Phasors, Complex Impedance & Admittance

A phasor discards the repeated carrier $e^{j\omega t}$ and keeps only a sinusoid's complex amplitude. For RMS convention,

$$x(t)=\sqrt2\,\Re\{\underline X e^{j\omega t}\},\qquad \underline X=X_{rms}e^{j\phi}.$$

All phasors in one equation must refer to the same frequency and convention.

Why differentiation disappears #

If $x(t)$ is represented by $\underline X$, then

$$\frac{dx}{dt}\leftrightarrow j\omega\underline X.$$

The RLC element laws become $\underline V=Z\underline I$ with

Resistance is the real part of impedance; reactance is the imaginary part. Conductance and susceptance are the corresponding parts of admittance.

Rectangular and polar forms #

Use rectangular form $Z=R+jX$ for addition and matrix equations. Use polar form $Z=|Z|\angle\theta$ for multiplication, division, and interpreting magnitude/phase:

$$|Z|=\sqrt{R^2+X^2},\qquad \theta=\mathop{\mathrm{atan2}}(X,R).$$

The atan2 form is essential because an ordinary arctangent loses the quadrant.

Frequency belongs to the model #

$Z_L$ grows with frequency and $|Z_C|$ falls. A phasor solution is therefore not a time-domain circuit solution in compressed notation; it is the steady-state response at one specified $\omega$.

Once every element is replaced by its impedance or admittance, KCL, KVL, nodal analysis, mesh analysis, and Thevenin/Norton reduction work exactly as in a DC linear circuit, but with complex coefficients.

Always return to the declared time convention. Under $e^{j\omega t}$, inductive impedance is $+j\omega L$; adopting $e^{-j\omega t}$ flips every reactive sign consistently.