The frequency of oscillations of a phase shift oscillator is :
Phase Shift Oscillator Frequency: Understanding the Formula
A phase shift oscillator is an electronic circuit that generates a sinusoidal output waveform. It is composed of an amplifying device (like an operational amplifier or a transistor) and a feedback network, which typically consists of resistors (R) and capacitors (C). The feedback network provides the necessary phase shift to sustain oscillations.
Oscillator Principles: Barkhausen Criterion
For an oscillator to produce sustained oscillations, it must satisfy the Barkhausen criterion. This criterion has two main conditions:
- Loop Gain Condition: The magnitude of the loop gain (\(A\beta\)) must be equal to or greater than unity (1). Here, \(A\) is the gain of the amplifier and \(\beta\) is the gain of the feedback network. For stable oscillations, it is typically set to exactly 1.
- Phase Shift Condition: The total phase shift around the feedback loop must be 0 degrees or an integer multiple of 360 degrees.
In a typical RC phase shift oscillator, the amplifier often provides a 180-degree phase shift (for example, if it's an inverting amplifier configuration). To meet the Barkhausen phase shift condition, the RC phase shift network must provide the remaining 180 degrees of phase shift.
RC Phase Shift Network Analysis
A common configuration for an RC phase shift oscillator uses three cascaded RC stages. Each RC stage contributes a certain amount of phase shift, and together they are designed to provide a total of 180 degrees of phase shift at the desired oscillation frequency. Each stage, at the oscillation frequency, provides approximately 60 degrees of phase shift, summing up to 180 degrees for three stages.
Frequency of Oscillation Formula
For a standard three-stage RC phase shift oscillator where all resistors (\(R\)) are equal and all capacitors (\(C\)) are equal in the phase shift network, the frequency of oscillations (\(f\)) is given by the following formula:
\[f = \frac{1}{2\pi RC\sqrt{6}}\]
Let's break down the components of this formula:
- \(f\): Represents the frequency of oscillation, measured in Hertz (Hz).
- \(\pi\): Is the mathematical constant pi, approximately 3.14159.
- \(R\): Is the resistance of each resistor in the RC network, measured in Ohms (\(\Omega\)).
- \(C\): Is the capacitance of each capacitor in the RC network, measured in Farads (F).
- \(\sqrt{6}\): Is the square root of 6, which is approximately 2.449. This term arises from the derivation of the phase shift and gain requirements for the three-stage RC network.
Comparing Options for Phase Shift Oscillator Frequency
Let's compare the derived formula with the given options to identify the correct one for the frequency of oscillations of a phase shift oscillator:
| Option | Expression | Analysis |
|---|---|---|
| 1 | \(\frac{1}{2\pi \sqrt{6RC}}\) | This expression is incorrect. The \(\sqrt{6}\) term should be outside the square root and multiply \(RC\), not be under the square root with \(RC\). |
| 2 | \(\frac{1}{2\pi \sqrt{RC}}\) | This expression is incorrect. It lacks the \(\sqrt{6}\) factor and misplaces the square root over \(RC\). |
| 3 | \(\frac{1}{2\pi RC}\) | This expression is incorrect. It represents the cut-off frequency of a simple RC circuit but does not include the \(\sqrt{6}\) factor specific to a phase shift oscillator's oscillation frequency. |
| 4 | \(\frac{1}{2\pi RC\sqrt{6}}\) | This expression is the correct and standard formula for the frequency of oscillation of a three-stage RC phase shift oscillator. |
Based on the standard formula for an RC phase shift oscillator, the frequency of oscillations is indeed \(\frac{1}{2\pi RC\sqrt{6}}\).
Important Questions from Oscillators
In an RC phase shift oscillator, the phase of the feedback voltage is shifted by ______ with a three stage RC phase shift network
Which is a fixed frequency oscillator?
Oscillators operate on the principle of
- The frequency of oscillation of a Hartley Oscillator is given as :
- In the circuit for relaxation oscillator, with R BB = 5 kΩ, η = 0.6, V v= 1 V, I v= 10 mA and I p= 10 μA, the value of R B1 is:
