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Phasor Analysis

A technique used in AC circuit analysis to represent sinusoidal signals (voltage and current) as rotating vectors (phasors) in the complex plane, simplifying differential equations to algebraic ones.
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The statement of the theorem

Let the circuit be described by a set of linear differential equations in the time domain, L(v(t),i(t),t)=0\mathcal{L}(\mathbf{v}(t), \mathbf{i}(t), t) = 0. For sinusoidal steady-state analysis, we assume v(t)=Re(Vejωt)\mathbf{v}(t) = \text{Re}(\mathbf{V} e^{j\omega t}) and i(t)=Re(Iejωt)\mathbf{i}(t) = \text{Re}(\mathbf{I} e^{j\omega t}), where V\mathbf{V} and I\mathbf{I} are complex phasors. The transformation maps the time-domain differential operator ddt\frac{d}{dt} to multiplication by jωj\omega. The complex impedance Z(jω)\mathbf{Z}(j\omega) of a component is defined as the ratio of the voltage phasor V\mathbf{V} to the current phasor I\mathbf{I} across it: Z(jω)=VI\mathbf{Z}(j\omega) = \frac{\mathbf{V}}{\mathbf{I}}. Specifically, for a series RLC branch, the impedance is given by: \nZ(jω)=R+jωL+1jωC \mathbf{Z}(j\omega) = R + j\omega L + \frac{1}{j\omega C} \nApplying Kirchhoff's Voltage Law (KVL) in the frequency domain yields the nodal admittance matrix Y(jω)\mathbf{Y}(j\omega) such that the phasor relationship between nodal voltages Vnodes\mathbf{V}_{nodes} and source currents Isources\mathbf{I}_{sources} is: \nNodal Analysis: diag(Y(jω))Vnodes=Isources \text{Nodal Analysis: } \text{diag}(\mathbf{Y}(j\omega)) \mathbf{V}_{nodes} = \text{I}_{sources}