# RC Oscillator Circuit

RC Oscillators use a combination of an amplifier and an RC feedback network to produce output oscillations due to phase shift between stages.

In amplifier training, we found that a single-stage transistor amplifier can produce 180o phase shifts between output and input signals when connected as a common emitter type amplifier, and is connected to the input signal injected completely into the output signal along the collector load.

However, we can configure the transistor stages to work as oscillators by placing resistance-capacitor (RC) networks around the transistor to provide the necessary regenerative feedback without the need for a tank circuit. Frequency selective RC coupling amplifier circuits are easy to make and can be oscillated at any desired frequency by selecting the appropriate resistance and capacitance values.

In order for an RC oscillator to maintain its oscillations indefinitely, adequate feedback of the correct phase, i.e. positive (in-phase) Feedback, must be provided in contingent with the voltage gain of the single transistor amplifier used to inject sufficient loop gain into the closed circuit. Loop circuit to protect oscillations that allow it to be continuously released at the selected frequency.

In an RC Oscillator circuit, the input is shifted back to 180o and again to 180 o through an inverted amplifier stage during the feedbackcircuit, which rotates the signal out of phase to produce the necessary positive feedback. This gives us the "180o + 180o = 360o " phase shift, which is then the same as 0o,and thus gives us the necessary positive feedback. In other words, the total phase shift of the feedback loop must be any multiple of "0o" or 360o toachieve the same effect.

In the Resistance-Capacitance Oscillator briefly in the RC Oscillator, we can take advantage of the fact that a phase shift occurs using interconnected RC elements in the feedback branch between the input to an RC network and the output from the same network, for example:

The circuit on the left indicates a single network of resistance-capacitors that "direct" the output voltage input voltage at an angle of less than 90o. In a pure or ideal unipolar RC network. Precisely 90o maximum phase shift will produce and at least two unipolar networks should be used in the design of an RC oscillator, since 180o phase shifts are required for oscillation.

But in reality it is difficult to achieve exactly 90o phase shifts for each RC stage, so together we must gradually use more RC stages to achieve the necessary value in the oscillation frequency. The actual amount of phase shift in the circuit depends on the values of resistance (R) and capacitor (C) at the selected oscillation frequency, and the phase angle (φ) is given as follows:

Where: XC is capacitive reactance of capacitor, R is resistance of resistance and ε is Frequency.

In our simple example above, the R and C values are selected to direct the output voltage input voltage at an angle of about 60oat the required frequency. Then, the phase angle between each successive RC section increases by 60o, giving a phase difference of 180 o (3 x 60o)between input and output, as shown inthe following vector diagram.

Thus, by serially combining three such RC networks, we can produce a total phase shift of 180oin the circuit at the selected frequency, which forms the basis of an "RC Oscillator" known as the Phase Shift Oscillator when the phase angle is shifted. Then, phase shift occurs with the phase difference between the individual RC stages. Suitable op-amp circuits are available in quad integrated packages. For example, LM124 or LM324, etc. can also use all four RC stages to produce the required 180o-phase shifts at the required oscillation frequency.

We know that in an amplifier circuit using bipolar transistor or reversing opamp configuration, it will produce a phase shift of 180obetween input and output. If a three-stage RC phase shift network is connected as a feedback network between the output and input of an amplifier circuit, the total phase shift created to produce the necessary regenerative feedback is shown as: 3 x 60o + 180o = 360o = 0o.

Three RC stages are cascaded together to achieve 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 jω = 2piε = 1/1.732RC (tan 60o = 1.732). Next, it is necessary to use multiple RC phase shift networks, such as the circuit below, to achieve the required phase shift in an RC oscillator circuit.

The basic RC Oscillator, also known as a phase shift oscillator, produces a sinus wave output signal using regenerative feedback from the resistance-capacitor (RC) stair network. This regenerative feedback from the RC network is due to the capacitor's ability to store an electrical charge (similar to the LC tank circuit).

This resistance-capacitor feedback network can be connected as shown above to produce a leading phase shift (phase progression network) or modified to produce a delayed phase shift (phase delay network).

By changing one or more resistance or capacitors in the phase shift network, the frequency can be changed, and usually this is done using a 3-way variable capacitor, as it keeps the resistors the same and changes with capacitive reassurance (XC).

If the three resistances are equal in R value, that is, R1 = R2 = R3 and capacitors are equal in the value in the phase shift network, C1 = C2 = C3, then the frequency of oscillations produced by RC is simply given as follows:

Where:
εr is the oscillator output frequency in Hertz
R, the feedback resistance in Ohm is
C, the feedback capacitance in Farad is
the number of RC feedback stages.

This is the frequency at which the phase shift circuit is released. In our simple example above, the number of stages is given as three, that is, N = 3 (√(2*3) = √6). For a four-stage RC network, N = 4 (√(2*4) = √8),

Since the resistance-capacitor combination in the RC Oscillator ladder network also acts as a debilitating, that is, the signal decreases somewhat each time it passes through passive stage. It can be assumed that the three phase scrolling sectors are independent of each other, but this is not the case as the total accumulated feedback weakening is -1/29th (Vo/Vi = β = -1/29) in all three stages. Therefore, the voltage gain of the amplifier should be high enough to overcome these passive RC losses. Clearly then, in order to produce a total of -1 cycle gain in our above three-stage RC network, amplifier gain must also be equal to or greater than 29 to compensate for the weakening of the RC network.

The loading effect on the amplifier's 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 operated from a high impedance output source and fed with a low impedance load, such as a common emitter transistor amplifier, but better is to use a Transactional Amplifier, opamp, as it perfectly meets these conditions.

## RC Oscillator Circuit Using OPAMP

When RC oscillators are used, RC Oscillators used in Operational Amplifiers are more common than their bipolar transistor counterparts. The oscillator circuit consists of a negatively gained operational amplifier and a three-part RC network that produces 180o phase shifts. The phase scroll network connects from the op-amp output to the "invert" input, as shown below.

Since the feedback depends on the inverted input, the transactional amplifier therefore connects to the "inverter amplifier" configurationthat produces the required 180 o phase shift, while the RC network produces another 180 o phase shifts at the required frequency (180o + 180o). This type of feedback connection, made with serially connected capacitors and resistances due to soil (0V) potential, is known as phaseconfiguration. In other words, the output voltage leads to the input voltage, which produces a positive phase angle.

However, we can also create a phase-delayed configuration by simply changing the positions of RC components so that the resistors are serially connected and the capacitors are connected to the soil (0V) potential as shown. This means that the output voltage lags behind the input voltage and produces a negative phase angle.

### Phase Delayed OPAMP RC Oscillator Circuit

However, due to the reversal of the feedback components, the original equation for the frequency output of the RC oscillator is changed as follows:

Although it is possible to cascade only two unipolar RC stages together to ensure the required 180o phase shift (90o + 90o), the stability of the oscilator at low frequencies is usually poor.

One of the most important features of an RC Oscillator is its frequency stability, which is capable of providing constant frequency sinus wave output under changing load conditions. By cascading together three or even four RC stages (4 x 45o), the stability of the oscilator can be greatly improved.

Four-stage RC Oscillators are often used because commonly available transactional amplifiers come in quad IC packages, so it is relatively easy to design a 4-stage oscillator with 45o phase shifts relative to each other.

RC Oscillators are stable and provide a well-formed sinus wave output with a frequency proportional to 1/RC, and therefore a wider frequency range is possible when using a variable capacitor. However, RC Oscillators are limited to frequency applications due to bandwidth limitations to produce desired phase shift at high frequencies.

### RC Oscillator Example 1

A 3-stage RC Phase Shift Oscillator based on a transactional amplifier is required to produce a 4 kHz sinusoidal output frequency. If 2.4nF capacitors are used in the feedback circuit, calculate the value of frequency determination resistors and the value of the feedback resistance required to maintain oscillations. Also draw the circuit.

The standard equation for the phase-shifting RC Oscelator is:

The circuit will therefore be a 3-stage RC oscitor consisting of equal resistors and three equal 2.4nF capacitors. Since the oscillation frequency is given as 4.0kHz, the value of the resistors is calculated as follows:

Transactional amplifier gain must be equal to 29 to maintain oscillations. The resistance value of the oscillation resistors is 6.8kΩ, so the value of the op-amp feedback resistance Rε is calculated as follows: