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I'm studying signal analysis, and we reached a topic of correcting and enhancing a system with the following transfer function :

$$ \frac{Y(s)}{U(s)} = \frac{2}{s^2 + 5s + 6} $$

with \$U(s) = \frac{1}{s}\$ (unit step input)

When we calculated the steady state error (ess) we obtained 67 % >> 5 %. With 5% being the maximum acceptable value.

In order to reduce the error we used a unity negative feedback with an amplification k (amplifies after the summing point of the feedback).

Therefore, $$ \frac{Y(s)}{R(s)} = \frac{2k}{s^2 + 5s + 6 + 2k} $$

Where \$R(s)\$ is the new input but still a unit step.

Why we are using negative feedback, while (mathematically) we could only use an amplifier k which will simplifies the analysis ? Is it impractical ?

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It's the stability that's impractical in the real world.

Mathematically, numbers look very stable, and yes, of course we can tweak a coefficient here and get the answer we want.

In the real world, complicated amplifier components are very temperature sensitive, and vary from batch to batch.

Simple feedback components, a pair a resistors for example, are orders of magnitude more stable and easier to adjust. That's why the world builds amplifiers with 'too much' gain, then knocks it back down with negative feedback.

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It is almost impossible to make (nearly) linear amplifiers without feedback, because all of the electronic components are strongly nonlinear. Transistors can be approximated by a nonlinear characteristic like an exponential or quadratic function and amplifiers are base on these devices. Amplifiers are therefore non-linear.

To get linear amplifiers a high-gain amplifier is used in a feedback configuration, such that the closed behavior depends mainly on the feedback network.

The feedback network is often composed of passive devices like resistors and capacitors which can be made very linear.

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