mathematics//complex number

A complex number is a number of the form \(\sigma+j\omega\), a real part plus a real multiple of the imaginary unit \(j\) with \(j^2=-1\), and in engineering it is the compact way to write anything that rotates or oscillates: one complex number carries both how fast an oscillation grows or decays and how fast it turns. Engineers write \(j\) where mathematicians write \(i\), because \(i\) is already the current. Geometrically it is a point in a plane, so it has a magnitude (its distance from the origin) and an angle (its phase), and multiplying two of them multiplies the magnitudes and adds the angles.

A complex number is a number of the form σ+jω\sigma+j\omegaσ+jω, a real part plus a real multiple of the imaginary unit jjj with j2=−1j^2=-1j2=−1, and in engineering it is the compact way to write anything that rotates or oscillates: one complex number carries both how fast an oscillation grows or decays and how fast it turns. Engineers write jjj where mathematicians write iii, because iii is already the current. Geometrically it is a point in a plane, so it has a magnitude (its distance from the origin) and an angle (its phase), and multiplying two of them multiplies the magnitudes and adds the angles.

The reason it appears all over control is one identity, Euler's formula, applied to the exponential of a complex rate λ=σ+jω\lambda=\sigma+j\omegaλ=σ+jω:

eλt=eσt (cos⁡ωt+jsin⁡ωt).e^{\lambda t}=e^{\sigma t}\,(\cos\omega t+j\sin\omega t).eλt=eσt(cosωt+jsinωt).

The real part σ\sigmaσ is an envelope, decaying if negative and exploding if positive; the imaginary part ω\omegaω is an oscillation at ω\omegaω rad/s. A linear system's modes evolve as eλte^{\lambda t}eλt with λ\lambdaλ an eigenvalue, so reading an eigenvalue is reading a behaviour. A mass on a spring with λ=−0.2±1.99j\lambda=-0.2\pm1.99jλ=−0.2±1.99j oscillates at about 2 rad/s (0.32 Hz) and dies out with a time constant of 1/0.2=51/0.2=51/0.2=5 s.

Physical systems produce complex eigenvalues in conjugate pairs, σ±jω\sigma\pm j\omegaσ±jω, and the imaginary parts cancel in every real signal. The pair is two numbers standing for one real, damped sinusoid; a complex eigenvalue alone, with no partner, means the matrix itself has complex entries.

The complex plane is where stability is drawn. In continuous time a mode is stable when σ<0\sigma<0σ<0, the left half-plane; sampled every Δt\Delta tΔt, the same mode becomes μ=eλΔt\mu=e^{\lambda\Delta t}μ=eλΔt and stability means ∣μ∣<1|\mu|<1∣μ∣<1, the inside of the unit circle (discretization, pole). A numerical integrator can move a mode out of that region: the Euler method multiplies an undamped oscillation by 1+jωΔt1+j\omega\Delta t1+jωΔt, whose magnitude always exceeds one.

A sinusoid through a linear system comes out as the same sinusoid scaled and shifted, and the complex number that does it at each frequency is the system's frequency response (transfer function, Bode plot): its magnitude is the gain and its angle the phase lag. The Fourier analysis of a vibration signal returns one complex number per frequency for the same reason.

Nothing imaginary is measured. A sensor reads real numbers; the complex form is bookkeeping that turns differential equations with sines and cosines into algebra with exponentials, and the Laplace transform extends that bookkeeping to every linear ODE.