Using Magnitude Condition to Determine System Gain (K) - Explained with Example

Suppose we have a system with an input signal x(t) and an output signal y(t), connected by a transfer function H(s) given by:

H(s) = K / (s + 1)

where K is the gain of the system.

To determine the value of K, we can use the magnitude condition, which states that the magnitude of the transfer function at a specific frequency ω is equal to the ratio of the output amplitude to the input amplitude at that frequency. Mathematically, this can be expressed as:

|H(jω)| = |y(jω)| / |x(jω)|

where |H(jω)| is the magnitude of the transfer function at frequency ω, and |y(jω)| and |x(jω)| are the magnitudes of the output and input signals, respectively, at frequency ω.

Suppose we have an input signal x(t) = sin(ωt), and we measure the output signal y(t) to be y(t) = A sin(ωt + φ), where A is the amplitude of the output signal and φ is the phase shift. Then, we can calculate the magnitude of the transfer function at frequency ω as:

|H(jω)| = |y(jω)| / |x(jω)| = A / 1 = A

Now, we can use the expression for the transfer function H(s) to find the magnitude of H(jω) as:

|H(jω)| = |K / (jω + 1)| = |K| / |jω + 1|

Substituting |H(jω)| = A and simplifying, we get:

A = |K| / √(ω^2 + 1)

Squaring both sides and rearranging, we get:

K = A √(ω^2 + 1)

So, by measuring the amplitude of the output signal and the frequency of the input signal, we can use the magnitude condition to determine the value of the gain K of the system.