# The RC Oscillator Circuit

In the amplifiers tutorial we saw that a single stage amplifier will produce 180o 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 180o through the amplifier stage and 180o again through a second inverting stage giving us “180o + 180o = 360o” of phase shift which is effectively the same as 0o 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.

### RC Phase-Shift Network

The circuit on the left shows a single resistor-capacitor network whose output voltage “leads” the input voltage by some angle less than 90o. An ideal single-pole RC circuit would produce a phase shift of exactly 90o, and because 180o 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 90o 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:

### RC Phase Angle

Where: XC 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 60o. Then the phase angle between each successive RC section increases by another 60o giving a phase difference between the input and output of 180o (3 x 60o) as shown by the following vector diagram.

### Vector Diagram

Then by connecting together three such RC networks in series we can produce a total phase shift in the circuit of 180o 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 180o 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 60o + 180o = 360o 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 -180o when the phase shift of each stage is -60o. This occurs when ω = 2πƒ = 1.732/RC as (tan 60o = 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.

### Basic RC Oscillator Circuit

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 360o.

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:
• ƒr  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.

## The Op-amp RC Oscillator

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 180o phase shift. The phase shift network is connected from the op-amps output back to its “inverting” input as shown below.

### Op-amp RC Oscillator Circuit

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

Although it is possible to cascade together only two single-pole RC stages to provide the required 180o of phase shift (90o + 90o), 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 45o), 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 45o 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.

## RC Oscillator Example No1

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 Rf is calculated as:

### RC Oscillator Op-amp Circuit

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.