Ive bin using the 555 timer IC for years and ive never questioned the formulas. I blindly plug what i need into the formula and get the results. Most on this forum are already familiar with the formulas in question but for the novices, these are the ones im referring to:
Astable Mode: charge time T1 = 0.693(Ra + Rb)C Discharge time T2 = 0.693RbxC
Monostable Mode T = 1.1RC
Can someone fill me in on where the numbers .693 and 1.1 come from ?
I believe these factors are the times in relation to the RC fomulae against the difference between the top and bottom switching thresholds. Have to look at an RC curve and plot 1/3 to 2/3 (I think) voltage change.
When you're discharging a capacitor, the factor is exp(-t/T) where t is time and T is the RC product. Since exp(-ln(2)) is exactly 1/2, at t = .693RC is the point where the voltage has fallen to 1/2 of it's initial value. On the 555, the internal comparitors are set at
2/3 and 1/3 of Vcc, so the "initial" value here is 2/3 Vcc and the halfway point is 1/3 Vcc.
The 1.1 comes from the internal "turn off" comparitor's setpoint at
2/3 Vcc. The trigger "discharges" the comparitors (turning on the output) so we need to get back to the "turn off" point. The charge time curve is (1 - exp(-t/T)) and so we need 2/3 = (1 - exp(-t/T)) or t = -ln(1/3)RC.
Hmmm... case where having fancy fonts and formatting would have come in handy in a usenet posting. We'll do this without HTML, tho...
That expression is (voltage at time t) = V0 * e to the (- t/T). To switch states, the voltage has to fall (rise) from 2/3 Vcc to 1/3 Vcc (or vicy versa), a factor of 2. So, for the falling voltage case, (e ^ (-t/T)) must equal 1/2.
Manipulate that to (-t/T) = (log base n) of 1/2, or t = - ln(2) * T. The T is the RC time constant.
Thanx again Rich, that has cleared up a few misunderstandings and blunderings on my part.
Im beginning to see why ive never seen the question raised about the formulae used in the 555 timer. Appears one would require some knowledge of calculus and differencial equations to get a full grasp on the concepts involved. The last time ive even looked at a calculus book (or algebra for that matter) was over 13 years ago, so im abit rusty ( no surprise ;-) )
Anyway after an exhaustive search and postings to various news groups and forums, heres what ive got ( appart from a pounding headache ):]
FYI: Your equations show 1n (one-n) instead of ln (L-n).
It seems like you can't see the forest from the trees. The concept of the 555 is dirt simple if you keep in mind the internal voltage divider. All 3 internal resistors are the same value (or damn near the same values). Because of this the it forms a voltage divider of two types: an upper limit (for the upper voltage cutoff) and a lower limit (for the lower voltage cutoff). Since the lower voltage divider is
[R/(R+R+R)]V = 1/3V
and the upper voltage divider is
[(R+R)/(R+R+R)]V = 2/3V
these form the comparator 'window' of operating voltages.
The external capacitor and resistor form the charging and relaxation circuit. The time constant as mentioned (t) is the the time of charging and decay of the external resistor and capacitor circuit. The time constant is only important in respect to how long it takes for the capacitor to charge up to the upper limit to trigger the output to go low. Once the output is low, it drains off the capacitor to the lower voltage (1/3)V and is then set high (to V) to recharge the capacitor. The time constant is *only* in respect to the external resistor and capacitor values, since that is what determines the time it will take to reach the upper limit 2/3V and the time that it will take to reach the lower limit 1/3V.
If the internal resistors were not equal, then the numbers 1.1 and 0.7 are totally meaningless!!!