I'm trying to find the output expression of a relaxation oscillator in the Laplace domain, to then convert it to the time domain to be able to trace its output. However, my circuit analysis isn't giving me an expression for
\$V_o\$. Here is the circuit that I am trying to analyze as well as the work I've done so far.
simulate this circuit – Schematic created using CircuitLab
\begin{align}
V^- &= \frac{\frac{0}{\frac{1}{Cs}}+ \frac{V_o}{R}}{\frac{1}{\frac{1}{Cs}}+ \frac{1}{R}} = \frac{V_o}{CRs+1} \tag{Millmans}\\[0.7em]
V^+ &= \frac{\frac{0}{6k}+ \frac{V_o}{6k}}{\frac{1}{6k}+ \frac{1}{6k}} = \frac{V_o}{2} \tag{Millmans} \\[0.7em]
V^+ &= V^- \\[0.7em]
\frac{V_o}{2} &= \frac{V_o}{CRs+1} \\[0.7em]
1 &= \frac{2}{CRs+1}\\
\end{align}
As you can see, \$V_o\$ gets canceled.
Could someone please help me find where I am going wrong?
Thanks.
Best Answer
Ouch! The opamp is used in non-linear mode. Linear opamp circuit calculation practices nor transfer functions are not applicable for it.
You should divide this to a Schmitt-trigger and a charging and discharging RC circuit. Then you can find how long it takes for the RC circuit to charge and discharge between the tresholds of the Schmitt-trigger.
ADD you have explained the idea of your calculation properly in the comments. You wrote an equation for the condition that the opamp regenerates from the feedback voltages whatever it happens to output and stays at the same time in linear mode. The equation has two solutions
1) the output voltage Vo = 0 constantly
2) s=1/(RC)
Solution 1 is the highly unstable balance where an ideal opamp would give 0V. It's like a needle standing on its sharp tip - a theoretical only possibility.
Refining solution 2 in s-domain is beyond my math knowledge, but it obviously presents something which allows a class of infinitely growing exponential functions. It's not an explicit formula for the output voltage but a criteria for possible voltage functions just like the oscillation criteria of a sinewave oscillator would give the possible oscillation frequency or frequencies.
The explanation for solution 2 can be found easily in time domain as a limit process where the gain of the opamp grows to infinite. The possible solutions for Vo are all exponentially growing voltages A*exp(t/RC) where A is the intial value - positive or negative or zero. Due the high gain a little noise in the input kicks the calculated Vo beyond the clipping limit, so it's purely theoretical.