03.transfer function - Transfer Transfer Function Dongik...

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Transfer Function Dongik Shin Transfer Function Transfer function is the ratio of the output to the input in s domain. G ( s ) Y ( s ) U ( s ) () Ys Gs Us = Transfer function is a complex rational function 1 11 0 1 mm nn bs b s bs b ps qs sa s a s a - - - ++ + + == + + where the complex independent variable s is interpreted as a differential ti t i d i 0 n - operator in time domain: d d s t = 2 Copyright © Dongik Shin, 2010
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DC Gain Transfer function can be written in the form 1 1 m bs b s ++ +  where 1 () 1 m n n Gs K as = + 0 00 0 ,, i i i i b ab K aab === Note that we assume a 0 ¹ 0 and b 0 ¹ 0. K is DC gain , which means the steady-state gain when the DC (or step) input u ¥ 1( t ) is applied. 3 Copyright © Dongik Shin, 2010 The steay-state response to a step input is, by the final value thoerem , lim ( ) 1( ) lim ( ) lim ( ) ts s u yg t u t s G s u G s K u s ¥ ¥¥ ¥ ¥ ¥ =* = = = y ¥ Response Input y K u ¥ ¥ = u ¥ Once the system reach steay-state, the differential operator s plays no role, that is, s will disappear in the transfer: s 0. (0) KG = 4 Copyright © Dongik Shin, 2010
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System Type and DC Gain Generally b 0 ¹ 0. Otherwise, we had better adopt rather than u ( t ) as the input. Therefore, the TF can be written, very generally, as ( () ) ut = 1 1 1 1 1 m m kn k nk bs b s Gs K sa s a s - - ++ + = +  The system has k pure integrators and the number k is called system's type number . o The DC gain of type-0 system is G (0). o The DC gain of type-1 or higher system is infinite. 5 Copyright © Dongik Shin, 2010 Transient and Steady-State System If the input U ( s ) is rational, so is the output Y ( s ). m 1 1 1 1 1 m k b s b s K s - - + = + ´ Steady state system Transien system Steady-state This governs the steady- state. If k = 0, K is the Transient At steady-state, s will disappear, so will this U (s) Y (s) DC gain. whole term as 1. 6 Copyright © Dongik Shin, 2010
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Poles and Zeros In complex analysis, o The poles of a complex rational function G ( s ) are isolated points in the complex plane , where G ( s ) is not defined, or has singularity. o The zeros of G ( s ) are isolated points in , where the function values are zero values are zero.
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This note was uploaded on 10/16/2011 for the course MECHATRONI 111 taught by Professor Jung during the Spring '11 term at Hanyang University.

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03.transfer function - Transfer Transfer Function Dongik...

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