## General equation for the capacitor voltage Assignment Help

Assignment Help: >> Alternative Astable Multivibrator - General equation for the capacitor voltage

General equation for the capacitor voltage:

The general equation for the capacitor voltage may be written as

Vc (t ) = A + B e- t/ T0

where

T0   =   C0  RA  RB /(RA   + RB  )

If there were no control, at t = ∞ ,

Vc (t ) =   (RB /(RA  + RB ) )Vcc

Substitution of this in Eq. (105) gives

A + B = (2/3 )Vcc

Also, in infinite time (that meanst = ∞ ) capacitor voltage shall attain a steady value of RB /(RA  + RB ) Vcc , because of which from Eq. (105), we obtain, A =         (RB ) /(RA  + RB ) Vcc , therefore, one finds

B = (2/3 ) V cc   - (RB )/ (RA   + R B )   V                  cc = (2RA  - RB ) Vcc / 3 (R A + R B  )

The required equation is, thus,

V  (t ) = (RB )  / (RA+ R B ))      Vcc = ((2RA  - RB ) Vcc / 3 (RA+ R B )) e - t /T0

Now, for t = TL, Vc (t) =   Vcc /3 , therefore, substituting these values in Eq. (105), we determine

V cc {(1/3) -  (RB ) / ( RA+ R B   )     =   (2RA   - RB  ) Vcc / 3( RA+ R B   ) e- TL/ T0

which simplifies to

(RA  - 2RB ) =(2R A  + RB   ) e - t/T0

e - t /T0  = 2RA  - RB / RA  - 2RB

∴          T L  = (C0 RA RB  /CRA + RB ) ln (2RA   - RB  / RA  - 2RB) ; R A> 2RB

Furthermore, TH = C0 RA ln 2 (as usual), therefore, 50% duty cycle shall be attained if

(RB )  / (RA+ R B )) ln (2RA   - RB  / RA  - 2RB) =     ln 2

(2RA   - RB  / RA  - 2RB) = 2 (1+ (RA/RB))

2 ((RA  /  RB  -1))/ ((RA  /  RB  -2)) = 2 (1+ (RA/RB))

2K - 1/ K - 2  = 21+ K

where

RA/ RB    = k

Clearly, Eq. (7.110) does not have a closed solution, but it can be solved by hit and trial, by picking up some value of k and then seeing that RHS equals LHS. This happens at k = 2.3, hence, if we take RB = R then Ra should be 2.3 R. 