amplifiercapacitoroscillatorresistors
I am trying to design a phase shift oscillator, using a CE Amplifier.
I have a Vc of approximately 6V, gain of approximately 40(>29, as required).
I aim for an oscillation frequency of 1.5 kHz.
But whenever I feedback the CR network, no oscillations take place. Can anyone tell me what is wrong with the circuit?
simulate this circuit – Schematic created using CircuitLab
I think that writing the loop equations would be easier.
The loop equations for first two loops: $$I_1(Z+R) = V_{in}$$ $$- I_1R +I_2(Z+2R) -I_3R = 0$$
Where \$Z=\dfrac{1}{Cs}\$. From this:
$$I_2(Z+2R) -I_3R = V_{in}\frac{R}{R+Z}\tag1$$ The remaining two loop equations: $$- I_2R +I_3(Z+2R) -I_4R = 0\tag2$$ $$- I_3R +I_4(Z+2R) = 0\tag3$$
Expressing in matrix form:
$$\left[\begin{array}{ccc} &Z+2R &-R &0 \\ &-R &Z+2R &-R \\ &0 &-R & Z+2R \end{array}\right] \left[\begin{array}{c} I_2\\ I_3\\ I_4 \end{array}\right] = \left[\begin{array}{c} \frac{V_{in}R}{R+Z}\\ 0\\ 0 \end{array}\right]$$
Now by Cramer's rule: $$ I_4 = \frac{\left|\begin{array}{ccc} Z+2R &-R &\frac{V_{in}R}{R+Z} \\ -R &Z+2R &0 \\ 0 &-R & 0 \end{array}\right|}{\left|\begin{array}{ccc} Z+2R &-R &0 \\ -R &Z+2R &-R \\ 0 &-R & Z+2R \end{array}\right|}$$
$$V_{out} = I_4\times R$$
From this the transfer function can be calculated. Gain and phase shift can be calculated from transfer function. (substitute \$Z=\frac{1}{jwC}\$)
The crystal + 2 caps form a PI network:
simulate this circuit – Schematic created using CircuitLab
Lets examine the loop gain and loop phaseshift; here in Signal Chain Explorer, the XTAL model is embedded as Z(feedback) of the Inverting OpAmp stage; crystal inductance is 58.9766 Henry; capacitance series/resonance is 0.8pf; the plate-to-plate capacitance is also 0.8pF; Rloss is 1Kohm.
I reduced the two capacitors of the PI network, here before and after the OpAmp, from 22pF to 4.7pF. I've read the low frequency oscillators need small capacitors, so the energy is not shorted to GND.
Key point is the gain and bandwidth of the OpAmp. With gain of 10dB (3.16x), shown in the far right BODE, the gain is barely above 0dB.
In contrast, examine the 40dB gain, flat to 100KHz, and see a healthy peak (30+dB gain) at the correct frequency.
Does the value of R4 (in the schematic) matter? Lets examine that next.
Best Answer
Works for me, if you make a few tweaks to the circuit:
Adding R7 and C4 let you get more gain out of the circuit, while keeping the DC operating point approximately the same. Your circuit wasn't oscilating because the collector of Q1 is "loaded down" by the parallel impedance of the phase shift network, therefore the actual gain of the stage was significantly less than the standard \$\frac{R_c}{R_e}\$ formula would indicate. Most analyses of the phase shift oscillator assume that the output of the gain stage is low impedance and isn't affected by the phase shift network, and also that the phase shift stages themselves don't load each other.
Sometimes when simulating an oscillator circuit it's also necessary to start the supply voltage at 0 volts, to "kick start" the oscillation, as I've done here.