Circuit Theory 2 · Complex numbers and phasor notation

#02 Complex-plane coordinates, polar and rectangular forms, Euler's formula, and the 10∠30° example

Write one phasor in polar and rectangular form, then convert 10∠30° with exact intermediate values and a magnitude check.

Question

Reviewed reference card from the original English lesson on complex forms and phasor notation.
Polar and rectangular forms represent the same phasor. Exact geometry needs equally scaled axes; the notebook distinguishes exact values from rounded decimals.

Describe a phasor as a complex number at a specified sinusoidal frequency, using one consistent amplitude convention. Explain the real coordinate a, imaginary coordinate b, magnitude M, and phase φ; j is the imaginary unit. Geometric lengths and angles require equal axis scales, and the phase is measured counterclockwise from the positive real axis. Connect polar and rectangular forms with Euler's formula. Convert 10∠30° step by step, retain the exact value 5√3 before rounding it to 8.66, and verify the magnitude. The example uses degrees; use radians in exponential formulas. Phase must retain the correct quadrant and is undefined at zero magnitude. A phasor represents a sinusoidal quantity; it need not be a physical spatial vector. No volt/ampere unit is assigned to the source's bare numerical example. The retained frames are reviewed reference cards, not corrected versions of the source diagrams.

Written solution and narration transcript(shows the full solution)

Below are all the lines written in the notebook together with the full narration transcript.

  1. 1. Turn phasor intuition into notation

    Reviewed reference card from the original English lesson on complex forms and phasor notation.
    Polar and rectangular forms represent the same phasor. Exact geometry needs equally scaled axes; the notebook distinguishes exact values from rounded decimals.
    Previously: a time waveform can be represented using a rotating vector and a phasor.
    Now give the phasor a notation that supports calculation at the chosen frequency.
    In alternating-current (AC) analysis, j denotes the imaginary unit, not a current:
    j2=−1\displaystyle j^{2}=-1

    Narration transcript

    Last lesson gave the picture: time waveform, rotating vector, phasor. In this lesson, we give that phasor a notation we can calculate with. Complex numbers are the writing system of A C analysis.

  2. 2. Identify real and imaginary components

    Reviewed reference card from the original English lesson on complex forms and phasor notation.
    Polar and rectangular forms represent the same phasor. Exact geometry needs equally scaled axes; the notebook distinguishes exact values from rounded decimals.
    Represent the phasor V as a point or a vector from the origin on the complex plane.
    Use real coordinates a horizontally and b vertically; the complex number is:
    V=a+jb\displaystyle V=a+j b
    With equally scaled axes, length gives magnitude; phase is measured counterclockwise from the positive real axis:
    M=a2+b2\displaystyle M=\sqrt{a^{2}+b^{2}}

    Narration transcript

    Place the phasor on the complex plane. The horizontal axis is the real part, and the vertical axis is the imaginary part. The vector length is the magnitude, and the angle from the real axis is the phase.

  3. 3. Connect polar and rectangular forms

    Reviewed reference card from the original English lesson on complex forms and phasor notation.
    Polar and rectangular forms represent the same phasor. Exact geometry needs equally scaled axes; the notebook distinguishes exact values from rounded decimals.
    Polar and rectangular notation describe one complex phasor.
    Polar form explicitly gives magnitude M and phase φ:
    V=M∠φ\displaystyle V=M∠\varphi
    Rectangular form gives the real and imaginary components:
    V=a+jb\displaystyle V=a+j b
    Euler's formula connects the forms; for a radian angle:
    V=M(cos⁡φ+jsin⁡φ)\displaystyle V=M\left(\cos \varphi +j \sin \varphi \right)

    Narration transcript

    So one phasor has two common forms. Polar form shows magnitude and phase directly. Rectangular form writes the same point as a plus j b. Euler's identity is the bridge between those two descriptions.

  4. 4. Convert 10∠30° step by step

    Reviewed reference card from the original English lesson on complex forms and phasor notation.
    Polar and rectangular forms represent the same phasor. Exact geometry needs equally scaled axes; the notebook distinguishes exact values from rounded decimals.
    Take magnitude 10 and an angle of 30 degrees:
    V=10∠30∘\displaystyle V=10∠30^{\circ}
    Substitute the degree-mode trig values into each component:
    a=10cos⁡30∘=53≈8.66,b=10sin⁡30∘=5\displaystyle a=10 \cos 30^{\circ}=5\sqrt{3}\approx 8.66, b=10 \sin 30^{\circ}=5
    Keep exact and rounded values distinct:
    V=53+j5≈8.66+j5\displaystyle V=5\sqrt{3}+j5\approx 8.66+j5
    Changing notation preserves the magnitude; check the exact components:
    M=(53)2+52=10\displaystyle M=\sqrt{\left(5\sqrt{3}\right)^{2}+5^{2}}=10

    Narration transcript

    Take ten angle thirty degrees. The real part is ten cosine thirty, and the imaginary part is ten sine thirty. So the same phasor becomes eight point six six plus j five. Same phasor, different writing.

  5. 5. Choose a useful form without changing the phasor

    Reviewed reference card from the original English lesson on complex forms and phasor notation.
    Polar and rectangular forms represent the same phasor. Exact geometry needs equally scaled axes; the notebook distinguishes exact values from rounded decimals.
    Polar form is convenient for magnitudes, angles, multiplication and division; rectangular form for addition and subtraction.
    Next: impedance relates voltage and current phasors for resistors, inductors and capacitors.

    Narration transcript

    Keep this rule: polar form is great for geometry, and rectangular form is great for algebra. In the next lesson, we will attach these phasors to resistors, inductors, and capacitors, and define impedance.

Source video: Circuit Theory-2 #02 Complex Form and Phasor Notation (1:23)