# lab_2_f08 - EE 350 CONTINUOUS-TIME LINEAR SYSTEMS FALL 2008...

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EE 350 CONTINUOUS-TIME LINEAR SYSTEMS FALL 2008 Laboratory #2: Sinusoidal Steady-State Analysis The objectives of the second laboratory are to Experimentally measure the frequency response function H ( ω ) of two networks. Use MATLAB to plot the magnitude and angle of H ( ω )in decibels as a function of frequency ω . 1 Background If a bounded-input bounded-output (BIBO) stable linear time-invariant (LTI) system is driven by a sinusoid of frequency ω , then the steady-state response of the circuit is also a sinusoid of frequency ω . Figure 1 shows a BIBO stable LTI system with input f ( t ) and response y ( t ). Figure 1: BIBO stable LTI system with input f ( t ) and output y ( t ). Suppose a sinusoidal input f ( t )= F m cos( ωt + θ f ) , where F m and θ f are constants, is applied to the system. The resulting zero-state response can be expressed as the sum of a natural response y n ( t ) and forced response y φ ( t ) y ( t y n ( t )+ y φ ( t ) . Because the system is BIBO stable, the natural response y n ( t ) will exponentially relax towards zero so that the steady-state response of the circuit is the forced response y ss ( t y φ ( t ) . As the forcing function is a sinusoid of frequency ω , the steady-state response is also a sinusoid of frequency ω y ss ( t Y m ( ω ) cos( + θ y ( ω )) . It is important to note that the amplitude Y m and phase θ y of the response depends on the frequency ω of the input. The sinusoidal input f ( t ) and the sinusoidal steady-state response y ss ( t ) can be represented by phasors ˜ F ( ω F m e θ f ˜ Y ( ω Y m ( ω ) e θ y ( ω ) . Given this representation, the frequency response function H ( ω ) is de±ned as H ( ω ˜ Y ( ω ) ˜ F ( ω ) . Once we know H ( ω ), we can ±nd the sinusoidal steady-state response y ss ( t ) for any sinusoidal input f ( t ) using the relationships ˜ Y ( ω H ( ω ) ˜ F ( ω ) and y ss ( t )=Re { ˜ Y ( ω ) e ωt } . In order to understand the eﬀect of the circuit on a sinusoidal input, it is useful to plot the magnitude of H ( ω ) as a function of ω . The magnitude of the frequency response function can vary over many orders of magnitude as frequency is varied, and so it is useful to plot the logarithm of | H ( ω ) | . Electrical engineers typically express the magnitude of the frequency response function in decibels (abbreviated dB) that are de±ned as 20 log 10 | H ( ω ) | .

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## This note was uploaded on 10/13/2008 for the course EE 350 taught by Professor Schiano,jeffreyldas,arnab during the Fall '07 term at Pennsylvania State University, University Park.

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lab_2_f08 - EE 350 CONTINUOUS-TIME LINEAR SYSTEMS FALL 2008...

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