Find Poles Calculator
This calculator helps you find the poles of a second-order system given by the denominator of its transfer function: as² + bs + c = 0. Enter the coefficients 'a', 'b', and 'c' to determine the system's poles.
System Coefficients
Results
Discriminant (Δ): –
Pole 1: –
Pole 2: –
For as² + bs + c = 0, the poles are given by (-b ± sqrt(b² – 4ac)) / 2a.
Poles on the s-Plane
The s-plane showing the real (Re) and imaginary (Im) axes, with poles marked as 'X'.
Pole Information
| Item | Value | Nature |
|---|---|---|
| Coefficient a | – | – |
| Coefficient b | – | – |
| Coefficient c | – | – |
| Discriminant | – | – |
| Pole 1 | – | – |
| Pole 2 | – | – |
Table summarizing the coefficients and calculated poles.
What is a Find Poles Calculator?
A find poles calculator is a tool used primarily in control systems engineering and signal processing to determine the poles of a system's transfer function. The transfer function describes the relationship between the input and output of a system in the frequency domain (or more specifically, the s-domain for continuous-time systems). The poles are the roots of the denominator polynomial of this transfer function.
This calculator specifically helps you find the poles for a second-order system, whose denominator is typically a quadratic equation of the form as² + bs + c = 0. The values of 's' that satisfy this equation are the system's poles.
Anyone studying or working with control systems, electronics, signal processing, or any field involving dynamic systems can use a find poles calculator. Understanding the location of the poles in the s-plane (complex plane) is crucial for analyzing system stability and response characteristics (like damping and oscillation).
A common misconception is that poles are just abstract mathematical concepts. In reality, the location of the poles directly dictates how a physical system will behave when subjected to an input – whether it will be stable, oscillate, or respond sluggishly.
Find Poles Calculator Formula and Mathematical Explanation
For a second-order system with a transfer function whose denominator polynomial is:
as² + bs + c = 0
We need to find the values of 's' that make this equation true. These values are the roots of the quadratic equation, which are the poles of the system. We use the quadratic formula:
s = [-b ± sqrt(b² - 4ac)] / 2a
The term inside the square root, Δ = b² - 4ac, is called the discriminant. The nature of the poles (real or complex) depends on the value of the discriminant:
- If Δ > 0: There are two distinct real poles.
- If Δ = 0: There is one real pole (or two repeated real poles).
- If Δ < 0: There are two complex conjugate poles.
When the poles are complex, they are expressed as s = σ ± jω, where σ = -b / 2a is the real part and ω = sqrt(4ac - b²) / 2a is the imaginary part.
Variables Table
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| a | Coefficient of s² | Depends on system | Non-zero real numbers |
| b | Coefficient of s | Depends on system | Real numbers |
| c | Constant term | Depends on system | Real numbers |
| Δ | Discriminant (b² – 4ac) | Depends on system | Real numbers |
| s | Complex frequency (Poles) | Radians/second or complex | Complex numbers |
| σ | Real part of pole (Damping factor) | Radians/second | Real numbers |
| ω | Imaginary part of pole (Natural frequency) | Radians/second | Real numbers |
Practical Examples (Real-World Use Cases)
Let's see how the find poles calculator works with some examples.
Example 1: Overdamped System
Consider a system with the denominator s² + 5s + 6 = 0.
Here, a=1, b=5, c=6.
Using the find poles calculator (or manually): Δ = 5² – 4*1*6 = 25 – 24 = 1. Since Δ > 0, we have two distinct real poles: s1 = (-5 + sqrt(1)) / 2 = (-5 + 1) / 2 = -2 s2 = (-5 – sqrt(1)) / 2 = (-5 – 1) / 2 = -3
The poles are at s = -2 and s = -3. Both are in the left-half of the s-plane, indicating a stable, overdamped system (no oscillations).
Example 2: Underdamped System
Consider a system with the denominator s² + 2s + 5 = 0.
Here, a=1, b=2, c=5.
Using the find poles calculator: Δ = 2² – 4*1*5 = 4 – 20 = -16. Since Δ < 0, we have complex conjugate poles: Real part: σ = -2 / (2*1) = -1 Imaginary part: ω = sqrt(16) / (2*1) = 4 / 2 = 2 The poles are at s = -1 + 2j and s = -1 - 2j.
The poles are in the left-half of the s-plane (real part is negative), indicating a stable, underdamped system that will oscillate before settling.
How to Use This Find Poles Calculator
Using the find poles calculator is straightforward:
- Enter Coefficient 'a': Input the value for 'a', the coefficient of s² in your denominator polynomial
as² + bs + c. Note that 'a' cannot be zero for a second-order system. - Enter Coefficient 'b': Input the value for 'b', the coefficient of s.
- Enter Coefficient 'c': Input the value for 'c', the constant term.
- Calculate: Click the "Calculate Poles" button or simply change any input value after the first calculation. The results will update automatically if you change inputs after the first click.
- Read Results: The calculator will display:
- The primary result: The calculated poles (s1 and s2).
- Intermediate values: The discriminant (Δ).
- The s-plane plot showing the location of the poles.
- A table summarizing the inputs and results.
- Interpret: If the real parts of the poles are negative, the system is stable. If they are positive, it's unstable. If they are zero (and no repeated poles on the jω axis), it's marginally stable. Complex poles indicate oscillatory behavior.
- Reset: Use the "Reset" button to go back to default values.
- Copy: Use "Copy Results" to copy the main findings.
Key Factors That Affect Find Poles Calculator Results
The location of the poles is entirely determined by the coefficients 'a', 'b', and 'c' of the system's characteristic equation (denominator polynomial).
- Coefficient 'a': Scales the equation. While it doesn't change the nature of the roots as much as 'b' and 'c' relative to 'a', it affects the magnitude of the poles. Often, 'a' is normalized to 1.
- Coefficient 'b' (Damping Term): This coefficient is closely related to the damping in the system. A larger positive 'b' (relative to 'a' and 'c') generally leads to more damping, moving poles further left (if real) or reducing the oscillatory part (if complex).
- Coefficient 'c' (Stiffness/Inertia Term): This relates to the natural frequency of the system. Larger 'c' values (relative to 'a') tend to increase the natural frequency or move poles further from the origin along the real axis or increase the imaginary part if complex.
- The Ratio b²/4ac: The relationship between b² and 4ac determines whether the discriminant is positive, zero, or negative, thus dictating whether the poles are real and distinct, real and repeated, or complex conjugate.
- Sign of 'b': A positive 'b' (assuming 'a' is positive) is usually associated with energy dissipation or damping, leading to stable poles if 'c' is also positive. A negative 'b' can lead to unstable poles (positive real part).
- Signs of 'a' and 'c': If 'a' and 'c' have opposite signs, the discriminant b²-4ac will be b²+4|ac|, which is always positive, leading to two real poles, one positive and one negative (unstable). For stability in typical physical systems, 'a', 'b', and 'c' are often positive.
The find poles calculator uses these coefficients directly to find the roots and thus the pole locations.
Frequently Asked Questions (FAQ)
- What are poles of a transfer function?
- Poles are the values of the complex variable 's' (from the Laplace transform) that make the denominator of the system's transfer function equal to zero. They are crucial in determining the system's behavior.
- Why is it important to find the poles?
- The location of the poles in the complex s-plane dictates the stability and transient response of the system. Poles in the left-half plane (LHP) correspond to stable behavior, while poles in the right-half plane (RHP) indicate instability.
- What is the s-plane?
- The s-plane is a complex plane where the horizontal axis represents the real part (σ) of the complex variable 's', and the vertical axis represents the imaginary part (jω). Poles and zeros of a system are plotted on this plane.
- How do poles relate to system stability?
- If all poles of a system lie strictly in the left-half of the s-plane (real parts are negative), the system is stable. If any pole is in the right-half plane (positive real part), or if there are repeated poles on the imaginary axis, the system is unstable. Poles on the imaginary axis (and not repeated) suggest marginal stability or sustained oscillations.
- What do complex poles mean?
- Complex conjugate poles (e.g., -a ± jb) indicate an oscillatory component in the system's response. The real part (-a) determines the damping (decay rate of oscillations), and the imaginary part (b) relates to the frequency of oscillations.
- What if the discriminant is zero?
- If the discriminant (b² – 4ac) is zero, there is one real pole of multiplicity two (a repeated real pole). The system is critically damped.
- Can I use this calculator for higher-order systems?
- This specific find poles calculator is designed for second-order systems (denominator is a quadratic). For higher-order systems, you would need to find the roots of a higher-degree polynomial, which generally requires numerical methods beyond the simple quadratic formula.
- What if coefficient 'a' is zero?
- If 'a' is zero, the equation
as² + bs + c = 0reduces tobs + c = 0, which is a first-order system with one pole at s = -c/b (if b is not zero). Our calculator requires 'a' to be non-zero for a second-order interpretation.