# Block diagrams to transfer functions

• Dustinsfl
In summary, the conversation discusses writing a block diagram as a transfer function, specifically looking at the closed-loop transfer function and how it is calculated. It is clarified that the transfer function in the forward path appears in both the numerator and denominator, while the transfer function in the feedback path does not. This explains why the book's formula may differ from the calculated one.
Dustinsfl

## Homework Statement

I am trying to write a block diagram as a transfer function.

## The Attempt at a Solution

Let ##G = \frac{1}{s}##, ##H = \frac{K}{Js + a}##, and ##L= K_f##. Then wouldn't the closed loop transfer function be written as
$$\frac{\frac{HLG}{1+HL}}{1 + \frac{HLG}{1+HL}} = \frac{KK_f}{s^2J + (KK_f + a)s + KK_f}$$
I only ask because the book has it as
$$\frac{K}{s^2J + (KK_f + a)s + K}$$
and I don't see how that was obtained.

Last edited:
The transfer function for your inner feedback loop is $\frac{H(s)}{1 + H(s) L(s)}$, not $\frac{H(s) L(s)}{1 + H(s) L(s)}$.

I think that should sort it out.

1 person
milesyoung said:
The transfer function for your inner feedback loop is $\frac{H(s)}{1 + H(s) L(s)}$, not $\frac{H(s) L(s)}{1 + H(s) L(s)}$.

I think that should sort it out.

Can you explain why that is?

Dustinsfl said:
Can you explain why that is?
It's an easy mistake to make. The transfer function in the forward path appears in both the numerator and denominator of the closed-loop transfer function but the transfer function in the feedback path does not:
http://en.wikipedia.org/wiki/Closed-loop_transfer_function

I appreciate your attempt at solving the problem and your use of block diagrams and transfer functions to represent a system. However, it is important to note that the closed loop transfer function can be written in different forms, as long as they are mathematically equivalent. In this case, your solution and the one provided by the book are both correct representations of the closed loop transfer function. It is possible that the author of the book chose to simplify the transfer function by factoring out the K term, which does not change the overall behavior of the system. It is also possible that they used a different method to derive the transfer function. Therefore, it is important to understand the concept and principles behind block diagrams and transfer functions, rather than just focusing on the specific form of the equation. Keep up the good work!

## 1. What is a block diagram?

A block diagram is a graphical representation of a system or process that uses blocks to represent its components and their interactions. It is commonly used in engineering and science to visualize complex systems and their functions.

## 2. What is a transfer function?

A transfer function is a mathematical representation of the relationship between the input and output of a system. It describes how the system responds to different inputs and is typically represented as a ratio of the output to the input.

## 3. How are block diagrams used to determine transfer functions?

Block diagrams are used to represent the different components and their interactions in a system. By analyzing the connections between the blocks and their transfer functions, the overall transfer function of the system can be determined.

## 4. What is the purpose of converting block diagrams to transfer functions?

Converting block diagrams to transfer functions allows for a more simplified representation of a system. It also allows for easier analysis and understanding of the system's behavior and response to different inputs.

## 5. Are there any limitations to using block diagrams to transfer functions?

While block diagrams can be useful in visualizing and understanding complex systems, they may not accurately represent all aspects of the system. They also do not take into account external factors or non-linear relationships, which can limit their accuracy in certain situations.

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