In an RC phase shift oscillator the frequency of oscillation is given by
\(\dfrac{0.065}{RC}\)
The standard result for a three-section RC phase-shift oscillator is
\(f=\dfrac{1}{2\pi RC\sqrt{6}}\)
and evaluating the constant gives the answer in the form the options use:
\(\dfrac{1}{2\pi\sqrt{6}}=\dfrac{1}{2\times3.1416\times2.4495}=\dfrac{1}{15.39}=0.0650\)
\(f=\dfrac{0.065}{RC}\)
which is option 2.
Where the \(\sqrt{6}\) comes from. The Barkhausen criterion requires the loop phase shift to be 360°. The inverting amplifier supplies 180°, so the RC network must supply the other 180°. A single RC section can approach but never reach 90°, so at least three sections are needed, each contributing about 60°. Working out the transfer function of three cascaded sections,
\(\beta=\dfrac{1}{1-5\alpha^{2}-j\left(6\alpha-\alpha^{3}\right)}\qquad\text{where}\ \alpha=\dfrac{1}{\omega RC}\)
Setting the imaginary part to zero gives \(\alpha^{2}=6\), hence \(\omega=\dfrac{1}{RC\sqrt{6}}\).
The gain condition follows from the same algebra. Substituting \(\alpha^{2}=6\) into the real part gives \(\beta=-\dfrac{1}{29}\), so the amplifier must supply
\(\left|A\right|\ge29\)
— the famous figure that goes with this oscillator, and the second half of the Barkhausen condition \(\left|A\beta\right|=1\).
| Oscillator | Frequency | Gain needed |
|---|---|---|
| RC phase shift | \(\dfrac{1}{2\pi RC\sqrt{6}}\) | 29 |
| Wien bridge | \(\dfrac{1}{2\pi RC}\) | 3 |
Option 4 is the trap — \(\dfrac{1}{2\pi RC}\) is the Wien bridge frequency, where a lead-lag network gives zero phase shift and a non-inverting amplifier of gain 3 completes the loop. Mixing the two formulas up is the commonest error in this topic. Option 1, 0.65/RC, is simply the correct constant with the decimal point misplaced by a factor of ten.
Hence, the frequency of oscillation is 0.065/RC.
Similar Questions
The phase locked loop (PLL) is one of the interesting applications of the lock-in amplifier. Apart from FM stereo decoders, tracking filters, motor speed control, FM demodulators, etc. it has found wide applications in generation of local oscillator frequencies in house-hold TV and FM tuners as automatic frequency control (AFC). Indeed, PLL has emerged as one of the fundamental building blocks in electronics and it is commercially available as a single package. Basically, a PLL is a lock-in amplifier in which the reference signal is provided by its own output, converted to frequency by a voltage controlled oscillator (VCO). When locked to the input frequency the dc output is small but sufficient to drive the VCO to produce a frequency which is equal to that of the signal. In this tracking situation, the input signal and the VCO output are almost in phase quadrature and the lock-in amplifier produces a small dc voltage which is often referred to as error voltage. The moment input signal is fed, the VCO frequency starts changing and the PLL is said to be in the capture mode. The VCO continues to change its frequency until it equals that of the input and stays there ; the PLL is then in the phase-locked state. In this state, if there is any change in the input frequency, the loop automatically tracks it through its repetitive action.
Consider the following statements regarding an RC phase shift oscillator :
i. amplifier gain is positive.
ii. amplifier gain is negative.
iii. phase shift introduced by the feedback network is 180°.
iv. phase shift introduced by the feedback network is 360°.Which is correct ?
Assertion (A) : In applications such as FM and FSK, VCO plays an important role.
Reason (R) : The frequency control is easily possible by varying d.c. voltage.
Which of the following oscillations makes use of both positive and negative feedback ?
For a FET based phase shift oscillator, what should be the value of capacitor (C) for oscillator operation at 1 kHz. The resistor (R) in the feedback network is 20 kΩ.
The current amplification factor in radian square of Colpitts oscillator is :
The voltage controlled oscillator is used for :
The PLL is in the free-running state when :
Assertion (A) : A monostable multivibrator can be used to alter the pulse width of a repetitive pulse train.
Reason (R) : Monostable multivibrator has a single stable state.
Select your answer using the codes given below :
Important Questions from Types of Oscillators
Electronic ohmmeter uses OP-AMP as a/an:
Which of the following statements about the Wien Bridge Oscillator is CORRECT?
Hartley Oscillator is a:
Which of the following is the fixed frequency oscillator?
If R = 51 kΩ and C = 0.001 μF, the resonant frequency of a Wien Bridge oscillator is:
