pházor

A pházor is a mathematical tool used in electrical engineering and physics to represent sinusoidal signals, such as alternating currents and voltages. It simplifies the analysis of circuits containing resistors, inductors, and capacitors by converting differential equations into algebraic equations.

Essentially, a pházor is a complex number that encapsulates both the amplitude and phase of a sinusoidal waveform. The magnitude of the complex number represents the amplitude (often RMS value) of the sinusoid, while its angle represents the phase shift relative to a reference sinusoid. The time-varying part of the sinusoid is implicitly understood and is not explicitly carried in the pházor representation itself.

To convert a time-domain sinusoidal signal, like $v(t) = V_m \cos(\omega t + \phi)$, into a pházor, one typically uses the relationship where the pházor $V$ is given by $\frac{V_m}{\sqrt{2}} e^{j\phi}$. Here, $V_m$ is the peak amplitude, $\omega$ is the angular frequency, $\phi$ is the phase angle, and $j$ is the imaginary unit. The factor of $\sqrt{2}$ is used when dealing with RMS values.

In circuit analysis, components are represented by their pházor impedances. For a resistor, the impedance is simply R. For an inductor, it is $j\omega L$, and for a capacitor, it is $\frac{1}{j\omega C}$ or $-j\frac{1}{\omega C}$. Kirchhoff's laws, when applied in the pházor domain, become simple algebraic equations, making calculations much easier than solving differential equations in the time domain. Once the pházor solution is obtained, it can be converted back to the time domain to find the time-dependent response of the circuit.