Complex amplitudes & AoN notation
Master rectangular, exponential and polar forms before impedance. Fix the cosine/peak convention, convert with atan2, choose the right form for each operation, normalize phase and negative amplitudes, distinguish peak from RMS phasors, track time-delay phase and avoid confusing phasor multiplication with time-domain waveform multiplication.
- Apply AoN's underlined-lowercase peak-amplitude phasor convention without mixing it with RMS notation.
- Convert exactly between rectangular, exponential and polar complex forms using atan2 for quadrant-safe phase recovery.
- Choose rectangular form for addition/subtraction and polar form for multiplication/division.
- Normalize equivalent phases and absorb negative amplitudes into a 180° phase shift.
- Convert sine-reference waveforms into AoN's cosine-reference phasors.
- Relate time delay to phase through −ωτ and explain why phase requires a reference.
- Convert consistently between peak and RMS phasors while separating DC offset from the sinusoidal component.
- Explain why multiplication of phasor coefficients is not generally the phasor of the product of two real waveforms.
Chapter 01 justified phasors. The next question is operational: once a sinusoid has been replaced by one complex amplitude, how do we calculate with that amplitude without losing the waveform convention, units, phase reference, or physical meaning?
This chapter fixes the AoN notation and the complex-number discipline that every later AC calculation will inherit.
02.1The AoN mapping is a declared convention, not hidden notation
Unless a lesson explicitly says otherwise, AC-01 uses a cosine reference and a peak-amplitude phasor. The phasor is lowercase and underlined; the real waveform remains lowercase with its time argument.
The phasor has the same physical unit as the sinusoidal quantity: volts for voltage, amperes for current. The angle is dimensionless; engineering displays may use degrees, while the exponential form is mathematically interpreted in radians.
02.2Rectangular, exponential and polar forms are the same number
Rectangular form stores the real and imaginary coordinates directly. Polar form stores magnitude and angle. Conversion is exact apart from numerical rounding.
Use the two-argument function atan2(b,a), not merely arctan(b/a), because the signs of both coordinates are required to recover the correct quadrant.
The naive value arctan(4/−3)=−53.13° points to the wrong quadrant unless the missing 180° correction is restored.
Figure 2.1 — One complex amplitude, two equivalent descriptions
The one question is: how do rectangular coordinates and magnitude-phase describe the same complex number?
The complex point has coordinates (a,b). Its distance from the origin is M and its oriented angle is φ. The forms a+jb, Me^{jφ} and M∠φ represent exactly the same complex number.
02.3Add and subtract in rectangular form
This is why same-frequency sinusoid addition becomes so simple: the common rotating factor has already been removed, so only the complex coefficients are added.
A reliable workflow is polar → rectangular → add/subtract → polar only if the final magnitude and phase are useful.
02.4Multiply and divide most naturally in polar form
These rules are coefficient algebra. If a linear sinusoidal relation later has the form y=Kx, multiplication by the complex coefficient K scales magnitude and shifts phase.
02.5Normalize phase before comparing answers
AoN reports phasor magnitude as non-negative. A negative sinusoidal coefficient is absorbed into a 180° phase shift. For display, a principal interval such as (−180°,180°] is convenient, but equivalent angles describe the same complex number.
Because AoN uses cosine as the reference, a sine-form waveform is converted before its phasor is written. This prevents hidden 90° errors.
02.6Phase is relative to the time reference
A positive time delay produces a negative phase shift under the AoN cosine convention. Phase therefore has meaning only relative to a stated time origin or another same-frequency signal.
Changing the chosen time origin changes every absolute phase consistently; physically meaningful phase differences remain invariant when the same reference change is applied to all signals.
02.7Peak and RMS phasors differ by √2, not by phase
For a zero-offset sinusoid, peak and RMS phasors carry the same phase and differ only in magnitude by √2. Ratios between voltage and current phasors are therefore unchanged if both use the same convention.
02.8Conjugation and relative phase
Complex conjugation mirrors a point across the real axis. It will become useful later in power calculations, but it is an algebraic operation on the complex coefficient; do not assign it a time-domain physical meaning without deriving one.
The angle of a phasor ratio directly gives the relative phase between two same-frequency quantities.
02.9A disciplined conversion workflow
Example at one common frequency:
At this stage that last quantity is only a complex voltage/current ratio. Chapter 03 will derive when such a ratio is legitimately called an element impedance.
02.10Mastery exercises
- Convert 6−j8 to magnitude-angle form.
- Convert 5∠150° to rectangular form.
- Add 4∠30° and 3∠−60°.
- Normalize −7∠20° using a non-negative magnitude.
- Convert 2 sin(ωt+35°) A to an AoN peak phasor.
- A signal is delayed by T/8. Find its phase change.
- Convert a 12 V peak phasor to RMS convention without changing phase.
- Explain why the product of two phasors at ω does not generally represent the product of the two real waveforms.
Solutions
1. Magnitude 10 and angle atan2(−8,6)=−53.13°, so 10∠−53.13°.
2. 5(cos150°+j sin150°)=−4.330+j2.500.
3. 4∠30°=3.464+j2; 3∠−60°=1.5−j2.598; sum 4.964−j0.598=5.00∠−6.87° approximately.
4. −7∠20°=7∠200°=7∠−160°.
5. 2 sin(ωt+35°)=2 cos(ωt−55°), hence i=2∠−55° A.
6. τ=T/8 gives −ωτ=−2π/8=−π/4=−45°.
7. 12/√2=8.485 V RMS with the same phase.
8. Time-domain multiplication generates sum and difference frequencies; for equal frequencies these are DC and 2ω, so the result is not a single component at ω.
Chapter 03 can now derive capacitor and inductor impedance from their time-domain laws without changing notation or complex-number rules.