Transfer Function

The transfer function is the foundational mathematical tool used to describe the input-output relationship of a linear time-invariant (LTI) system in the frequency domain. Every control system analysis begins with deriving or identifying the transfer function, making it one of the most important concepts for GATE and university examinations.

Core Concept Explanation

A transfer function is defined as the ratio of the Laplace transform of the output to the Laplace transform of the input, with all initial conditions set to zero. Mathematically, if R(s) is the input and C(s) is the output, then G(s) = C(s) / R(s). The assumption of zero initial conditions is critical because it isolates the system's inherent dynamic behavior from any stored energy at the start.

The transfer function is always expressed as a rational polynomial in s, meaning it takes the form G(s) = N(s) / D(s), where N(s) is the numerator polynomial and D(s) is the denominator polynomial. The variable s is the complex frequency variable from the Laplace transform, defined as s = σ + jω. This representation converts differential equations in time domain into algebraic equations in the s-domain, making analysis far more tractable.

Transfer functions are only valid for linear time-invariant (LTI) systems. Nonlinear systems do not have a single transfer function because their behavior depends on the operating point. For linearized models around an operating point, a transfer function can be approximated and used for small-signal analysis.

Mathematical Expression

The general form of a transfer function is G(s) = (b_m s^m + b_{m-1} s^{m-1} + ... + b_0) / (a_n s^n + a_{n-1} s^{n-1} + ... + a_0). The order of the system is determined by the highest power of s in the denominator, which is n. In physical systems, n is always greater than or equal to m, making G(s) a proper rational function.

The roots of the denominator D(s) = 0 are called poles of the system, and the roots of N(s) = 0 are called zeros. Poles determine the natural response modes of the system. If any pole has a positive real part, the system is unstable. Zeros affect the amplitude and phase of the frequency response but do not directly determine stability.

The DC gain of a system is found by substituting s = 0 into G(s), giving G(0) = b_0 / a_0. This value tells how much the system amplifies or attenuates a steady-state (DC) input signal. For systems with a pole at origin, G(0) becomes infinite, indicating integrating behavior.

Practical Understanding

To derive a transfer function from a physical system, the differential equation governing the system is first written. Then the Laplace transform is applied to both sides with zero initial conditions. The ratio of output transform to input transform directly gives G(s). For electrical circuits, impedance methods using s-domain representations (sL for inductor, 1/sC for capacitor) are used.

Consider a simple RC circuit with input voltage V_in(s) and output across the capacitor V_out(s). Using voltage divider in s-domain: V_out(s) / V_in(s) = (1/sC) / (R + 1/sC) = 1 / (1 + sRC). This is a first-order transfer function with a single pole at s = -1/RC, confirming stable behavior since the pole is in the left half of the s-plane.

Given:
R = 10 kΩ = 10,000 Ω
C = 10 µF = 10 × 10⁻⁶ F
Input: Step voltage of 5V

Why this formula applies:
For an RC low-pass filter, voltage divider in s-domain gives the transfer function directly.

Formula:
G(s) = 1 / (1 + sRC)

Substitution:
RC = 10,000 × 10 × 10⁻⁶ = 0.1 s
G(s) = 1 / (1 + 0.1s)

Calculation:
Pole location: s = -1/0.1 = -10 rad/s
Time constant τ = RC = 0.1 s
DC Gain = G(0) = 1 / (1 + 0) = 1

Final Answer:
G(s) = 1 / (1 + 0.1s), pole at s = -10 rad/s, system is stable.

Exam Tip: GATE frequently asks to identify poles and zeros from G(s) and determine system order. Remember: order = highest power of s in denominator. If G(s) = K/[s(s+a)], this is a second-order system with poles at s=0 and s=-a, and no finite zeros.

Mechanism: Properties and Characteristics

• Transfer function is uniquely defined for LTI systems and is independent of the input signal type.
• The characteristic equation is obtained by setting the denominator of G(s) to zero: D(s) = 0. Its roots are the system poles.
• A system is BIBO stable if and only if all poles of G(s) lie strictly in the left half of the s-plane (negative real parts).
• The degree of D(s) gives the order of the system. Higher order means more complex dynamic behavior.
• For a physical system, degree of N(s) cannot exceed degree of D(s). Such systems are called proper systems.
• Gain K in G(s) = K / (s+a) scales the magnitude of response without changing pole or zero locations.

Quick Revision

• Transfer function G(s) = C(s)/R(s), defined with all initial conditions zero.
• G(s) = N(s)/D(s), a rational polynomial in s.
• System order = highest power of s in D(s).
• Poles = roots of D(s) = 0. Zeros = roots of N(s) = 0.
• Stability condition: all poles must have negative real parts (left half s-plane).
• DC gain = G(s) evaluated at s = 0.
• GATE trap: Transfer function does NOT apply to nonlinear systems. It is only valid for LTI systems.