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03.transfer function

# 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 ) ( ) ( ) ( ) Y s G s U s = Transfer function is a complex rational function 1 1 1 0 1 ( ) ( ) ( ) m m m m n n b s b s b s b p s G s q s s a s a s a - - - + + + + = = + + + + where the complex independent variable s is interpreted as a differential t i ti d i 1 1 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 b s b s + + + where 1 ( ) 1 m n n G s K a s a s = + + + 0 0 0 0 , , i i i i b a b a b K a a b = = = 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 , 0 0 lim ( ) 1( ) lim ( ) lim ( ) t s s u y g t u t sG s u G s Ku 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) K G = 4 Copyright © Dongik Shin, 2010
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 ( ( ) ) u t u t = 1 1 1 1 ( ) 1 m m k n k n k b s b s G s K s a 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). Th DC i f t 1 hi h t i i fi it 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 ). 1 m b b 1 1 1 ( ) 1 m k n k n k b s b s G s K s a s a s - - + + + = + ´ + + Steady-state system Transient system This governs the steady- state. If k = 0, K is the 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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03.transfer function - Transfer Transfer Function Dongik...

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