Please explain working of the circuit

Greetings,
Could anyone help me explaining the following circuit


Actually i have been assigned to build a circuit with pi controller to maintain the temperature set point of 40C of an incubator. the above is proposed to be used as the first stage before pi controller.
Explanation with mathematical relations would do great.

The circuit is not clear enough to show the inverting and non-inverting pins.
Please do not be offended, which material is this? If it is Malvino, tell me the page.

The circuit is not clear enough to show the inverting and non-inverting pins.

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Clear enough, I think. The circuit is implementing the claimed transfer function only if R1 >> R2.

It's a differential amplifier with variable gain and a non-inverting "integrator + 1" circuit, in other words representing a PI controller.
R2/R2' sets KP, R3*C TI parameter. Unlike with the often used parallel controller topology, integrator time constant TI is kept konstant when varying the gain KP .

FvM said:

Clear enough, I think. The circuit is implementing the claimed transfer function only if R1 >> R2.

It's a differential amplifier with variable gain and a non-inverting "integrator + 1" circuit, in other words representing a PI controller.
R2/R2' sets KP, R3*C TI parameter. Unlike with the often used parallel controller topology, integrator time constant TI is kept konstant when varying the gain KP .

Click to expand...

I am using lm35 to sense the temperature inside the incubator , the output of the lm35 is to be fed at 'r' point or 'y' point?
And i presume with reference to your explanation that the above stated circuit output can be directly fed to transistor which is controlling relay to switch the bulb on/off.

the output of the lm35 is to be fed at 'r' point or 'y' point?

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y is the usual formula sign for process value (the measured temperature in this case)

that the above stated circuit output can be directly fed to transistor which is controlling relay to switch the bulb on/off

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Doing so creates a simple on/off controller which will show an oscillating process value. You get better controller performance by feeding the controller output to a linear actuator or a PWM circuit.

FvM said:

Doing so creates a simple on/off controller which will show an oscillating process value. You get better controller performance by feeding the controller output to a linear actuator or a PWM circuit.

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it means we can add a threshold level to the output to refrain from oscillating the actuator

An on/off controller will result in an oscillating process value by nature. In case of a temperature controller, the control process time constants and the heating power will determine the oscillation band. E.g. when heating, the controller might switch-off the heater at the set point. But energy stored in the heater will further increase the temperature.

A linear controller in contrast can adjust the heating power to the exact demand. An application like an incubator surely demands for a linear controller.

PWM (pulse width modulation) is a way to achieve effectively linear control with an on-off operated actuator. The switching period is chosen sufficiently short so that (almost) no temperature variations arrive at the location of interest.

FvM said:

A linear controller in contrast can adjust the heating power to the exact demand. An application like an incubator surely demands for a linear controller.

PWM (pulse width modulation) is a way to achieve effectively linear control with an on-off operated actuator. The switching period is chosen sufficiently short so that (almost) no temperature variations arrive at the location of interest.

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okay, i got the point, rather from ur point of view, the pwm technique is a more suitable approach.

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FvM said:

R2/R2' sets KP, R3*C TI parameter. Unlike with the often used parallel controller topology, integrator time constant TI is kept konstant when varying the gain KP .

Click to expand...

Also could u clarify more that the Ti u are referring is
as we have for pi controller block Kp+Ki/s so by taking common Kp
we have Kp(1 + Ki/Kp s)
which means Ki/Kp = 1/Ti
so the above is the ti u are refering too?

which means Ki/Kp = 1/Ti
so the above is the ti u are refering too?

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Yes. You'll find a derivation of both PID forms e.g. here https://en.wikipedia.org/wiki/PID_controller

Thank u very much @FvM.
But now i have opted the following circuit

The above circuit doesnt switches relay can u help me in detecting the problems

Reference: