S 0 , k 1 = ∑ k = 0 k 1 q k = 1 − q k 1 + 1 1 − q {\displaystyle S_{0,k_{1}}=\sum _{k=0}^{k_{1}}q^{k}={\frac {1-q^{k_{1}+1}}{1-q}}} e i x = cos x + i sin x {\displaystyle e^{ix}=\cos x+i\sin x} e − i x = cos x − i sin x {\displaystyle e^{-ix}=\cos x-i\sin x} cos x = 1 2 ( e i x + e − i x ) {\displaystyle \cos x={\frac {1}{2}}(e^{ix}+e^{-ix})} sin x = 1 2 i ( e i x − e − i x ) {\displaystyle \sin x={\frac {1}{2i}}(e^{ix}-e^{-ix})} e − i x = cos x − i sin x {\displaystyle e^{-ix}=\cos x-i\sin x} | q | < 1 {\displaystyle \left\vert q\right\vert <1} k 1 → ∞ {\displaystyle k_{1}\rightarrow \infty } S 0 , ∞ = ∑ k = 0 ∞ q k = 1 1 − q {\displaystyle S_{0,\infty }=\sum _{k=0}^{\infty }q^{k}={\frac {1}{1-q}}} L [ e − a t f ( t ) ] = F ( s + a ) {\displaystyle {\mathcal {L}}[e^{-at}f(t)]=F(s+a)}
L [ f ( t − θ ) ] = e − θ s F ( s ) {\displaystyle {\mathcal {L}}[f(t-\theta )]=e^{-\theta s}F(s)}
Z [ f ( t − m T ) ] = z − m F ( z ) {\displaystyle {\mathcal {Z}}[f(t-mT)]=z^{-m}F(z)} e x = 1 + x + x 2 2 ! + x 3 3 ! + . . . . . . {\displaystyle e^{x}=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+......} 1 1 + x = 1 − x + x 2 − x 3 + . . . . . . {\displaystyle {\frac {1}{1+x}}=1-x+x^{2}-x^{3}+......} e − θ s ≈ 1 1 + θ s {\displaystyle e^{-{\theta }s}\approx {\frac {1}{1+{\theta }s}}} θ {\displaystyle \theta } y [ k ] = − ∑ m = 1 N a m y [ k − m ] + ∑ m = 0 M b m x [ k − m ] , k > 0 {\displaystyle y[k]=-\sum _{m=1}^{N}a_{m}y[k-m]+\sum _{m=0}^{M}b_{m}x[k-m],k>0} y [ k ] = ∑ m = 0 M b m x [ k − m ] {\displaystyle y[k]=\sum _{m=0}^{M}b_{m}x[k-m]} f s ( t ) = ∑ k f ( k T s ) δ ( t − k T s ) {\displaystyle f_{s}(t)=\sum _{k}f(kT_{s})\delta (t-kTs)} F s ( s ) = L [ f s ( t ) ] = ∑ k f ( k T s ) L [ δ ( t − k T s ) ] = ∑ k f ( k T s ) e − k T s s {\displaystyle F_{s}(s)={\mathcal {L}}[f_{s}(t)]=\sum _{k}f(kT_{s}){\mathcal {L}}[\delta (t-kT_{s})]=\sum _{k}f(kTs)e^{-kT_{s}s}} z = e T s s {\displaystyle z=e^{T_{s}s}} L [ f s ( t ) ] = ∑ k f ( k T s ) z − k = ∑ k f ( k ) z − k ≜ Z [ f ( k ) ] {\displaystyle {\mathcal {L}}[f_{s}(t)]=\sum _{k}f(kTs)z^{-k}=\sum _{k}f(k)z^{-k}\triangleq {\mathcal {Z}}[f(k)]} Z [ δ [ k ] ] = ∑ k = 0 ∞ δ [ k ] z − k = ∑ k = 0 ∞ δ [ k ] z 0 = δ [ 0 ] = 1 {\displaystyle {\mathcal {Z}}[\delta [k]]=\sum _{k=0}^{\infty }\delta [k]z^{-k}=\sum _{k=0}^{\infty }\delta [k]z^{0}=\delta [0]=1} Z [ u [ k ] ] = ∑ k = 0 ∞ u [ k ] z − k = ∑ k = 0 ∞ z − k = 1 1 − z − 1 {\displaystyle {\mathcal {Z}}[u[k]]=\sum _{k=0}^{\infty }u[k]z^{-k}=\sum _{k=0}^{\infty }z^{-k}={\frac {1}{1-z^{-1}}}} ω ( s ) V ( s ) = K t L J s 2 + ( L b + R J ) s + R b + K e K t {\displaystyle {\frac {\omega (s)}{V(s)}}={\frac {K_{t}}{LJs^{2}+(Lb+RJ)s+Rb+K_{e}K_{t}}}} ω ( s ) V ( s ) = K ( τ e s + 1 ) ( τ m s + 1 ) {\displaystyle {\frac {\omega (s)}{V(s)}}={\frac {K}{(\tau _{e}s+1)(\tau _{m}s+1)}}} K = K t R b + K e K t {\displaystyle K={\frac {K_{t}}{Rb+K_{e}K_{t}}}} τ e = L R {\displaystyle \tau _{e}={\frac {L}{R}}} τ m = R J R b + K e K t {\displaystyle \tau _{m}={\frac {RJ}{Rb+K_{e}K_{t}}}}
![{\displaystyle {\mathcal {L}}[e^{-at}f(t)]=F(s+a)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65c6bd3d01f45718b9319889a233490ef1dd26d6)
![{\displaystyle {\mathcal {L}}[f(t-\theta )]=e^{-\theta s}F(s)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b804040e4cd29ad17e553ab8296b11797dc0840b)
![{\displaystyle {\mathcal {Z}}[f(t-mT)]=z^{-m}F(z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/27de3c0286dd5bf922c50974cc49cf6b2bf5d34f)
![{\displaystyle y[k]=-\sum _{m=1}^{N}a_{m}y[k-m]+\sum _{m=0}^{M}b_{m}x[k-m],k>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2f3c2938585b30148922fe573f27261b052250a)
![{\displaystyle y[k]=\sum _{m=0}^{M}b_{m}x[k-m]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf481ef17e03019839d362ff7f5d91cce23a8fdb)
![{\displaystyle F_{s}(s)={\mathcal {L}}[f_{s}(t)]=\sum _{k}f(kT_{s}){\mathcal {L}}[\delta (t-kT_{s})]=\sum _{k}f(kTs)e^{-kT_{s}s}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49a80c40e197f6acbd59d391039851391b6466fb)
![{\displaystyle {\mathcal {L}}[f_{s}(t)]=\sum _{k}f(kTs)z^{-k}=\sum _{k}f(k)z^{-k}\triangleq {\mathcal {Z}}[f(k)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/34409148a34b06c445f1e9f48a113080073300b7)
![{\displaystyle {\mathcal {Z}}[\delta [k]]=\sum _{k=0}^{\infty }\delta [k]z^{-k}=\sum _{k=0}^{\infty }\delta [k]z^{0}=\delta [0]=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46eb5a43a229076564612468467299e756835821)
![{\displaystyle {\mathcal {Z}}[u[k]]=\sum _{k=0}^{\infty }u[k]z^{-k}=\sum _{k=0}^{\infty }z^{-k}={\frac {1}{1-z^{-1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f55f8632a78954cc793ac4426444291a0daa16d7)
