Why analyze frequency and phase?
A complex system can be probed one sine wave at a time.
First, picture it
Push a swing slowly and it follows; push near its natural rhythm and it moves more; push too quickly and it may barely respond. The amplitude ratio and timing lag depend on the input frequency.
What the model says
For a stable LTI system in sinusoidal steady state, input sin(ωt) produces an output at the same ω with magnitude |G(jω)| and phase ∠G(jω). A phase of −90° means a quarter-cycle lag; it is not a physical angle of the motor. Frequency analysis helps separate tracking of slow commands, rejection of disturbances, sensor-noise amplification and stability near crossover. It does not replace transient or nonlinear analysis.
G(jω)=|G(jω)| e^{jφ(ω)}Magnitude tells how much; phase tells how late, for each frequency ω.
For a first-order low-pass plant, high frequencies are attenuated and increasingly delayed. This is a shape sketch, not numerical data from the lab.
Make it concrete
A motor speed loop can track a slow target change yet fail to follow rapid oscillations. That can be desirable if rapid variations are measurement noise.
Phase is frequency dependent. A single 'delay angle' cannot describe an entire system.
Check your understanding+
Does −90° phase mean a shaft physically rotated backward by 90°?
No. It means the sinusoidal output lags the input by one quarter of that signal's period.