find the laplace transform h s voutvin sSolutionHs Vouts Vi

find the laplace transform h (s)= vout/vin (s)

Solution

H(s) = V_out(s) /V_in(s)

Let s = j

Then, if v_in(t) = V sin(₁t),

v_out(t) = V · |H(j₁)|sin[ ₁t +angle of H(j₁) ]

where V , sin and ₁ are given by the input.

Let us take v_in(t) = V sin(₂t)

H(s) = Vout(s) /Vin(s) = RCs /RCs + 1

Let s = j₂,

H(j₂) = jRC₂ /(1 + jRC₂)

Find |H(j2)|:

|H(j₂)| = |jRC₂| /|1 + jRC₂| = RC₂/ (1 + (RC₂)²)^1/2

Find Angle of H(j2):

LH(j₂) = LjRC₂ L(1 + jRC₂) = / 2 tan¹(RC₂)

Then,

V_out(t) = V · [RC₂/ (1 + (RC₂)²)^1/2]. sin [₂t + /2 tan¹ (RC₂) ]