Capacitor Charge within a time period

[Regis Tro]

Homework Statement

This is one of my oscillator circuits, that I am trying to develop a simple formula to calculate RC for Oscillation Frequency, based on the fact the oscillation is done with the Capacitor charge changing from 1.1V to 1.68V back and forth, for a set of physical numbers.

What I know so forth?

Having:

• +5V power supply.
• 100nF Capacitor.
• Capacitor lead (a) is connected to Ground
• Capacitor lead (b) is connected to a 10kΩ (1%) Resistor.
• Resistor other lead is connected to the output of a 74HCT14 Schmitt-Trigger gate.
• This auto-oscillating gate delivers +5V or Ground to the Resistor, so it charges or discharges the Capacitor via the Resistor.

Measured using Tektronix 2246 oscilloscope:

I don't know the time required for the Capacitor first charge from zero to 1.1V, this happens at Power On, and I don't care, but I know the Capacitor charges from 1.1V to 1.68V in 160μs, with +5V applied to the Resistor, then the Resistor is switched to Ground and the Capacitor discharges back to 1.1V after 413μs, when Resistor is again switched to +5V, and the cycle repeats forever, generating a sawtooth wave of 1745Hz.​

Homework Equations

Vcharge / VCC = 1 - [ e^(-t/RC)]

The Attempt at a Solution

Considering the cap took some unknown time to first charge up to 1.1V, but a known time (160μs) to charge from there to 1.68V, I eliminated (offset) this first unknown charge and its time, and also reduced it from Vcc and started all over from zero.

So, what I have now:

Vcc = (5V - 1.1V) = 3.9V
VcapInitial = 1,10V - 1,1V = 0V
VcapFinal = 1.68V - 1.1V = 0.58V
time = 160μs (from VcapInitial to VcapFinal)
Capacitor = 100nF
Resistor = 10kΩ​

Then, the math:

0.58 / 3.9 = 1 - [e^(-t/RC]
1 - 0.1487 = e^(-t/RC)
-t/RC = Ln(0.8512)
-0.00016 / RC = -0.1610
RC = 0.00016 / 0.1610 = 0.0009937
R = 0.0009937 / 0.0000001 = 9937Ω​

So far so good?
I accept the Resistor as 9937 instead of 10000, "physical variables measuring errors".

Now, what I have for the discharging time:

VcapInitial = 1.68V
VcapFinal = 1.10V
VcapDelta = 0.58V
time to discharge from VcapInitial to VcapFinal = 413μs
Capacitor = 100nF
Resistor = 10kΩ​

Math:

0.58 / 1.68 = 1 - [e^(-t/RC]
1 - 0.3452 = e^(-t/RC)
-t/RC = Ln(0.6548)
-t = -0.4234 * RC
RC = -0.000413 / -0.4234
RC = 0.0009754
R = 0.0009754 / 0.0000001
R = 9754 Ω​

Hmmm, not very close to 10k, but I will take in consideration several possible lab measurement errors.

Question? All i did was correct? is this the way?

The first assumption

So far I have, charge time:160μs, discharge time:413μs. May I assume that the ratio (413μs/160μs) = 2.58 as somehow a constant for different values of the Resistor?

Okay, having the lab available, right now I replaced the 10kΩ resistor for a 2kΩ one, the charge time now is 31.9μs and discharge is 82.3μs, with a ratio of 2.579, wow, the assumption is true, valid and verified. Need to confirm with a precise Frequency/Period meter, and much more resistors and capacitors.

What I wish I could calculate

How can I transform all this data into a simple formula, where the following is constant:

• VCC = 5V (will never change)
• Vcap1 = 1.1V (triangle wave bottom voltage = will never change)
• Vcap2 = 1.68V (triangle wave top voltage = will never change)
• Ratio Discharge/Charge = 2.58 (I will assume that as valid)
• Oscillation time = 3.58 * n

and the following may change:

• R
• C
• n (as Charging time)

So, R=10kΩ, C=100nF, f=1745, period=573μs, n=period/3.58, n=160μs.

But I can't see how to use R, C, 5V, 1.1V and 1.68V and end up with 160μs

All my knowledge about the above math I did learn tonight, producing this post.
What I wish I could do, is get to a small formula where with RC you get "n".

Okay, being bold I tried pure arithmetics.
RC = 100nF * 10k = 0.001
The charge time (n) = 0.00016
A ratio between them = 6.25, it could means nothing.

Now, somewhere above I replaced the resistor by a 2k and produced a charging time of 31.9μs, right?
now RC = 100nF * 2k = 0.0002
0.0002 / 0.0000319 = 6.27, it could means nothing, but it pretty close to the 6.25 above.

This is getting to close to 2∏, and I am starting to think...

n = RC / 2∏
n = 10k * 100nF / 6.28
n = 0.001 / 6.28 = 159.1μs
t = 1 / 3.58 * n = 1755 Hz.
That is pretty close to what I have on scope's screen.

The formula f = 2∏/ RC*3.58 = 1.755/RC, fits like a glove, and it has nothing to do with the voltages, charge and discharge times... perhaps I already grossly found it?

1.755 / 10k * 100nF = 1755 Hz, I got 1745 on scope
1.755 / 2k * 100nF = 8770 Hz, I got 8790 on scope

Found out 2kΩ resistor is in real 1976Ω, 10kΩ is 10060Ω, what somehow justify the such differences on scope.

Is there any rational and logical way to get to this simple f = 1.755/RC ?

Any help will be very much appreciated.
Thank you for allowing me in this forum and for this first post.

Regis Tro
Orlando Florida USA

 

[gneill]

Hi Regis Tro, Welcome to PF.

You've already done most of the math required to find a simplified expression for the frequency. For a given cycle of the waveform you've found the time to charge (call it Δt1) and the time to discharge (call it Δt2). The period of a cycle is then T = Δt1 + Δt2. Frequency is f = 1/T.

If you assign variable names to all your constants and carry them through the calculations symbolically, you should be able to arrive at an expression for T that involves all the parameters. The time constant $\tau$ should be easily extracted from this leaving an expression that, with the given real-life values of the other parameters, should yield your desired multiplier for your period and frequency expression.

You might want to represent the IC's actual upper and lower output voltages as variables, since not every IC will "hit the rails" with its output.

EDIT: PS: You might want to examine the data sheet for your 74HCT14 Schmitt-Trigger. They provide a simple RC oscillator configuration and a suggested formula for determining the frequency