Theoretical formulation (I) without external storage capacitor

(i) Notation

Let us define some notation first.

Cpp(i,j) - pixel capacitance matrix. If fcj, it denotes mutual capacitance between pixel electrode i and pixel electrode j, if i=j, it denotes pixel electrode self capacitance, i j

= 1,2,.. .np, np, the number of pixel electrodes in the cell;

Cbb(i,j) - bus capacitance matrix. Likewise, if tej, it denotes mutual capacitance

between bus electrode i and bus electrode j, if i=j, it denotes bus electrode self capacitance, i j =0,l,2,...«b, rib, the number of bus electrodes; If no bus, i.e., i,j=0, this matrix does not exist.

CPb(i,j) - pixel-bus mutual capacitance matrix between pixel electrode i and bus

electrode j. i=l,2,...np,j=0,l,2,...rib. If no bus, i.e.,y=0, this matrix does not exist.

Cbp(i,j) ~ bus-pixel mutual capacitance matrix between bus electrode i and pixel

electrode j. i=0,l ,2,.. .rib, j=l ,2,.. .np. If no bus, i.e., i=0, this matrix does not exist. Vp(/) - floating voltage to be found for pixel electrode i (i= 1,2,...np).

Vb(/) - voltage that is applied to the bus electrode i (i=l,2,...Wb) and is known.

Qp(i) - charge that is deposited in pixel electrode i (i=\,2,...np) at the time

T n

t = k ~ Y + tc (here Jc=0,1,2,..., Tp is the full frame time), immediately before the electrode de-links to the source. It is assumed not varying with the time until the source voltage changes polarity.

Qb{i) - charge that is deposited in bus electrode i (i= 1,2,.. ,«b) and is known.

Not that both indices i and j appearing above are local or subspace (pixels or buses)- referred, i.e., they are defined in the spaces (npxnp) or («bx«b) or («px«b)

C(iJ), V(i), and <2(0 - global or integrated capacitance, voltage and charge matrices defined with respect to the whole space consisting of «p+wb. Therefore the indices i j

now take the values of \,2,...np,np+l,...np+rib, in a way that the number of bus electrodes rcb is appended to the number of the pixel electrodes np.

(ii) General formulation

Assuming that there are np pixel and nb bus electrodes in the LCD cell in question, then the problem concerning how to find the floating voltages that will be applied to the pixel electrodes np turns to solve the matrix equation

C(t)V(t)=Q(t) (100)

or

'C (l,l) C(1,2) C(l,3) _{"[>(1)1 W}

C(2,1) C(2,2) C(2,3) ■ 1/(2) _{2(2 )}

C(3,l) C(3,2) C(3,3) • V(3) 2(3)

Here the array indices i and j in C(ij) are defined in the whole space («p+«b)x(«p+wb) and are arranged such that the indices for the buses follow the indices for the pixels, i.e., ij=\,2,...np,np+l,np+2,...np+rib. Since (1) all voltages applied to the bus electrodes are known, (2) the charges deposited in the latter can be calculated, and (3) the charges in pixel electrodes are known also when the pixel electrodes become disconnected to the data line, then the floating voltage that will appear in the pixel electrodes can be calculated as long as all self- and mutual capacitances are given. However a quick and intuitive examination of expression (101) reveals that the capacitance matrix C can be partitioned or decomposed into blocks that are defined in the relevant subspaces according to the known data and unknowns. The benefit from doing so is that only relevant elements of the C matrix are needed to be calculated in order to obtain the complete information about the floating voltages. That means computing time will be saved as calculating the elements of C is a very expensive task. Following this, equation (101) can be rewritten as

1 o _{< v} ■ ■ 1 ■Q 1 ■ f i/ i o ■§" c “ . 1 ■■ 1 a _ (102a) or

‘ M 1-1)

Cpp(1-2)

'

c ^ i u ) c pb(1,2)

■K ( l ) l

'fi,(O '

C j 2 , l ) _{Cpp( 2,2) Cph(2,1)} Cpb(2,2)

■

Vp(2)

e ,(2 )

' C „(l.l) Cbp{

1,2)

_{C * (U )}

C** (1.2) •

v b(i) = Q„(

i)

Cbp(

2,1)

C j 2,2) c j

2,1)

C j

2,2)

•

n (2 )

a (2)

Here, as mentioned before, all block matrices in the above expression are confined to their respective subspaces [(npxnp), {n\>xn\>), (npx/ib)]. Expanding the above matrix gives rise to:

C ppV p + C pbVb = Q p (103)

CbpV p +C„„Vh = Q b (104)

Clearly, both the above equations lead to the solution for the Vp. However, equation (103) is chosen for this purpose because of two reasons: (a) it is directly related to the constant charge condition through Qp and (b) due to small physical dimension of bus electrodes, there are inevitably some numerical errors occurring in calculating Cbb

and Qb if Eq. (104) were used. From Eq. (103), we have

v P = c ; l ( Q p - c pbv t ) (105) or >„(!)■ ‘ a , ( u ) _{■ ■'}-1 ' QP ( 0 - Cpb ( u K (1)- c pb(1,2K (2) - Vp (2) - Cpp{2,1) Cpp(2,2) ■ ■ Q„ (2 )- Cpb (2 ,lK (1)- Cpb (2,2 y b ( 2 ) - • • • (106) here C~l denotes the inverse of the matrix Cpp. It can be seen that the existence of bus voltages and mutual capacitance between pixel and bus gives rise to a correction or

superposition to the situation where there does not exist any bus. Therefore, the behaviour of the bus voltage will be reflected in the pattern of the floating voltage of the pixel as will be seen below.

(iii) One pixel and no buses

This simplest electrode geometry normally appears in a test cell. In this case, the formula for Vp from the above Eq. (106) takes the simplest form:

qAi)

v,(i)=

(107)

(iv) One pixel and two buses

This is the most commonly seen electrode configuration. According to Eq. (106), the expression for the floating pixel voltage takes the form of

v ,

(O’= - ^ j j j 1

QP

(i)-

c pb

( u K ( 0 -

c pk

(

1 2

K (

2)1 (108)

Note that indices i and j in the Cpb(iJ) are local. The first index i labels pixel electrode (i=l in this case) and the second index j labels bus electrode 0=1,2 in this case). It can be seen that in order to get the floating voltage Vp for the pixel, not only the pixel self-capacitance is needed, but the mutual capacitances Cpb between pixel and the two bus electrodes need to be calculated as well.

(v) Two pixels and one bus

This model is a simplified electrode configuration where the physical dimension of each pixel electrode in a periodic arrangement is too big to be convenient for a complete computer modelling. By keeping only part of each pixel electrode and one

Pixel electrode 1 Bus Pixel electrode 2

Neumann Boundary Condition Neumann Boundary Condition Ground electrode

Figure 28 A simplified model for the geometry with very large pixel electrode periodically aligned

complete bus electrode between two pixel electrodes and using a Neumann boundary condition, the modelling by computer is then feasible with the geometry as shown in Figure 28.

In this case, the explicit expression for pixel voltages according to Eq. (106) is given by

v

cj2ah^)-cpb{uyM-cpp(i2hP(2)-cpb(2,iyM

, i n n ,

V 'A l ) - C ( l , l ) C (2,2) - C ( l , 2 ) C (2,l) (109)

,,

- cJ 2’lM ) - cAuyM+cjuiQP(2)-cpb{2,iyM

pK>~ C ( U ) C ( 2 , 2 ) - C ( 1 2 ) C ( 2 , 1 ) U W )

In document Computer modelling of liquid crystal displays (Page 90-94)