system are given below: i(t) i₁(t)/ i(t) =ii(t) + i2(t) dv(t) dt i₂(t) = = R Show that the input-output plant transfer function is: K(s): di₁(t) v(t) = i(t)R + L dt Assume zero initial conditions, i.e., i₁(0) = 0 and v(0) = 0. = C 2 Problem 3 Consider now the same circuit, but now with known values of the circuit com nents (seeing your cousin's excitement, you helped them build a new one with known values!), C-1, L= 1, R=1 iz(t) s+1 v(t)

Introductory Circuit Analysis (13th Edition)
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system are given below:
i(t)
i₁(t)
R
i(t) = i(t) + i2(t)
i₂(t) = dv(t)
dt
di₁(t)
v(t) = i₁(t)R + L dt
Assume zero initial conditions, i.e., i₁(0) = 0 and v(0) = 0.
Show that the input-output plant transfer function is:
K(s)
i₂(t)
Problem 3
Consider now the same circuit, but now with known values of the circuit compo-
nents (seeing your cousin's excitement, you helped them build a new one with known values!),
C = 1, L = 1, R = 1
s+1
s² + s +1
C v(t)
Transcribed Image Text:system are given below: i(t) i₁(t) R i(t) = i(t) + i2(t) i₂(t) = dv(t) dt di₁(t) v(t) = i₁(t)R + L dt Assume zero initial conditions, i.e., i₁(0) = 0 and v(0) = 0. Show that the input-output plant transfer function is: K(s) i₂(t) Problem 3 Consider now the same circuit, but now with known values of the circuit compo- nents (seeing your cousin's excitement, you helped them build a new one with known values!), C = 1, L = 1, R = 1 s+1 s² + s +1 C v(t)
Expert Solution
Step 1: Summarize the details given:

Given Data:

An RLC circuit with the system of equations as,

  • Supply current, i open parentheses t close parentheses equals i subscript 1 open parentheses t close parentheses plus i subscript 2 open parentheses t close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space...... open parentheses 1 close parentheses
  • Capacitor voltage, v open parentheses t close parentheses equals i subscript 1 open parentheses t close parentheses R plus L fraction numerator d i subscript 1 open parentheses t close parentheses over denominator d t end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space...... open parentheses 2 close parentheses and
  • Capacitor current, i subscript 2 open parentheses t close parentheses equals C fraction numerator d v open parentheses t close parentheses over denominator d t end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space...... open parentheses 3 close parentheses

To Find:

Proving the transfer function when R equals 1 space text Ω end text comma space L equals 1 space text H end text comma space C equals 1 space text F end text is K open parentheses s close parentheses equals fraction numerator s plus 1 over denominator s squared plus s plus 1 end fraction.


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