Question 15.11: Objective: Design the 555 IC as an astable multivibrator for...

The frequency of oscillation, as given by Equation (15.102), is

f = \frac{1}{T} = \frac{1}{T_{C}  +  T_{D}} = \frac{1}{0.693(R_{A}  +  2R_{B})C}          (15.102)
f = \frac{1}{0.693(R_{A}  +  2R_{B})C}
Therefore,
R_{A} + 2R_{B} = \frac{1}{(0.693) f C} = \frac{1}{(0.693)(50  ×  10^{3})(1  ×  10^{−9})} ⇒ 28.9  k \Omega               (15.104)
The duty cycle, given by Equation (15.103), is

Duty cycle = \frac{T_{C}}{T} × 100% = \frac{R_{A}  +  R_{B}}{R_{A}  +  2R_{B}} × 100%     (15.103)
Duty cycle = 0.75 = \frac{R_{A}  +  R_{B}}{R_{A}  +  2R_{B}}
which yields
R_{A} = 2R_{B}             (15.105)
Combining Equations (15.104) and (15.105), we find that
R_{A} = 14.5  k \Omega         and          R_{B} = 7.23  k\Omega

Trade-offs: If standard-valued resistors are required, then R_{A} = 13  k \Omega and R_{B} = 7.5  k\Omega would provide a frequency of 51.5 kHz and a duty cycle of 73.2 percent.
Comment: A wide range of oscillation frequencies can be obtained by changing the resistance and capacitance values.