# Astable timer: Halve frequency while maintaining the same "up" pulse width?

Disclaimer: I am not an electrical engineer, but an environmental science student. I'm trying my best here. I have a basic astable 555 timing circuit with the following components:

$R_1 = 10k\Omega,~~R_2 = 1k\Omega,~~C = 0.1 \mu F$

$T = 0.7 \cdot (R_1 + 2R_2) \cdot C = 0.7 \cdot 12k\Omega \cdot 0.1 \mu F = 0.84 ms \\ f = \frac{1}{0.84 ms} = 1.19 kHz \\ t_{up} = 0.7 \cdot (R_1 + R_2) \cdot C = 0.7 \cdot 11k\Omega \cdot 0.1 \mu F = 0.77 ms \\ t_{down} = 0.7 \cdot R_2 \cdot C = 0.7 \cdot 1k\Omega \cdot 0.1 \mu F = 0.07 ms$

I am being asked to halve its frequency but maintain the same "up" pulse width.

I know that doubling the reactance of the capacitor from $0.1 \mu F$ to $0.2 \mu F$ will halve the frequency, as it will take roughly twice as long for it to charge. However, my "up" pulse width $t_{up}$ is also being doubled.

$T = 0.7 \cdot 12k\Omega \cdot 0.2 \mu F = 1.68 ms \\ f = \frac{1}{1.68ms} = 595Hz \\ t_{up} = 0.7 \cdot 11k\Omega \cdot 0.2 \mu F = 1.54 ms \\ t_{down} = 0.7 \cdot 1k\Omega \cdot 0.2 \mu F = 0.14 ms$

The desired output would have a frequency of 595Hz, but an "up" pulse width of 0.77 milliseconds.