Tech:loadshape - Loadshape technical support document

The basic principle underpinning loadshape is that the total energy [math]E[/math] consumed by a series of [math]N[/math] pulses

- [math]E = N P \Delta t \![/math]Eq. 1

where

- [math]P[/math] is the power of each pulse
- [math]\Delta t[/math] is the duration of each pulse.

At least one of the variables E, N, P, [math]\Delta t[/math] must remain unspecified by the user in order for Eq. 1 to not be overspecified. Thus

- [math]N = \frac{E}{P \Delta t} \![/math]Eq. 1a

is used to determine the number of pulses,

- [math]\Delta t = \frac{E}{P N} \![/math]Eq. 1b

is used to determine the pulse duration, and

- [math]P = \frac{E}{P \Delta t} \![/math]Eq. 1c

is used to determine the power of each pulse.

In cases where the standard deviation [math]\sigma[/math] is provided, the power of the pulse is adjusted so that

- [math]P_{adjusted} = Normal[P,\sigma][/math]Eq. 2

## Analog loadshapes

When the analog loadshape power [math]P[/math] is given, the loadshape amplitude [math]y_t[/math] at the time [math]t[/math] shall be computed from schedule value [math]x_t[/math] as

- [math]y_t = P x_t \![/math].

When the analog loadshape energy [math]E[/math] is given, the loadshape amplitude [math]y_t[/math] at the time [math]t[/math] shall be computed from the schedule values [math]x_t[/math] in the interval [math]t=(0,T([/math] as

- [math]y_t = \frac{E x_t}{\sum_{t=0}^T x_t \Delta t}[/math]

where

- [math]\Delta t[/math] is the time interval over which the schedule value [math]x_t[/math] is given.

When neither the energy nor the power are given, the loadshape amplitude [math]y_t[/math] at the time [math]t[/math] shall be the schedule value [math]x_t[/math].

## Pulsed shapes

When a pulse loadshape is specified, the power amplitude [math]y_t[/math] at the time [math]t[/math] based on the schedule value [math]x_t[/math] is given by

- [math]y_t = \frac{E}{N \Delta t} x_t[/math].

When the energy is not given, the power amplitude may be computed from the power and time interval

- [math]y_t = \frac{P T}{N \Delta t} x_t[/math].

When both the energy and power are given, the power amplitude may be computed as

- [math]y_t = \frac{P^2 T}{E}x_t[/math]

## Modulated shapes

TODO: describe how modulated shapes are generated from schedules

## Queued shapes

TODO: describe how queued shapes are generated from schedules

## Scheduled shaped

TODO: describe how pulse shapes are generated from schedules