Peak amplitude phasor ($v(t)=\mathrm{Re}(\hat{V}{e}^{i\omega t})$)

A peak amplitude phasor is the complex amplitude that reconstructs a real single-frequency sinusoid without an extra amplitude multiplier:Its modulus is the sinusoid's peak amplitude, and its complex argument is the wave phase relative to the chosen cosine reference. This is the default convention in phasor. MIT's sinusoidal notes explicitly describe the cosine reference and phase/amplitude parameters.^[ref]

For an original example, take $v(t)=3\cos \omega t-4\sin \omega t$ volts. Expanding the real part shows that $\hat{V}=3+4i$ volts, with peak amplitude $5$ volts and wave phase $\arctan (4/3)\approx 53.13^{\circ }$. The corresponding root mean square phasor is $(3+4i)/\sqrt{2}$ volts, whose magnitude is about $3.536$ volts. The physical signal is the same under both conventions.

The peak phasor versus positive-frequency Fourier coefficient distinction follows fromThus the positive-frequency coefficient in this two-exponential expansion is $\hat{V}/2$, not $\hat{V}$. A real sinusoid requires its conjugate negative-frequency term as well. This algebra does not assign ordinary Fourier-transform function values to an infinite-duration sinusoid; those require the corresponding distributional interpretation.

A common time-origin change multiplies both voltage and current peak phasors by the same unit-modulus factor, leaving their relative phase and impedance ratio unchanged. Peak-versus-RMS normalization in complex electrical power is still essential when an amplitude convention changes rather than just the time origin.

Peak phasor versus positive-frequency Fourier coefficient distinguishes a real sinusoid's reconstruction amplitude from one coefficient in its two-sided exponential expansion. For $v(t)=\mathrm{Re}(\hat{V}{e}^{i\omega t})$,Thus the positive-frequency Fourier coefficient is half the peak amplitude phasor. The negative-frequency coefficient is its conjugate. A real sinusoid needs both; a complex exponential input has a different one-sided reconstruction.

The distinction also controls a sampled record. For $N$ evenly spaced samples of a real sinusoid with exactly $k$ cycles in the record and $0<k<N/2$, use the unscaled forward discrete Fourier transformOrthogonality of the discrete exponentials gives $X_k=N\hat{V}/2$. ConsequentlyThe default SciPy forward FFT has exactly this unscaled convention; its forward and orthonormal modes have different factors.^[ref]

For an original example, sample $v_m=3\cos (2\pi 5m/64)-4\sin (2\pi 5m/64)$ volts. Then $\hat{V}=3+4i$ volts and $X_5=96+128i$ volts. Dividing by 64 gives the positive-frequency coefficient $1.5+2i$ volts; doubling it recovers the peak phasor. The RMS amplitude is $5/\sqrt{2}\approx 3.536$ volts.

These exact relations assume coherent sampling of the stated sinusoid without aliased extra components. DC and, for even $N$, the Nyquist bin do not have distinct conjugate partners and must not be doubled. With a noninteger number of cycles or a windowed record, finite-record phasor extraction with coherent sampling must be distinguished from a generic bin readout; spectral leakage and the window's amplitude normalization alter the estimate. The parent peak amplitude phasor specifies the continuous-time convention before any sampled transform is chosen.

Finite-record phasor extraction with coherent sampling recovers a sinusoid's complex amplitude by arranging its frequency on a discrete Fourier bin. Let samples be taken at $t_m=t_0+m\Delta t$, with $N\Delta t$ containing exactly $k$ cycles, where $k$ is an integer and $0<k<N/2$. Forthe constant baseline cancels from bin $k$. The peak phasor versus positive-frequency Fourier coefficient relation then gives the estimateThe factor $e^{-i\omega t_0}$ assigns wave phase to the stated physical time origin; without it the coefficient refers to the record's start. The default forward FFT is unscaled, while other FFT normalization modes require different prefactors.^[ref]

For an original example, sample at $80\mathrm{kHz}$ with $N=80$ and use a $7\mathrm{kHz}$ tone, so $k=7$ and the cyclic record duration is $1\mathrm{ms}$. Let its physical peak phasor be $2e^{i\pi /6}\mathrm{V}$ and let $t_0=50\mu \mathrm{s}$. The record-start wave phase is $30^{\circ }+126^{\circ }=156^{\circ }$, and the unscaled bin isApplying both $2/N$ and the time-origin correction recovers the physical $2e^{i30^{\circ }}\mathrm{V}$ phasor. Its RMS amplitude is $\sqrt{2}\mathrm{V}$. Omitting the start-time correction would give a correct magnitude but a different wave phase reference.

With independent zero-mean real noise of variance ${\sigma }^2$ on each sample, the extracted phasor remains unbiased. Orthogonality gives noise variance $2{\sigma }^2/N$ in each real and imaginary component and zero covariance between them. For $\sigma =0.01\mathrm{V}$ here, each component's standard deviation is about $1.581\mathrm{mV}$. These are complex-amplitude uncertainties; taking a magnitude is a nonlinear operation and need not retain unbiasedness.

Coherent sampling must describe the actual source/clock relationship, not just the rounded frequency entered into software. Frequency drift, extra tones, aliasing or a changed window invalidate the simple exact-bin identity; spectral leakage bias in harmonic distortion measurements shows one consequence. A coherent gain of a window can correct an on-bin amplitude scale in suitable models, but cannot in general turn a noncoherent one-bin reading into an exact phasor.