DC, time-domain and harmonic notation
V and I, then v(t) and i(t), then underlined harmonic quantities: one notation system from DC to AC.
- Use uppercase for constant/DC quantities.
- Use lowercase functions for instantaneous variables.
- Distinguish a phasor from a Fourier transform.
- Read Z(omega), Y(omega) and H(j omega) without confusing them with phasors.
Why does AoN refuse to use the same symbol for a DC bias, an instantaneous waveform, a single-frequency complex amplitude and a Fourier transform?
Because these are different mathematical objects. A notation system should reveal the regime before the equation is solved.
04.1Constant/DC quantities: uppercase real variables
AoN uses uppercase italic symbols for constant real electrical quantities:
The uppercase letter signals that the quantity is not being treated as an explicit function of time in the present model.
04.2Instantaneous time-varying quantities: lowercase functions
For example, a real sinusoidal voltage may be written
The lowercase symbol plus explicit time argument makes it impossible to mistake the waveform for a DC operating value.
04.3Single-frequency harmonic quantity: underlined lowercase phasor
In sinusoidal steady state, AoN writes the associated complex amplitude as an underlined lowercase quantity:
The real waveform remains v(t); the underlined object is a complex representation used for steady-state calculation. If one angular frequency is fixed throughout the derivation, (ω) may be omitted after the convention is stated.
04.4A phasor is not the real waveform
The waveform v(t) is real and varies in time. The phasor v(ω) is a complex amplitude associated with one sinusoidal frequency. Replacing one by the other without stating the representation is a category error.
AC-01 will derive the actual mapping and amplitude convention used in calculations. EF-01 only fixes the names of the objects so later equations cannot silently switch meanings.
04.5Fourier transform: a function of frequency, not one phasor
A general signal can contain many frequencies. Its Fourier-domain representation is therefore a function of ω. AoN writes that transform without the phasor underline, for example
This distinction prevents a single-frequency complex amplitude from being confused with a frequency spectrum.
04.6Impedance, admittance and transfer functions
AoN writes
without a phasor underline because these are frequency-dependent complex functions by definition in their later circuit context. Their arguments already tell the reader what domain they belong to.