Thesis - Department of Electronic & Computer Engineering
Thesis - Department of Electronic & Computer Engineering
Thesis - Department of Electronic & Computer Engineering
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Case II:0 > v i() t ≥ v j() tτ---- d [ vdt i() t – v j() t ] = – [ v i() t – v j() t ] + ( I i– I j)⎛v i () t – v j () t = ( I i – I j ) ⎜1 – e⎝t– -τ⎞⎟⎠+ [ v i ( 0) – v j ( 0)]e – t / τ > 0Case III:v i () t ≥ 0,v j () t ≤ 0τ---- d [ vdt i () t – v j () t ] = – [ v i () t – v j () t ] + ( I i – I j ) + ( a + b)v i () tτ---- d [ vdt i () t – v j () t ] > – [ v i () t – v j () t ] + ( I i – I j ) ⇒ v i () t – v j () t > 0∴v i() t > v j() tI i– I jLemma 3: ∀i,j ∈ { 1 , …,N }, if v i( 0) ≥ v j( 0)and ------------- ≥ δ ,∃T j< ∞,∀t > T j , v j () t < 0 .I iPro<strong>of</strong>:τ---- d [ vdt i () t – v j () t ] = – [ v i () t – v j () t ] + ( I i – I j ) + ( a + b) [ f i () t – f j () t ]There are three cases to be considered.Case I:v i () t ≥ v j () t > 0τ---- d [ vdt i() t – v j() t ] = ( a+b – 1) [ v i() t – v j() t ] + ( I i– I j)70