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# A draw the nyquist contour and mapping for ghs b

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(a) Draw the Nyquist contour and mapping for GH(s) (b) Determine whether the system is stable by utilizing the Nyquist criterion. GH(s) = ) 1 ( + s s k τ (a) The origin of the S-plane s=E e j Ф 0 lim E GH(s) = 0 lim E φ i Ee k = 0 lim E φ i e E k ) ( -90< Ф <+90 Ф =90 ׁ at ω =0 + GH(s) = -90 jv u 1 j 1 j + 0 1

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Ф =-90 ׁ at ω =0 - GH(s) =-+90 Magnitude of GH(s) is infinite. (b) The portion from ω =0 + to ω =+ GH(s) ω j s = = GH(j ω ) α ω + lim GH(j ω ) = α ω + lim ) 1 ( + ωτ ω j j k = α ω + lim ω π τω 1 2 tan 2 k ω = + magnitude = 0 phase = -180 ׁ ω = τ 1 , magnitude=K τ phase = -135 ׁ (c) The portion from ω = + to ω = - α r Lim GH(s) = α r Lim 2 r K τ e -2j Ф S= re j Ф at ω =+ , Ф ( ω ) = -180 at ω =- , Ф ( ω ) = +180 Radius E s-plane Radius r σ ω j * τ 1
Nyquist contour Z =N+P =0+0 = 0 This system is stable. Chapter (13) 18. Determine the z transform of f(t)=sin wt for t 0 (16 Marks) Sin wt = j e e jwt jwt 2 . sin ω t =e j ω t - e –j ω t 2j sin ω t = { e j ω t - e –j ω t } 2j 2j f(z) = 1 ( __z__ - ____z __ ) 2j z-e j ω t z-e –j ω t = 1 ( z ( z-e –j ω t ) - z (z-e j ω t ) ) 2j z 2 - ze –j ω t - ze j ω t +1 = 1 ( z ( z-e –j ω t ) - z (z-e j ω t ) ) 2j z 2 - ze –j ω t - ze j ω t +1 Radius= α * α ω = α ω + = τ ω 1 = + = 0 ω = 0 ω u

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