Op-amps can be used to generate various kinds of signals and one of them is sinusoidal waveforms. Sinusoidal waveforms can be generated using operational amplifier using LC circuit. These oscillators are called op-amp based LC oscillators. They are also called Hartley or Colpitts oscillators depending upon the network configuration of the L and C components. Here it is shown how sine wave can be generated using **Colpitts oscillator** with **LM358** and **TL072** operational amplifiers. Also their performance is tested.

### Colpitts oscillator with Op-Amp

The circuit diagram of Colpitts oscillator with Op-Amp is shown below.

A practical oscillator consist of an amplifying network and a feedback network. The operational amplifier along with the gain setting resistors R1 and Rf is the **amplifier network**. The amplifier is biased for mid-voltage using the resistors R2, R3 and capacitor C3.

The **feedback network** is made using the two capacitors C1, C2 and inductor L1. The C1,C2 and L1 forms an LC tank circuit which provides the feedback necessary for oscillation. With the given values for C1, C2 and L1 the tank circuit becomes resonant at that particular frequency.

The **resonant frequency** is given by,

\(f_o = \frac{1}{2 \pi \sqrt{LC_T}}\)

where, \(C_T = \frac{C_1C_2}{C_1+C_2} \)

For example if L =220uH, C1=10nF and C2=0.1uF then the resonant frequency of oscillation is fo=112.54 KHz. The values of L1 and C1,C2 can be calculated using the online oscillator calculator.

In the above circuit, the feedback voltage is across the capacitor C1. The feedback fraction \(\beta\) is the fraction of the output voltage that is feed back into the amplifier. This **feedback fraction** \(\beta\) is given by the following equation,

\(\beta = \frac{C1}{C2}\)

With C1=10nF and C2=0.1uF,

\(\beta = \frac{C_1}{C_2}=\frac{10nF}{0.1uF}=0.1\)

In the above op-amp oscillator circuit, the gain of the amplifier is set using the resistors R1 and Rf. This operational amplifier is configured in inverting configuration and for inverting configuration, the **voltage gain** is given by the following equation.

\(A_v = \frac{-R_f}{R1}\)

For creating oscillation the following criteria must be abided by,

\(A_v \beta \geq 1\)

This condition is one of the condition of **Barkhausen criteria** for oscillation. The other condition that must be met is that the phase change from output to the input must be 0 degree or 360 degree. When the signal passe through an inverting amplifier such as the one used here, there will be phase shift of 180 degree. But the feedback network which is in fact a filter, changes the incoming signal phase by another 180 degree so that the total phase shift in the loop is 360 degree(or 0 degree). So the second criteria for oscillation is also satisfied.

Hence from the above equation we can calculate the gain required from the operational amplifier which is,

\(A_v \geq \frac{1}{\beta}\)

Substituting the value of \(\beta\) we obtained above we have,

\(A_v \geq \frac{1}{0.1}\)

that is required gain is,

\(A_v \geq 10\)

We know that the voltage gain for the inverting op-amp is,

\(A_v = \frac{-R_f}{R1}\)

and so,

\(\frac{-R_f}{R1} \geq 10 \)

or, \(R_f \geq -10 \times R1\)

Negative sign just means that the phase is 180 degree out of phase with the input signal.

Lets choose R1=560Ohm then,

\(R_f \geq 10 \times 560Ohm \geq 5.6KOhm\)

The following shows the circuit schematic with the components value.

The biasing resistors R2 and R3 has to be equal and the value of the capacitor is just that the cutoff frequency provided by the biasing circuit is much much smaller than the resonant frequency of the oscillator. The cutoff frequency of the biasing network is,

\(f_c = \frac{1}{2 \pi (R2||R3)C_3}\)

Suppose, the cutoff frequency is 0.1 that of the resonant frequency fo=112.54 KHz. Then,

fc = 0.1*fo = 0.1*112.54 KHz = 1.12KHz

and,\( R2||R3 =\frac{R_2 R_3}{R_2 + R_3}=\frac{10KOhm\times 10KOhm}{10KOhm+10KOhm}=5KOhm\)

So that,

\(1.12KHz= \frac{1}{2 \pi (5KOhm)C_3}\)

or, \(C_3= \frac{1}{2 \pi (5KOhm)(1.12KHz)}\)

that is C3 = 0.0284uF = 28nF

Here the value of C3 chosen was 0.047uF.

### Colpitts Oscillator with TL072 and LM358 Operational Amplifiers

The above op-amp oscillator circuit was tested with TL072 op-amp and LM358 op-amp.

The following shows the TL072 based Colpitts oscillator on breadboard.

The sine signal waveform of the oscillator output is shown below.

As can be seen in the oscilloscope the signal frequency is 113KHz.

The next picture shows test with LM358 op-amp. The Colpitts oscillator with LM358 on breadboard is shown below.

Again as before the LM358 is configured as inverting amplifier with biasing resistor and capacitor(see How to design a Practical LM358 Op-Amp Inverting Amplifier) and the input is feed via the LC oscillator feedback network. The signal output from the LM358 colpitts oscillator is shown below.

The signal waveform from the LM358 oscillator does not appear sinusoidal and the frequency detected is 85KHz only.

#### LM358 vs TL072 Oscillator Outputs

As can be seen from the results above, the TL072 oscillator gives sinusoidal waveform and the frequency also is near to the expected value of 112KHz. Whereas, the signal waveform from the LM358 is not sinusoidal and the frequency also does not match. This is due to the inherent Gain Bandwidth Product(GBW) and slew rate of the operational amplifiers. The LM358 has a GBW of 0.7MHz and typical slew rate of 0.3V/us. The TL072 has a GBW of 3MHz and slew rate of 13V/us. So both the GBW and slew rate of TL072 is higher than LM358. Also TL072 or TL074 have lower noise compared to LM358.

So the frequency of operation is critical in selecting operational amplifier. To prove this, lets say we want to use the LM358 operational amplifier. The LM358 should work with lower frequency. Let's select the resonant frequency of 50KHz(half of what was used above). Then keeping the capacitor value the same, the inductor value required is 1.11mH. Let us use inductor with practical value of 1mH. Then the signal output of the oscillator on oscilloscope is shown below.

The oscilloscope shows better sine wave like signal and the frequency is 46KHz instead of 50KHz but this is nearer than at higher frequency experiment above. This frequency difference may be because for 50KHz 1.11mH inductor was required but we used 1mH and there is some parasitic effect of the breadboard. But we can say there is better response of LM358 at 50KHz than at 112KHz.

Also we can analyze the circuit during testing using a potentiometer replacing the feedback resistor. We can tune the potentiometer to get exact value of the feedback resistor for oscillation for the circuit that is build. The video below shows how the gain effects the oscillator output.