G(s) = 50/[s((s^2)+10s +50)(s + 5)]
Transfer function
A transfer function (also known as system function or network function) of a system, subsystem, or component is a mathematical function that modifies the output of a system in each possible input. They are widely used in electronics and control systems.
Convolution Integral
Among all the electrical engineering students, this topic of convolution integral is very confusing. It is a mathematical operation of two functions f and g that produce another third type of function (f * g) , and this expresses how the shape of one is modified with the help of the other one. The process of computing it and the result function is known as convolution. After one is reversed and shifted, it is defined as the integral of the product of two functions. After producing the convolution function, the integral is evaluated for all the values of shift. The convolution integral has some similar features with the cross-correlation. The continuous or discrete variables for real-valued functions differ from cross-correlation (f * g) only by either of the two f(x) or g(x) is reflected about the y-axis or not. Therefore, it is a cross-correlation of f(x) and g(-x) or f(-x) and g(x), the cross-correlation operator is the adjoint of the operator of the convolution for complex-valued piecewise functions.
![Transcription of the image for an Educational website:
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### Transfer Function Representation
The given transfer function \( G(s) \) is expressed in the Laplace domain as follows:
\[ G(s) = \frac{50} {s \left( s^2 + 10s + 50 \right) \left( s + 5 \right) } \]
This expression can be used in control system analysis to determine the behavior of the system in response to various inputs. The transfer function \( G(s) \) includes:
1. A constant numerator value of 50.
2. The denominator is composed of multiple polynomial terms:
- \( s \), a first-order term.
- \( s^2 + 10s + 50 \), a second-order polynomial.
- \( s + 5 \), another first-order term.
Understanding the poles and zeros of this transfer function is crucial for analyzing the stability and response characteristics of the dynamic system it represents.
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