A mathematical model was developed by integrating the flash discharge curve (flash intensity I vs. time t) in order to predict the clipped flash output at various shutter speeds. Because a flash unit employs a capacitor circuit, it is assumed that flash intensity during the discharge phase follows the exponential decay function as:
(1)
Where I0 is peak intensity at time t0, k is decay constant. The amount of flash energy (E) in time t can be calculated by integrating the flash intensity as:
(2)
Where the total flash energy (E0) of a complete discharge is:
(3)
The guide number (GN) of a flash measures its ability to illuminate the subject and is defined as:
(4)
Where d is distance to subject and F is aperture f-number which is related to a power-of-2 aperture step (Fs) defined as:
(5)
According to the inverse square law for a point source of light, GN is proportional to the square root of flash energy. Therefore the ratio of the clipped GN to the fully discharged GN0 is:
(6)
By combining the above equations, we have
(7)