Electrical – How to balance between frequency and duty cycle in this circuit

I have this astable circuit that generate square waves with independent control of frequency and duty cycle:

For this circuit:

• $$t_H=0.759\,C\,(R_f+R_y)$$
• $$t_L=0.759\,C\,R_x$$
• $$period = 0.759\,C\,(R_f+R_d)$$
• $$frequency = \frac{1}{period}=\frac{1.32}{C\,(R_f+R_d)}$$
• $$duty\; cycle =\frac{t_H}{period}=\frac{R_f+R_y}{R_f+R_d}$$

I want the duty cycle to range -at least- from 25% to 75%, and the frequency to range -at least- from 1 kHz to 10 kHz.

For these given values:

• Rd is a potentiometer whose whole resistance is 50 kOhm.
• Rf is a variable resistor whose maximum value is 50 kOhm.
• C = 10 nF.

, here are the possible ranges:

• The maximum frequency possible:
  $$freqency_{max}=\frac{1.32}{C\,(R_{f,min}+R_d)}=\frac{1.32}{10^{-8}(50k+10)}=2.64\,kHz$$
  where 10 Ohm is assumed as the minimum value that can be assigned to Rf, and 50 kOhm is Rd.

• The minimum frequency possible:
  $$freqency_{min}=\frac{1.32}{C\,(R_{f,max}+R_d)}=\frac{1.32}{10^{-8}(50k+50k)}=1.32\,kHz$$

You can see that the frequency range is very narrow.

On the other hand, calculating duty cycle and assuming that Rf is set to 50 kOhm:

• The maximum duty cycle possible:
  $$duty\; cycle =\frac{R_f+R_{y,max}}{R_f+R_d}=\frac{50k+50K}{50k+50k}=1$$

• The minimum duty cycle possible:
  $$duty\; cycle =\frac{R_f+R_{y,min}}{R_f+R_d}=\frac{50k+10}{50k+50k}=0.5$$

The maximum duty cycle possible is great but the minimum one is bad.

I can reduce the maximum value of the variable resistor Rf to improve the range of duty cycle but the range of frequencies will not improve noticeably. For example, if Rf(max) = 5 kOhm, duty cycle will range from 10% to 100%. But the frequency will range from 2.4 kHz to 2.64 kHz which is unacceptable.

Or I can reduce the whole resistance of the potentiometer Rd to improve the range of frequencies but the range of duty cycle will not improve noticeably. For example, if Rd is reduced to 5 kOhm, the maximum frequency becomes 25.9 kHz and the minimum frequency becomes 2.4 kHz. But the duty cycle will range from 90% to 100% which is so bad.

Even if I reduce both of Rf and Rd or change the value of C, I can't get to my goal.

How can I achieve the required range of frequency and duty cycle?