The Transfer Function of a Positive Feedback System

Oct 13, 2025 by magicalceylontours.com 20 views

Introduction

In the realm of control systems, stability is a crucial aspect to consider when designing a system. A stable system is one that can return to its equilibrium state after a disturbance or input. However, when it comes to positive feedback systems, the concept of stability becomes more complex. In this article, we will delve into the world of positive feedback systems and explore the possibility of achieving stability in such systems.

What is a Positive Feedback System?

A positive feedback system is a type of control system where the output is fed back to the input in a way that reinforces the original input. This means that the system's response to an input is amplified, rather than attenuated. Positive feedback systems are commonly used in applications such as oscillators, amplifiers, and control systems.

The Transfer Function of a Positive Feedback System

The transfer function of a positive feedback system is given by the equation:

\frac{H(s)}{1-G(s)H(s)}

where $H(s)$ is the transfer function of the feedback loop, and $G(s)$ is the transfer function of the system itself.

Approximating the Transfer Function

When the gain of the system, $G(s)H(s)$ , is much greater than 1, the transfer function can be approximated to be:

\frac{1}{-G(s)}

This approximation is valid when the system's gain is large, and the feedback loop is dominant.

Stability of Positive Feedback Systems

Now, let's discuss the stability of positive feedback systems. A system is considered stable if its output remains bounded for all time, regardless of the input. In other words, a stable system will not oscillate or grow indefinitely in response to an input.

The Problem with Positive Feedback Systems

The problem with positive feedback systems is that they can become unstable if the gain of the system is too high. When the gain is high, the feedback loop can become dominant, causing the system to oscillate or grow indefinitely. This is because the positive feedback loop reinforces the original input, causing the system's response to amplify over time.

Conditions for Stability

For a positive feedback system to be stable, the following conditions must be met:

The gain of the system, $G(s)H(s)$ , must be less than 1.
The phase margin of the system must be greater than 0.

Phase Margin

The phase margin is a measure of the stability of a system. It is defined as the difference between the phase angle of the system's transfer function and the phase angle of the system's gain. A positive phase margin indicates that the system is stable, while a negative phase margin indicates that the system is unstable.

Designing a Stable Positive Feedback System

To design a stable positive feedback system, the following steps can be taken:

Reduce the gain of the system, $G(s)H(s)$ , to a value less than 1.
Increase the phase margin of the system by adjusting the system's transfer function.
Use a feedback loop with a high gain, but with a phase shift that is opposite to the phase shift of the system's transfer function.

Conclusion

In conclusion, while positive feedback systems can be unstable, they can also be designed to be stable. By reducing the gain of the system, increasing the phase margin, and using a feedback loop with a high gain and opposite phase shift, a stable positive feedback system can be achieved. However, the design of such a system requires careful consideration of the system's transfer function and the feedback loop's gain and phase shift.

References

[1] Ogata, K. (2010). Modern Control Engineering. Prentice Hall.
[2] Franklin, G. F., Powell, J. D., & Emami-Naeini, A. (2014). Feedback Control of Dynamic Systems. Pearson Education.
[3] Dorf, R. C., & Bishop, R. H. (2011). Modern Control Systems. Pearson Education.

Appendix

A.1 Transfer Function of a Positive Feedback System

The transfer function of a positive feedback system is given by the equation:

\frac{H(s)}{1-G(s)H(s)}

where $H(s)$ is the transfer function of the feedback loop, and $G(s)$ is the transfer function of the system itself.

A.2 Approximating the Transfer Function

When the gain of the system, $G(s)H(s)$ , is much greater than 1, the transfer function can be approximated to be:

\frac{1}{-G(s)}

This approximation is valid when the system's gain is large, and the feedback loop is dominant.

A.3 Stability of Positive Feedback Systems

A system is considered stable if its output remains bounded for all time, regardless of the input. In other words, a stable system will not oscillate or grow indefinitely in response to an input.

A.4 Conditions for Stability

For a positive feedback system to be stable, the following conditions must be met:

The gain of the system, $G(s)H(s)$ , must be less than 1.
The phase margin of the system must be greater than 0.

A.5 Phase Margin

Q: What is a positive feedback system?

A: A positive feedback system is a type of control system where the output is fed back to the input in a way that reinforces the original input. This means that the system's response to an input is amplified, rather than attenuated.

Q: What is the transfer function of a positive feedback system?

A: The transfer function of a positive feedback system is given by the equation:

\frac{H(s)}{1-G(s)H(s)}

where $H(s)$ is the transfer function of the feedback loop, and $G(s)$ is the transfer function of the system itself.

Q: Can a positive feedback system be stable?

A: Yes, a positive feedback system can be stable if the gain of the system, $G(s)H(s)$ , is less than 1 and the phase margin of the system is greater than 0.

Q: What is the phase margin?

A: The phase margin is a measure of the stability of a system. It is defined as the difference between the phase angle of the system's transfer function and the phase angle of the system's gain. A positive phase margin indicates that the system is stable, while a negative phase margin indicates that the system is unstable.

Q: How can I design a stable positive feedback system?

A: To design a stable positive feedback system, you can follow these steps:

Reduce the gain of the system, $G(s)H(s)$ , to a value less than 1.
Increase the phase margin of the system by adjusting the system's transfer function.
Use a feedback loop with a high gain, but with a phase shift that is opposite to the phase shift of the system's transfer function.

Q: What are some common applications of positive feedback systems?

A: Positive feedback systems are commonly used in applications such as:

Oscillators
Amplifiers
Control systems
Medical devices

Q: What are some common challenges associated with positive feedback systems?

A: Some common challenges associated with positive feedback systems include:

Instability
Oscillations
Growth of the system's response over time

Q: How can I troubleshoot a positive feedback system that is not stable?

A: To troubleshoot a positive feedback system that is not stable, you can try the following:

Check the gain of the system, $G(s)H(s)$ , to ensure that it is less than 1.
Check the phase margin of the system to ensure that it is greater than 0.
Adjust the system's transfer function to increase the phase margin.
Use a feedback loop with a high gain, but with a phase shift that is opposite to the phase shift of the system's transfer function.

Q: What are some common mistakes to avoid when designing a positive feedback system?

A: Some common mistakes to avoid when designing a positive feedback system include:

Not reducing the gain of the system, $G(s)H(s)$ , to a value less than 1.
Not increasing the phase margin of the system.
Using a feedback loop with a low gain or a phase shift that is the same as the phase shift of the system's transfer function.

Q: What are some best practices for designing a positive feedback system?

A: Some best practices for designing a positive feedback system include:

Carefully selecting the system's transfer function to ensure that it is stable.
Using a feedback loop with a high gain, but with a phase shift that is opposite to the phase shift of the system's transfer function.
Regularly testing and verifying the stability of the system.

Conclusion

In conclusion, a positive feedback system can be stable if the gain of the system, $G(s)H(s)$ , is less than 1 and the phase margin of the system is greater than 0. By following the steps outlined in this article, you can design a stable positive feedback system that meets your needs. Remember to carefully select the system's transfer function, use a feedback loop with a high gain and opposite phase shift, and regularly test and verify the stability of the system.