In the amplifiers tutorial we saw that a single stage amplifier will produce 180^{o} of phase shift between its output and input signals when connected in a class-A type configuration.

For an oscillator to sustain oscillations indefinitely, sufficient feedback of the correct phase, that is “Positive Feedback” must be provided along with the transistor amplifier being used acting as an inverting stage to achieve this.

In an **RC Oscillator** circuit the input is shifted 180^{o} through the amplifier stage and 180^{o} again through a second inverting stage giving us “180^{o} + 180^{o} = 360^{o}” of phase shift which is effectively the same as 0^{o} thereby giving us the required positive feedback. In other words, the phase shift of the feedback loop should be “0”.

In a **Resistance-Capacitance Oscillator** or simply an **RC Oscillator**, we make use of the fact that a phase shift occurs between the input to a RC network and the output from the same network by using RC elements in the feedback branch, for example.

The circuit on the left shows a single resistor-capacitor network whose output voltage “leads” the input voltage by some angle less than 90^{o}. An ideal single-pole RC circuit would produce a phase shift of exactly 90^{o}, and because 180^{o} of phase shift is required for oscillation, at least two single-poles must be used in an *RC oscillator* design.

However in reality it is difficult to obtain exactly 90^{o} of phase shift so more stages are used. The amount of actual phase shift in the circuit depends upon the values of the resistor and the capacitor, and the chosen frequency of oscillations with the phase angle ( Φ ) being given as:

Where: X_{C} is the Capacitive Reactance of the capacitor, R is the Resistance of the resistor, and ƒ is the Frequency.

In our simple example above, the values of R and C have been chosen so that at the required frequency the output voltage leads the input voltage by an angle of about 60^{o}. Then the phase angle between each successive RC section increases by another 60^{o} giving a phase difference between the input and output of 180^{o} (3 x 60^{o}) as shown by the following vector diagram.

Then by connecting together three such RC networks in series we can produce a total phase shift in the circuit of 180^{o} at the chosen frequency and this forms the bases of a “phase shift oscillator” otherwise known as a **RC Oscillator** circuit.

We know that in an amplifier circuit either using a Bipolar Transistor or an Operational Amplifier, it will produce a phase-shift of 180^{o} between its input and output. If a three-stage RC phase-shift network is connected between this input and output of the amplifier, the total phase shift necessary for regenerative feedback will become 3 x 60^{o} + 180^{o} = 360^{o} as shown.

The three RC stages are cascaded together to get the required slope for a stable oscillation frequency. The feedback loop phase shift is -180^{o} when the phase shift of each stage is -60^{o}. This occurs when ω = 2πƒ = 1.732/RC as (tan 60^{o} = 1.732). Then to achieve the required phase shift in an RC oscillator circuit is to use multiple RC phase-shifting networks such as the circuit below.

The basic **RC Oscillator** which is also known as a **Phase-shift Oscillator**, produces a sine wave output signal using regenerative feedback obtained from the resistor-capacitor combination. This regenerative feedback from the RC network is due to the ability of the capacitor to store an electric charge, (similar to the LC tank circuit).

This resistor-capacitor feedback network can be connected as shown above to produce a leading phase shift (phase advance network) or interchanged to produce a lagging phase shift (phase retard network) the outcome is still the same as the sine wave oscillations only occur at the frequency at which the overall phase-shift is 360^{o}.

By varying one or more of the resistors or capacitors in the phase-shift network, the frequency can be varied and generally this is done by keeping the resistors the same and using a 3-ganged variable capacitor.

If all the resistors, R and the capacitors, C in the phase shift network are equal in value, then the frequency of oscillations produced by the RC oscillator is given as:

- Where:
- ƒ
_{r}is the Output Frequency in Hertz - R is the Resistance in Ohms
- C is the Capacitance in Farads
- N is the number of RC stages. (N = 3)

Since the resistor-capacitor combination in the **RC Oscillator** circuit also acts as an attenuator producing an attenuation of -1/29th ( Vo/Vi = β ) per stage, the gain of the amplifier must be sufficient to overcome the circuit losses. Therefore, in our three stage RC network above the amplifier gain must be greater than 29.

The loading effect of the amplifier on the feedback network has an effect on the frequency of oscillations and can cause the oscillator frequency to be up to 25% higher than calculated. Then the feedback network should be driven from a high impedance output source and fed into a low impedance load such as a common emitter transistor amplifier but better still is to use an Operational Amplifier as it satisfies these conditions perfectly.

When used as RC oscillators, **Operational Amplifier RC Oscillators** are more common than their bipolar transistors counterparts. The oscillator circuit consists of a negative-gain operational amplifier and a three section RC network that produces the 180^{o} phase shift. The phase shift network is connected from the op-amps output back to its “inverting” input as shown below.

As the feedback is connected to the inverting input, the operational amplifier is therefore connected in its “inverting amplifier” configuration which produces the required 180^{o} phase shift while the RC network produces the other 180^{o} phase shift at the required frequency (180^{o} + 180^{o}).

Although it is possible to cascade together only two single-pole RC stages to provide the required 180^{o} of phase shift (90^{o} + 90^{o}), the stability of the oscillator at low frequencies is generally poor.

One of the most important features of an **RC Oscillator** is its frequency stability which is its ability to provide a constant frequency sine wave output under varying load conditions. By cascading three or even four RC stages together (4 x 45^{o}), the stability of the oscillator can be greatly improved.

*RC Oscillators* with four stages are generally used because commonly available operational amplifiers come in quad IC packages so designing a 4-stage oscillator with 45^{o} of phase shift relative to each other is relatively easy.

**RC Oscillators** are stable and provide a well-shaped sine wave output with the frequency being proportional to 1/RC and therefore, a wider frequency range is possible when using a variable capacitor. However, RC Oscillators are restricted to frequency applications because of their bandwidth limitations to produce the desired phase shift at high frequencies.

A *3-stage RC Phase Shift Oscillator* is required to produce an oscillation frequency of 6.5kHz. If 1nF capacitors are used in the feedback circuit, calculate the value of the frequency determining resistors and the value of the feedback resistor required to sustain oscillations. Also draw the circuit.

The standard equation given for the phase shift RC Oscillator is:

The circuit is to be a 3-stage RC oscillator which will therefore consist of three resistors and three 1nF capacitors. As the frequency of oscillation is given as 6.5kHz, the value of the resistors are calculated as:

The operational amplifiers gain must be equal to 29 in order to sustain oscillations. The resistive value of the three oscillation resistors are 10kΩ, therefore the value of the op-amps feedback resistor R_{f} is calculated as:

In the next tutorial about Oscillators, we will look at another type of **RC Oscillator** called a Wien Bridge Oscillators which uses resistors and capacitors as its tank circuit to produce a low frequency sinusoidal waveform.

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Please mention on this oscillator working

Produces phase shift. The phase shift network is connected from the op amp out put feedback

Unlabelled identity’s made the algebra hard to follow. What does Xc refer too?

Xc is the Capacitive Reactance of the capacitor

Whole article is superb. But my question is why we need 3 RC stages only?

The circuit Is a feed back amp with a three stage RC network, my question is what starts the oscillations, is it external radio wave noise, if so then RC are acting as the frequency tuned circuits and would an aerial make any improvement.

Thank you for the very interesting article.

An RC oscillator does not use an aerial.

I need the derivation of the opamp phase shift oscillator..

Thank u so much for this outstanding article , it helped me a lot before the exam day, bless you .

Outstanding!!

Informative Articles.

Detailed explanation

There should be also some features like ,Circuit Picture in GIF .

This site is really informative ?

It is very useful .the explanation of one topic is very nice .I like it very much