Question

Calculate the poles of the rational function $$ G(s)=\frac{s+5}{s^{2}+3 s+2} $$

Short answer

Answer: The poles of the given rational function are \(s_1 = -1\) and \(s_2 = -2\).

Step-by-step solution

Factor the denominator

In order to find the roots of the denominator, we need to factor the polynomial \(s^2 +3s + 2\). We will look for two binomials whose product is equal to the given polynomial. This requires finding two numbers that multiply to 2 and add to 3, which are 1 and 2. So we can factor the denominator as follows: $$ s^2 +3s + 2 = (s+1)(s+2) $$ Note that the numerator doesn't influence the poles, so it doesn't need to be factored.

Find the roots of the factored denominator.

Next, we'll find the values of \(s\) where the factored denominator is equal to 0, i.e., \((s+1)(s+2)=0\). To do this, set each factor in the polynomial equal to 0 and solve for s. * For the first factor: \(s+1=0\), we find \(s=-1\). * For the second factor: \(s+2=0\), we find \(s=-2\).

State the poles of the rational function

Since the roots of the denominator polynomial are -1 and -2, the poles of the given rational function \(G(s)\) are: $$ s_1 = -1 \quad \text{and} \quad s_2 = -2 $$ The poles represent the values of \(s\) that make the rational function infinite. These are crucial in control systems and signal processing applications.

Key concepts

Factorization

Factorization is a process where a mathematical expression is decomposed into a product of simpler factors. In the context of polynomials, factorization involves breaking down a polynomial equation into its simplest components, which are usually binomials or other simpler polynomials. This makes it easier to find the polynomial’s roots. For instance, in the exercise provided, the polynomial denominator \(s^2 + 3s + 2\) was factored into \( (s+1)(s+2) \).

• Factorization helps in problem-solving by simplifying complex polynomials.
• By factoring, you prepare the ground for finding roots, which are vital in evaluating mathematical expressions.
• In the realm of rational functions, only the denominator needs factoring to find poles.

Whenever you are faced with a polynomial, consider breaking it down to its simplest factors. Identify pairs of numbers that multiply to the coefficient of the last term while adding to the middle coefficient. This practice enhances understanding and helps find solutions efficiently.

Roots of Polynomials

The roots of a polynomial are the values of the variable that make the polynomial equal to zero. These roots play a significant role in various applications across mathematics and engineering. In this specific context, we were interested in finding the roots of the polynomial \(s^2 + 3s + 2\), which involved first factoring it into \((s+1)(s+2)\). Once factored, finding the roots becomes straightforward. Set each factor to zero and solve for the variable:

• For \(s+1=0\), the root is \(s=-1\).
• For \(s+2=0\), the root is \(s=-2\).

These roots are critical in identifying the poles of the rational function. Hence, understanding how to calculate polynomial roots is beneficial, especially in fields like calculus and algebra where these calculations form the basis for more complex analyses.

Control Systems

Control systems are integral to engineering and technology as they manage and regulate the behavior of dynamic systems. Poles of rational functions are essential in control systems because they determine system stability. The location of poles can tell us whether a control system will perform efficiently or encounter issues like instability or unwanted oscillations. In control systems:

• Poles correspond to eigenvalues of the system matrix, providing insights into system response and behavior over time.
• Stable systems typically have poles with negative real parts, as they ensure that signals decay over time.
• Pole placement is a technique used in system design to achieve desired specifications for stability and performance.

Understanding the concept of poles ensures that engineers can design systems that meet safety and performance requirements. Mastering this concept empowers you to predict and manage the output behavior of control systems under various conditions.

Signal Processing

Signal processing involves manipulating signals to improve their quality, extract information, or transform them into a desired format. Poles of rational functions play a crucial role here as they affect the behavior and characteristics of filters and other signal processing systems. In signal processing:

• Poles help determine the frequency response of systems, influencing how different frequencies are amplified or attenuated.
• They affect the stability and causality of filters, crucial for ensuring that filters perform reliably in real-time applications.
• Location and distribution of poles can define the type of response (e.g., Low-pass, High-pass) of the system.

By understanding pole placement, you can design signal processing systems that meet specific performance criteria and operate optimally under anticipated signal conditions. This knowledge ensures effective signal enhancement or transformation, pivotal in modern communication, audio processing, and more.