Nodes 5 and 6 can be connected to form the node datum.
Properties of the ideal transformer.
1) Laplace transform of the input signal as a pulse of duration
2) Laplace transform of the input signal as a sinusoidal voltage of pulsation
Assume Rs as the internal resistance of the voltage source Vi.
Branches number:
Nodes number:
From the circuit inspection, I create the admittance matrix and then the Impedance matrix:
Branches admittance matrix:
Branches impedance matrix:
Branch-cur rent phasor:
Node to datum voltage phasor:
Branch-voltage phasor:
Sources:
Sources:
Initial conditions:
Define each element of the incidence matrix applying the rule:
Incidence Matrix :
eliminating the last row, I obtain the reduced incidence matrix:
R educed incidence matrix (no -1)xb :
Transpose of the R
educed incidence matrix:
branch voltage phasor:
Node Admittance matrix:
Node:
1
2
3
4
Node:
2
3
4
Branch equations:
V1
V2
V3
V4
V5
V6
I1
I2
I3
I4
I5
= 0
= 0
= 0
= 0
= 0
= vi
Matrix form of the Branch equations:
Verify:
Capacitors-matrix definition:
Coefficients matrix of the branch volta ges:
Inductances-matrix definition:
Coefficients matrix
D=Dimensionless
Branch equations:
Branch-voltage phasor definition:
Branch-current phasor definition:
Branch-current phasor:
Branch-voltage phasor:
Node to datum voltage phasor:
Zero's vector:
Zero's matrices creation:
Definition of the TABLEAU of a Linear time Invariant electric network based on the reduced incidence matrix :
Building the TABLEAU
Branch-cur rent phasor:
Node to datum voltage phasor:
Branch-voltage phasor:
Calculation of the node and branches voltages and currents:
Transfer function:
POLES
Denominator coefficients:
Representation of the contour of integration on the Gaussian plane of the complex variable s.
Radius of the circle that encloses all the poles of
BODE PLOTS
Pulse response:
Pulse response:
Given the Laplace transform obtained by
calculate the inverse Laplace transform of V5(s):
System answer to
Given the Laplace transform of the output signal obtained by
calculate the inverse Laplace transform of V55(s):
Branch 1 admittance:
Transformer primary
Transformer secondary
Branch 2 admittance:
Branch 3 admittance:
Branch 4 admittance:
Branch 5 admittance:
Branch 6 admittance:
Branch 1 impedance:
Branch 2 impedance:
Branch 3 impedance:
Branch 4 impedance:
Branch 5 impedance:
Branch 6 impedance: