phasor representation in rectangular and polar forms

phasor representation in rectangular and polar forms

1. What a Phasor Is

Phasor = rotating vector representing a sinusoid of fixed frequency $ω$ as a complex number (magnitude + phase), with time factor $e^{jωt}$ suppressed. It converts differential equations to algebraic in $jω$.

For $v(t)=V_m sin(ωt+φ)$ or $V_m cos(ωt+φ)$ (choose one, be consistent; many use $cos$ for phasor), the peak phasor (or RMS phasor $V_{rms}=V_m/√2$) is:

In rectangular (Cartesian) and polar:

where $j=√(-1)$ engineering, $∠φ$ means angle $φ$ from real axis.

2. Rectangular vs Polar — Conversion

• Polar to rectangular:

• Rectangular to polar:

(quadrant careful)

Forms:

• Polar: $V=|V|∠φ$ convenient for multiply/divide (magnitude multiply, angle add).
• Rectangular: $V=V_x+jV_y$ convenient for add/subtract.

3. Why Phasors Work

For $v(t)=Re[ \tilde{V} e^{jωt}]$, operations $d/dt → jω$, $∫ →1/jω$, so $v=L di/dt$ becomes $\tilde{V}=jωL·\tilde{I}$ (impedance).

4. Worked Examples

(a) v(t)=10 sin(314t+30°) ⇒ peak phasor \tilde{V}=10∠30° =10(cos30+j sin30)=8.66+j5

(b) Add v1=10∠0° and v2=10∠60°: rectangular v1=10+j0, v2=5+j8.66 ⇒ sum=15+j8.66 ⇒ polar =17.32∠30° ⇒ time domain 17.32 sin(314t+30°) ✓

(c) Rectangular 6+j8 ⇒ polar 10∠53.1° (since √(36+64)=10, tan^{-1}8/6=53.1°)

(d) For RMS, divide magnitude by √2: V_rms phasor=7.07∠30° if V_m=10.

5. Exam Points

• Define phasor $V=V_m∠φ =V_x+jV_y$ — 5 marks, very frequent.
• Show conversion rectangular↔polar — 5 marks, frequent.
• Add/subtract in rectangular, multiply/divide in polar — 3-5 marks, frequent.
• Mention phasor suppresses $e^{jωt}$ — 3 marks.