Printing System Models - Control of Color Imaging Systems: Analysis and Design - page 602

t p PR ( K p )

σ p = σ p0 + σ p1 e ikx

g

t d

Developer ( K d )

Effective developer
electrode position

FIGURE 10.45 Schematic of a development nip.

Similarly, the development process can be modeled completely using the
dot development models shown in Table 10.6 and the MTF equations shown in
Equation 10.65.
In a development nip, toners are recruited from the top of the magnetic brush to
the PR by electric fields due to the latent image. Figure 10.45 shows a schematic of a

development nip. The MTF for the normal electric field at the surface of the magnetic
brush may be written as

K d
A k þ B k cosh( kt d )

MTF( k ) ¼ ( 10 : 65 )

where A k and B k are given by

K d þ 1 K d 1

A k ¼ t p coth( kt p )

sinh[ k ( t d þ g )] sinh[ k ( t d g )]

2 2

( 10 : 66 )

K d þ K d

t K p 1 1

B k ¼ cosh[ k ( t d þ g )] þ cosh[ k ( t d g )]

2 2

k is the angular spatial frequency
t p is the PR thickness
K p is the PR dielectric constant
g is the air gap
t d is the effective developer layer thickness
K d is the dielectric constant of the developer
sinh(.), cosh(.), and coth(.) are hyperbolic sine, hyperbolic cosine, and hyper-
bolic cotangent functions, respectively

Note that Equation 10.65 is similar to the MTF derived by Neugebauer [33] and
others. The effective developer layer thickness may be expressed as