Laplace transform solution in time domain solution in s domain inverse Laplace transform problem in time domain transform solution in original way of thinking solution in transform way of thinking inverse transform problem in original way of thinking linear differential equation time domain solution Laplace transformed eq. (algebra eq.) Laplace solution Laplace transform Inverse Laplace transform easy time domain difficult frequency domain (Laplace domain) transform solution in original way of thinking solution in transform way of thinking inverse transform problem in original way of thinking System OutputInput () ()         Input flowrate Output flowrate R C   ()   () () inlet air signal   =    = flexible diaphragm flow direction spring with capacitance, C m stem and diaphragm plate are the moving mass, m seal with resistance, R m applied force Fluid temperature, T a (actual temperature) The liquid in the bulb expands as in its temperature increases, forcing more liquid up the small tube. The height of liquid is a measure of the liquid in the bulb Bulb temperature, T m (measured temperature) ()   ()   ()  Input flowrate Output flowrate ( 0 ) () Positive displa- cement pump T m T a inlet air signal   =    = flow direction applied force System OutputInput () () () () () 0.5 0 i ( t ), mA 1.0 1.5 2.5 3.52.0 3.0 4.0 2 4 6 8 10 12 14 time , s Input current i ( t ) 0.5 0 x ( t ), mm 1.0 1.5 2.5 3.52.0 3.0 4.0 2 4 6 8 10 12 14 time , s final value of x ( t )=8 mm control valve position x ( t )     = 8+8.04  (10+174.3 0 ) inlet air signal  0 period= 1  output,  input,      Frequency (rad/sec) Magnitude (dB) 10 20 30 40 50 60 0 10 -4 10 -3 10 -2 10 -1 Frequency (rad/sec) -90 -80 -70 -50 -30 -20 -10 0 -40 -60 Phase (deg) 0 10 -4 10 -3 10 -2 10 -1 gain phase angle . equation time domain solution Laplace transformed eq. (algebra eq.) Laplace solution Laplace transform Inverse Laplace transform easy time domain difficult frequency domain (Laplace domain) transform solution. angle