In our series of tutorials about Amplifiers , 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, ie, “Positive Feedback” must be provided with the amplifier being used as one 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:

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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Is it possible to get triangular output for RC PHASE SHIFT OSCILLATOR using OP-AMP???? I got triangular output. But i dont know whether its correct or not… plz reply me its very urgent….

You will get triangular output in case the frequency is too high. In this case it is the limited SLEW RATE of the opamp that does not allow a sinusoidal output.

More than that, the frequency is not as expected.

Try to use another opamp with a larger slew rate – for example in current-feedback topology.

thank u so much!!!!

relation value between capacitor,resistor & frequency.

The shown RC oscillator circuit with opamp will not work.

It is wrong. The most right grounded resistor R must NOT be connected to ground.

Instead it must be connectet to the inverting terminal of the opamp.

I have to correct my former statement as follows:

The circuit will work – however not as expected.

The grounded R of the right most CR section can be omitted because it is connected to a virtual ground node.

Thus, we have to CR highpass sections and on active inverting differentiator which contributes -90 deg to the phase shift. Hence, there will be an oscillator frequency at a point where both CR sectons contribute in total a phase shift of +90 deg. This satisfies the oscillation condition: Loop phase of zero deg.

that was helpful

thanks frd

hey please tell me how to plot the curve v(output)=f(v(input))in the range v(output)= 0,…,12v for wienbridge and phaseshift oscillators

Thanks. Helped a lot. ðŸ™‚

What is the reason why the application of RC Phase Oscillators is limited to generation of Audio Frequency(20Hz to 20kHz) signals?

The capacitor as low reactance for high freq. oscillations. A high frequency causes the 3 capacitors to get shorted – It provides no impedance to the current flow at all. Thus no phase shift occurs and hence the RC oscillator becomes useless

please,i’d like u to post the “equivalent circuit of the transistor and feedback network of the phase-shift oscillator” (not that of the operational amplifier-Opam) and to use it to find the frequency of the phase-shift oscillator. thanks.

please can u give me the exact part No. of the Transistor, especially the common type?

You can use an n-p-n BC-547/BC-548 transistor

Great article, I think the author perhaps made a mistake on the Op-amp RC Oscillator by stating that the feedback is connected to the non-inverting input. It is in fact connected to the inverting input which causes an addition 180 phase shift for a total of 360.

Thanks Pablo, changed the design but forgot to update the text.