Difficult Understanding Magnitude and Phase Shift of Transfer Function

  • #1
wellmoisturizedfrog
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1
TL;DR Summary
I am unsure if my current understanding of transfer functions is correct.
Hello,

My textbook offers the following transfer function as an example.

1701556509480.png


It then goes on to explain that the following equations represent the magnitude and phase shift of the transfer function.

1701556549125.png


However, I am having some difficulty jumping from the first equation to these equations. From my understanding, in order to find the magnitude of the transfer function, the magnitude of the complex number in the denominator is found. I'm not sure if this logic is correct.

I am also unsure about how the equation for the phase shift of the transfer equation has a negative sign in front. I understand the other aspects of it, though.

I would appreciate any clarifications.
 
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  • #2
Plotting the complex numbers graphically may help you understand why the denomiator is that way.

Multiply numerator and denominator by complex conjugate of denominator should help understand the angle.
 
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  • #3
scottdave said:
Multiply numerator and denominator by complex conjugate of denominator should help understand the angle.
And the magnitude as well......this is the standard way to manipulate complex numbers.
 
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  • #5
Ah I see, thank you for the insight! I appreciate the insight and resources.
 
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FAQ: Difficult Understanding Magnitude and Phase Shift of Transfer Function

What is a transfer function?

A transfer function is a mathematical representation that describes the relationship between the input and output of a linear time-invariant (LTI) system in the frequency domain. It is typically expressed as a ratio of two polynomials, where the numerator represents the output and the denominator represents the input. Transfer functions are used to analyze the behavior of systems in terms of magnitude and phase shift at different frequencies.

Why is it difficult to understand magnitude and phase shift in transfer functions?

Understanding magnitude and phase shift in transfer functions can be challenging due to the complex nature of the mathematical representations involved. The frequency response of a system is often not intuitive, requiring a grasp of concepts like complex numbers and logarithmic scales for magnitude. Additionally, the phase shift can vary non-linearly with frequency, making it difficult to predict system behavior without thorough analysis or simulation.

How do magnitude and phase shift relate to system stability?

Magnitude and phase shift are critical factors in determining the stability of a system. A system is considered stable if the output remains bounded for a bounded input. The Nyquist stability criterion relates the open-loop gain and phase shift to closed-loop stability. If the magnitude of the transfer function approaches unity (0 dB) while the phase shift approaches -180 degrees, the system may become unstable. Understanding these relationships is essential for designing stable control systems.

What tools can be used to analyze magnitude and phase shift?

Several tools can be used to analyze magnitude and phase shift, including Bode plots, Nyquist plots, and root locus diagrams. Bode plots provide a graphical representation of the magnitude and phase shift as functions of frequency, making it easier to visualize system behavior. Nyquist plots help assess stability and gain margins, while root locus diagrams are useful for determining how the poles of the transfer function change with varying system parameters.

How can I improve my understanding of magnitude and phase shift in transfer functions?

Improving your understanding of magnitude and phase shift in transfer functions can be achieved through a combination of theoretical study and practical application. Start with foundational concepts in control theory and complex analysis. Utilize simulation tools like MATLAB or Python to visualize Bode and Nyquist plots. Engage in hands-on projects or experiments that involve system modeling and analysis to reinforce your knowledge and gain practical experience.

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