Conversion between Forms

A phasor is a rotating vector that graphically represents a sinusoidal quantity (voltage or current). Its length represents the amplitude (typically the RMS value, though peak value can also be used, consistency is key). Its angle with respect to a reference axis (usually the positive real axis) represents the phase angle of the waveform at time t=0. Phasors are assumed to rotate counter-clockwise at the angular frequency ω. By "freezing" them at a specific time (usually t=0), we can represent the relative phase relationships between different quantities. Example: A voltage v(t)=Vm sin(ωt+ϕ) can be represented as a phasor V=(VRMS )∠ϕ, where VRMS =Vm /2.

Phasors are mathematically represented as complex numbers in the complex plane. A complex number Z can be expressed in: Rectangular Form: Z=x+jy, where x is the real part and y is the imaginary part. j is the imaginary unit, where j=−1. Polar Form: Z=∣Z∣∠θ, where ∣Z∣ is the magnitude (modulus) and θ is the angle (argument). Conversion between Forms: From Rectangular to Polar: ∣Z∣=x2+y2 θ=arctan(y/x) (paying attention to the quadrant of x and y) From Polar to Rectangular: x=∣Z∣cosθ y=∣Z∣sinθ.

In AC circuits, the total opposition to current flow is called impedance, denoted by Z. Impedance is a complex number that accounts for both energy dissipation (resistance) and energy storage (reactance). Ohm's Law for AC Circuits (Phasor Form): V=IZ, I=V/Z, Z=V/I. Here V and I are voltage and current phasors, and Z is the complex impedance. Impedance of a Resistor (ZR ): A resistor dissipates energy but does not store it. In a purely resistive circuit, voltage and current are always in phase. Formula: ZR =R∠0∘=R+j0 (Ohm's). The impedance is purely real. Impedance of an Inductor (ZL ): An inductor stores energy in its magnetic field. In a purely inductive circuit, the current lags the voltage by 90∘. Inductive Reactance (XL ): The opposition offered by an inductor to the change in current. Formula: XL =ωL=2πfL (Ohms). Complex Impedance: ZL =jXL =XL ∠90∘. The impedance is purely imaginary and positive. Impedance of a Capacitor (ZC ): A capacitor stores energy in its electric field. In a purely capacitive circuit, the current leads the voltage by 90∘. Capacitive Reactance (XC ): The opposition offered by a capacitor to the change in voltage. Formula: XC =1/(ωC)=1/(2πfC) (Ohms). Complex Impedance: ZC =−jXC =XC ∠−90∘. General Complex Impedance (Z): For a circuit containing a combination of R, L, and C, the total impedance is represented in rectangular form as: Z=R+j(XL −XC ). Where R is the net resistance and (XL −XC ) is the net reactance. In polar form: Z=∣Z∣∠θ. Magnitude of Impedance: ∣Z∣=R2+(XL −XC )2. Impedance Angle: θ=arctan((XL −XC )/R). This angle represents the phase difference between the total voltage across the impedance and the total current flowing through it. If θ is positive, the voltage leads the current (inductive circuit); if negative, the voltage lags the current (capacitive circuit).