Start from the exponential series
\[e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!}\]
The power series is valid for real and complex arguments.
Known Taylor expansion.
FORMULA / 001
Euler's formula and phasors
Derives Euler's identity from Taylor series and maps rotating complex vectors to sinusoidal waveforms.
\[e^{j\theta}=\cos\theta+j\sin\theta\]
Complex exponentials turn phase, frequency, and sinusoidal steady state into geometry.
Problem Definition
Problem statement and variable definition
On an oscilloscope, two AC nodes can have the same amplitude but arrive with a phase shift; the waveform looks delayed even when the circuit is in steady state.
Core variable legend
| Symbol | Quantity | Unit | Meaning |
|---|---|---|---|
| $j$ | Imaginary unit | 無 | Rotates a vector by 90 degrees. |
| $\theta$ | Phase angle | rad | Position of the rotating vector. |
| $\omega$ | Angular frequency | rad/s | Rotation speed in the complex plane. |
| $f$ | Frequency | Hz | Cycles per second. |
Analytical Derivation
Model assumptions and analytical derivation
Model simplifications
| Item | Assumption |
|---|---|
| Assumption 1 | Signal is sinusoidal and steady state. |
| Assumption 2 | Amplitude is normalized unless stated otherwise. |
| Assumption 3 | Only the real projection is measured as the physical waveform. |
Substitute a phase argument
\[e^{j\theta}=1+j\theta-\frac{\theta^2}{2!}-j\frac{\theta^3}{3!}+\cdots\]
Powers of j cycle through 1, j, -1, and -j.
Use j^2=-1.
Group real and imaginary terms
\[e^{j\theta}=\left(1-\frac{\theta^2}{2!}+\cdots\right)+j\left(\theta-\frac{\theta^3}{3!}+\cdots\right)\]
The two groups match cosine and sine series.
Identify Taylor series.
$\theta\to0$$e^{j\theta}\approx1+j\theta$Small phase error is almost a linear imaginary perturbation.
$\theta=\pi$$e^{j\pi}=-1$A half-cycle rotation flips signal polarity.
Parametric Verification
Parametric simulation and numerical verification
Use the sliders to change the physical parameters and watch the formula expose its behavior. The metric panel keeps sample count and compute latency visible so the visualization remains an engineering instrument, not only a picture.
INTERACTIVE MODULE
Rotating vector and projected waveform
Engineering Application
Engineering application and implementation notes
Design rules
- Use phasors only after the transient has died out.
- Track the sign convention before mixing SPICE, RF, and control formulas.
- Convert degrees to radians before numerical computation.
Core algorithm
python
import numpy as np
def phasor(amplitude, phase_deg, t, f):
phi = np.deg2rad(phase_deg)
return np.real(amplitude * np.exp(1j * (2*np.pi*f*t + phi)))