electrical network transfer function
TRANSCRIPT
Electrical Network Transfer Function
Engr. Aaron Don M. Africa
Voltage-current, voltage-charge, and impedance relationships for capacitors, resistors, and inductors
Mesh Nodal
Example 1
Find the transfer function relating to the capacitor voltage VC(s) to the input voltage, V(s)
)()()()(
)()(
)()(1)()(
)(1
)()(
2
2
2
2
tvtvcdt
tdvcRC
dt
tvcdLC
tCvctq
tvtqCdt
tdqR
dt
tqdL
diC
tRidt
tdiL
t
o
Changing variable from current to charge using i(t) = dq(t) / dt
Block diagram of series RLC electrical network Block diagram of series RLC electrical network
LCs
LR
s
LC
sV
sVc
sVsVcRCsLCs
1/1
)(
)(
)()()1(
2
2
Simplifying the solution
)()(
)(
)()(
)()(
)(1
)(
sZsI
sV
sLsIsV
sRIsV
sICs
sV
Defining the Transfer Function
)()(1
sVsICs
RLs
[Sum of Impedances]I(s)=[Sum of Applied Voltages]
Transfer function-single loop via transform method
CssIsVc
CsRLssV
sI
sVsICs
RLs
1)()(
/1
1
)(
)(
)()(1
The voltage across the capacitor Vc(s) is the product of the current and impedance of the capacitor
Example 2
Find the transfer function I2(s) / V(s) via mesh analysis
0)(1
)(
)()()()(
0)()(1
)()(
)()()()(
221
211
12222
2111
sICs
RLssLsI
sVsLsIsILsR
sLsIsICs
sIRsLsI
sVsLsIsLsIsIR
Recitation – Matrix Form
1212
21
22
)()()(
)()(
RsLCRRLCsRR
LCsLs
sV
sIsG
Block Diagram of Transfer Function
Example 3
Find the transfer function, Vc(s) / V(s) using nodal analysis
0)()()(
)()()(1
0)()(
)(
0)()()()()(
22
1221
2
21
sVCGsVG
GsVsVcGsVLs
GG
R
sVsVsVC
R
sVsV
Ls
sV
R
sVsV
CSL
L
LCCS
CLLL
Rearranging and expressing the resistances as conductance G1 = 1 / R1 and G2 = 1 / R2
LCG
sLC
CLGGsGG
sCGG
sV
sVc
221221
21
)()(
)(
Solving for the transfer function, Vc(s)/ V(s)
Example 4
Write the Equations only of the Electrical Network
Recitation
0)(1
14)(4)(
0)(4)()19()()12(
)()()()12()()22(
321
321
321
sIs
sssIsI
ssIsIssIs
sVsIsIssIs
Example 5
Find the Transfer function G(s) = VL(s) / V(S) for the network in the figure given VL(s) = sI2(s)
)()2()(
0)()2()(
)()()()1(
21
21
21
sIssI
sIssI
sVsIsIs i
Writing the Mesh Equations:
But:
Recitation
13)(
)(
)()(
)13/(1)(/)(
)()()()2)(1(
2
2
22
22
ss
s
sV
sV
ssIsV
sssVsI
sVsIsIss
i
L
L
i
i