I am going to describe on how mathematics can lead into verification of performance of a circuit at steady state. Combining ideas of Ohms law, Laplace transform and Matlab, we will try to see which circuit is better among the following having the same number of passive components . In modern systems, it is better to see what is the relationship between Output and Input . The response will determine if the system is stable or unstable. These two circuits can be part of complex systems for missiles, PLC, Robots, they are called compensators. Furthermore, the gain depends on the internal parameters of the system rather than the value at the input as we are going to see below. Suggestions for corrections and mistakes in the derivation are highly appreciated.
R1=R3=1kOhm
R2=R4=1.5kOhm
C=C1=4microFarad
From control theory, we will force our equations to be in standard form as below, Voltage across the capacitor is a variable parameter.
=
First circuit mathematical analysis:
, but
Rearranging these equations, we get
Apply Laplace Transform, make subject the ratio between and by omitting from above two equations
Let and
Then:
, and
and time constants determines characteristics of our circuit, the transient response for steady state depends on resistor 1 and 2. And it seems is always less than
Nyquist plot at different Gain(k) of the circuit at different values of resistor R2, The choice of R2 will tell you how fast the circuit will reach steady. state(Video). Which is better system, with gain k<1 going to -1 Real axis point? or Gain k>1 away from -1 Real axis? Study about Nyquist theorem.
Simulate Second circuit with the same values and see the differences.
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Nyquist criterion and Transient response in Electrical circuit
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