Electronic – Obtaining frequency of 550 Hz from Simple Triangle and Square Waveform Generator Circuit

There are a couple of unclear things in your schematic, which should be taken care of:

1. I assume that U0 is the output voltage which you want to measure with a high-ohmic oscilloscope. The way you depicted it, U0 is a 0 V voltage source which will defeat any functionality of your circuit.
2. The depicted circuit cannot work as a limiter. U0 is a full-swing output. I think you're confusing the terms limiter and limiting amplifier. This circuit is a limiting amplifier (at least the way you use it), to be more specific it's a common emitter stage.
3. To finally answer your question about the rise time: The main problem with this circuit is that it is not properly biased. You need to define the operating point of this common emitter stage so that the output voltage is in the middle between the rails for a 0V input signal.

I hope it helps! :)

^{simulate this circuit – Schematic created using CircuitLab}

Figure 1. Flying start 555 oscillator.

I've only used a 555 twice in the last few decades but this may do the trick for you.

• C3 is connected between V+ and the timing capacitor. On power up C3 and C1 form a capacitive potential divider. If the voltage rises smartly the junction of C3 and C1 will rise to \$V_+ \frac {C_3}{C_3+C_1} \$ thus priming the capacitor for you. So, to aim for 1/2 supply set C3 = C1. Bear in mind that R1 and R2 will be discharging C3 during power up so you may have some calculations or experiments to do.
• Once the oscillator starts up the timing will depend in C1 in parallel with C3 so use value \$C = {C_1+C_3} \$ in your timing calculations. The fact that C3 is connected to V+ rather than GND won't matter.