seen in the variable_tracker and execution_table.

Visual Quiz - 3 Questions

Look at the execution_table at step 1, what is the calculated output value?

A 3.0

B 2.0

C 0.6667

D 1.5

💡 Hint

Check the 'Output' column in execution_table row for step 1.

According to variable_tracker, what is the value of Output after step 2?

A undefined

B 0.6667

C 1.0

D 2.0

💡 Hint

Look at the 'Output' row under 'After Step 2' in variable_tracker.

If the numerator changes to [4], how would the output at step 1 change?

A Output would be 1.3333

B Output would be 0.3333

C Output would remain 0.6667

D Output would be 2.0

💡 Hint

Output = Numerator / Denominator * Input; doubling numerator doubles output (see execution_table step 1).

Concept Snapshot

Transfer function block syntax:
Output(s) = Numerator(s) / Denominator(s) * Input(s)
Numerator and Denominator are polynomial coefficients.
Evaluated in Laplace domain; steady state uses s=0.
Output signal is the system response to input signal.

Full Transcript

The transfer function block takes an input signal and applies a mathematical function defined by numerator and denominator polynomials in the Laplace variable s. The output is calculated by dividing the numerator polynomial by the denominator polynomial and multiplying by the input. For steady state analysis, s is set to zero. This process transforms the input signal into the output signal representing the system's response. Variables like Input, Numerator, Denominator, and Output change as the block processes the signal. Understanding these steps helps in modeling and simulating dynamic systems in Simulink.

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