a$)$ Design an astable multivibrator with f$0=4$ KHz and D $(\%)=55\%$ by using $555$ timer. The timing accuracy of the $555$ astable approaches $1\%,$ with a temperature stability of $0\mathrm{.}005\%/$C and a power supply stability of $0\mathrm{.}05\%/$V$.$

b$)$ Since TH$>$TL$,$ the circuit always gives D $(\%)>50\%.$ Find a better approach to get perfect symmetry.

c$)$Assuming Vcc$=6$V in the multivibrator, find the range of variation of f$0$ and D $(\%)$ if the voltage at the control input is modulated by ac coupling to it external sine wave with a peak amplitude of $1$V$.$

d$)$ The node in the $555$ timer at which the voltage is VTH $($i$.$e$.$ the inverting input terminal of comparator $1)$ is usually connected to an external terminal. This allows the user to change VTH externally $($i$.$e$.$ VTH no longer remains at $).$ Note, however that whatever the value of VTH becomes, VTL always remains at$.$ For the astable multivibrator, rederive the expressions for TH and TL$,$ expressing them in terms of VTH and VTL$.$