Single Pole System Explained

Single Pole System

A single-pole system (or first-order system) is a linear time-invariant system whose transfer function contains exactly one pole in the denominator and no zeros in the numerator that can cancel it.

Mathematical Definition

There are two primary ways to express a first-order single-pole transfer function:

1. Time Constant Form (Standard Form):

H(s) = K / (1 + sτ)

Where K is the DC Gain and τ (tau) is the time constant.

2. Pole-Location Form:

H(s) = A / (s + α)

Where -α is the pole location. In this form, the DC Gain is actually A/α.

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What is a Pole?

A pole is the value of s (the complex frequency) that makes the denominator of the transfer function zero, causing the gain to approach infinity.

For the system: H(s) = 10 / (s + 5)

Set denominator to zero: s + 5 = 0

The pole is at: s = -5

For stability in a continuous system, this pole must be in the Left-Half Plane (negative real part).

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Frequency Response

To find the frequency response, we substitute s = jω.

1. Magnitude Response

|H(jω)| = K / √(1 + (ωτ)²)

• Low Frequencies (ω << 1/τ): Gain is approximately K.
• Cutoff Frequency (ω = 1/τ): Gain is K / √2 ≈ 0.707K (the -3 dB point).
• High Frequencies (ω >> 1/τ): Gain drops at a rate of -20 dB/decade (or -6 dB/octave).

2. Phase Response

∠H(jω) = -arctan(ωτ)

• At low frequencies, phase shift is 0°.
• At the cutoff frequency, phase shift is -45°.
• At very high frequencies, phase shift approaches a maximum of -90°.

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Time Domain Response

For a unit step input, the output y(t) of a single-pole system is:

y(t) = K(1 - e^-t/τ)

Key milestones:

• At t = τ: Output reaches 63.2% of its final value.
• At t = 5τ: Output reaches 99.3% (considered "steady state").

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Practical Example: RC Low-Pass Filter

Given R = 1 kΩ and C = 1 μF:

• Time Constant (τ): RC = (1000)(1e-6) = 1 ms.
• Cutoff Frequency (f_c): 1 / (2πRC) ≈ 159.15 Hz.
• Behavior: Signals significantly below 159 Hz pass through; signals significantly above 159 Hz are attenuated.

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Summary Characteristics

Property	Specification
System Order	First-Order
Total Number of Poles	One (Real)
Roll-off Rate	-20 dB/decade
Max Phase Lag	-90°
Energy Storage	One element (e.g., 1 Capacitor OR 1 Inductor)
Stability	Stable if the pole is negative (s < 0)

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Applications

• Smoothing: Removing high-frequency noise from a sensor signal.
• Op-Amps: Internal compensation to ensure stability (Dominant Pole Compensation).
• Modeling: Describing thermal mass or motor velocity ramp-ups.
• Filters: Simple crossover networks for audio.

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