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Rose-Hulman ECE 205 - Steady State Frequency Response

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8.0 Steady State Frequency Response Consider the response of two LTI system with transfer functions 15()1sHs=+ 2280()1.2 10Hsss=++ to the inputs 0( ) sin( ) ( )xttutω= for0ω= 5, 10, and 15 radians/sec. Figure 8.1 displays the response of the first system to the three input sinusoids, while Figure 8.2 displays the response of the second system to the three input sinusoids. In both figures, the input sinusoid is displayed as a dashed line and the output is a continuous line. Both figures include heavy solid lines that bound the amplitude of the output sinusoid in steady state. As the figures demonstrate, both of the systems go through some initial transients, and then reach a steady state response. The first system has a pole at -1 and the second system has its poles at approximately 10. . j6 3±−, so the 2% setting times for the two systems are estimated to be 4 and 6.7 seconds which corresponds pretty well with the results in the figures. Once the systems come into steady state, the output of the system has a constant amplitude, and there is a constant relationship between the input signal and the output signal. It is important to note that the settling time for the system is not a function of the frequency of the input, but is a property of the system! At this point we know how to quickly estimate the settling time of a system based on the poles of the system, and we next want to be able to quickly estimate the steady state output of an asymptotically stable system with a sinusoidal input. In order to do this we first need to review Euler’s identities and write them in a different form than you are used to seeing. 8.1 Euler’s Identity and Other Useful Relationships The usual form of Euler’s identity is 000cos( ) sin( )jttj teωωω=+ We can also write this as 000cos( ) sin( )tjtj teωωω−=− If we add and subtract these two expressions we get 0000002cos( )2sin( )jt jtjt jtetee jeωωωωωtω−−+=−= Finally, we get our alternate forms of Euler’s identity 000000cos( )2sin( )2jt jtjt jteeteetjωωωωωω−−+=−= ©2009 Robert D. Throne 1This form of Euler’s identity is very useful in both this and in later courses. Figure 8.1. Response of the first system to sinusoids of 5, 10, and 15 radians/sec. The estimated settling time of the system is 4 seconds. The input signal is the dashed line and the output signal is the solid line. The thick solid line bounds the steady state amplitude. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1012y1(t)0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1-0.500.51y2(t)0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1-0.500.51y3(t)Time (sec) Next, let’s assume our usual case of a proper transfer function that is the ratio of two polynomials, 111111()smmmmnnnbs b s bs bHssa asa−−−−00+++ +=++++"" Since this is proper we have . Let’s also assume that we have a real valued system, so that if the input is a real valued function the output will be a real valued function. This means that all of the coefficients must be real values. To understand this, remember that this transfer function means the input and output are related by the differential equation mn≤ 1110 111 11() () () () () ()() ()mmnmmmmnnnnxxaaabdyt d yt dyt d t d t dxt0yttdt dt dt dt dtbdtbb−−−−−−++++=++++""x Hence, if the input is real and we want the output to be real all of the coefficients must also be real. ©2009 Robert D. Throne 2Figure 8.2. Response of the second system to sinusoids of 5, 10, and 15 radians/sec. The estimated settling time of the system is 6.7 seconds. The input signal is the dashed line and the output signal is the solid line. The thick solid line bounds the steady state amplitude. 0 1 2 3 4 5 6 7 8 9 10-10-50510y1(t)0 1 2 3 4 5 6 7 8 9 10-2-10123y2(t)0 1 2 3 4 5 6 7 8 9 10-1012y3(t)Time (sec) Next, let’s assume we want to evaluate the transfer function at0sjω= and also at 0sjω=− and if we can relate the two. Remember that as we raise j to increasing powers we cycle through the same four values, i.e., 1jj=,21j=− , 3jj=− , , and 41j =5jj= so we are right back where we started. To determine what is going on with transfer functions we will look at low order systems and build are way up. Finally, remember that to determine the complex conjugate of a number, you replace j with –j. ©2009 Robert D. Throne 3First Order System: *10 100 10000000 00))() ,( ,( (bjb jbHs H j H j H jsa j asbjbba0)ωωωωωωω−=−++−++=+=+= Second Order System: 22210 201002210 0 1002*20 10 00020100() , ( ,(()))bb b jbHs H jsasa ja abssjbHj Hjjabbabωωωωωωωωωωω+−++=+−+−−−===−− +++0++ Third Order System: 32 3 23210 3020100032 3 2210 02010032*30 20 10 00032020 100() , ()(),)(bbb jb b jbHs H jssasa ja jaajb b jbHj Hjja jsss bbabaaωω ωωωω ωωω ωωωωω ω++ ++=+++=+−−=+−−+−−−=−− ++ Fourth Order System: 432 4 3 243210 403020100432 4 3 24210 0 020100432*40 30 20 10 00043203 03020 1 0)() , ( ,(())bbbb b jbb jbHs H jsssasa jaa jaabjbbjbHj Hjja a ja assssb baabωωωωωωωωωωωωωωωωωωω+−−=+−++ + + +=+++++=−+ +−=−++−−−0) At this point it is pretty straightforward to generalize the relationship *() (Hj H jωω=− . The last thing we need to do is look at representing the transfer function in polar form to determine two important properties of transfer functions for real-valued systems. Assume that we have the complex function written in rectangular form as (() ) ()azjbωωω=+ where ()aω and ()bω are real valued functions. We can write this in polar form as () ()() () )||(jd j zzce ezωωωω ω==( where 221)()()|(()() )((tan ()abcbdza)|zωωω ωωωωω−=+=⎛⎞==⎜⎟⎝⎠( Now let’s assume we have a complex valued transfer function )(Hjω. We can write this in polar form as ()()|()|jHjHj Hj eωωω=( ©2009 Robert D. Throne 4Next, let’s look at *(H )jω and ()Hjω− to derive some useful relationships. * * () ()( ) | ( )| | ( )|jHj jHjHj Hj e Hj eωωωω ω−−==(( () ()( )|( )| |( )|jH j jH jHj Hj e Hj eωωωω ω−−=−=− −(( Now since we know that for a real valued system*() (Hj H j)ωω=− we can conclude the following useful information: • |)| ||((HHj j)ωω=− (the magnitude is an even function of frequency) • )(()HjHjωω=−((−( the phase is an odd function of frequency) Although we have only shown this to be true for transfer functions that are ratios of polynomials, it is true in general for any


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