Transfer Function

Transfer Function

General with order, linear, time invariant differential equation

an dn(t)dtn+ an-1 dn-1c(t)dtn-1+…a0ct= bmdmrtdtm+bm-1dm-1rtdtm-1+…b0r(t)

Where: c (t) is the output r (t) I is the input

By taking the Laplace transform of both sides

ansn cs+ an-1sn-1 cs+…a0cs+initial condition involving c(t)

=bmsmRt+bm-1sm-1Rt+…b0Rs+initial condition involving r(t)

If we assume that all initial condition are zero

ansn+ an-1sn-1….+…a0cs=bmsm+bm-1sm-1+…b0r(s)

Rs-→ bmsm+bm-1sm-1+…b0ansn+ an-1sn-1….+…a0--→c(s)

Gs=c(s)r(s)=bmsm+bm-1sm-1+…b0ansn+ an-1sn-1….+…a0

Transfer function ratio of output over input

Laplace Transfer Theorem

1. L f(t)=Fs=0∞f(t)e-stdt

2. L Kf(t)=KF(s)

3. L f1t+f2t=F1s+F2s superposition theorem

4. L e-atft=Fs+a complex shifting theorem

5. L ft-a=e-as F(s) real shifting theorem

6. L fat= 1aFsa similarity theorem

7. L dfatdt=sFs-f(0) derivative theorem

8. L d2fatdt2=s2Fs-sf'0-f(0) multiple derivative theorem

9. L 0τfτdτ=F(s)s integral theorem

Example

Find the transfer function represented by

* d(t)dt=2ct=r(t)

Gs=c(s)r(s)

First find the Laplace transform

L d(t)dt+2 L ct=L r(t)

scs+2cs=r(s)

s+2cs=r(s)

c(s)r(s)=1s+2 To find Gt the solve for inverse Laplace transform
Find the transfer function Gs=Y(s)u(s)

L d2Ytdt2+ L dYtdt+ L Yt=2 L dutdt+L ut

s2Ys+sYs+Ys=2s us+u(s)

s2+s+1Ys=2s+1 u(s)

Gs=Y(s)u(s)=2s+1s2+s+1

Electrical network transfer function component | Voltage-current | Current-voltage | capacitor | vt=1c0titdt | it=c dV(t)dt | resistor | vt=Ri(t) | it=1Rv(t) | inductor | vt=Ldi(t)dt | it=1Lotvtdt |

component | Voltage-charge | Impedance | capacitor | vt=1cq(t) | 1c | resistor | vt=Rdq(t)dt | R | inductor | vt=Ld2q(t)dt | LS |

* Find