EE422
Solution to Homework #9
1a) Before t=0, iL = 0 Amps vC = 5 Volts. For t > 0:
b) States are vC1 and vC2: 
. For t < 0, the circuit looks like:
For t < 0, Vc2(0)=-5-2Vc1(0) and Vc1(0)=-3Vc2(0). Therefore, Vc2(0)= 1 Volt and Vc1(0) = -3 Volts.
Therefore: 
Model for t > 0:
2a) Use P-Spice and replace the capacitors and inductors by voltage sources and current sources to help you find the state variable model for the following 4th order circuit
Let the states be vC1, vC2, iL1, iL2:
Here is the Pspice Schematic for t < 0 (note the dummy voltage sources in series with the inductors to measure current):
**** 09/30/98 15:27:44 ******** NT Evaluation PSpice (October 1996) **********
**** CIRCUIT DESCRIPTION
******************************************************************************
* Schematics Version 7.1 - October 1996
* Wed Sep 30 15:27:44 1998
** Analysis setup **
.OP
* From [SCHEMATICS NETLIST] section of msim.ini:
.lib nom.lib
.INC "hw13.net"
**** INCLUDING hw13.net ****
* Schematics Netlist *
R_R4 $N_0002 $N_0001 2
R_R2 $N_0004 $N_0003 3
R_R1 $N_0005 $N_0004 4
C_C1 $N_0003 $N_0001 0.25
V_V4 $N_0005 $N_0006 DC 0
V_V3 $N_0003 $N_0007 DC 0
I_I1 0 $N_0001 DC 5
C_C2 0 $N_0004 0.33333
V_V1 $N_0005 0 DC 10
R_R5 0 $N_0003 4
L_L1 $N_0006 $N_0008 0.2
R_R3 $N_0008 $N_0002 1
L_L2 0 $N_0007 0.5
**** RESUMING hw13.cir ****
.INC "hw13.als"
**** INCLUDING hw13.als ****
* Schematics Aliases *
.ALIASES
R_R4 R4(1=$N_0002 2=$N_0001 )
R_R2 R2(1=$N_0004 2=$N_0003 )
R_R1 R1(1=$N_0005 2=$N_0004 )
C_C1 C1(1=$N_0003 2=$N_0001 )
V_V4 V4(+=$N_0005 -=$N_0006 )
V_V3 V3(+=$N_0003 -=$N_0007 )
I_I1 I1(+=0 -=$N_0001 )
C_C2 C2(1=0 2=$N_0004 )
V_V1 V1(+=$N_0005 -=0 )
R_R5 R5(1=0 2=$N_0003 )
L_L1 L1(1=$N_0006 2=$N_0008 )
R_R3 R3(1=$N_0008 2=$N_0002 )
L_L2 L2(1=0 2=$N_0007 )
.ENDALIASES
**** RESUMING hw13.cir ****
.probe
.END
**** 09/30/98 15:27:44 ******** NT Evaluation PSpice (October 1996) **********
* C:\My Documents\Class\Ee422\fall98\hw13.sch
**** SMALL SIGNAL BIAS SOLUTION TEMPERATURE = 27.000 DEG C
******************************************************************************
NODE VOLTAGE NODE VOLTAGE NODE VOLTAGE NODE VOLTAGE
($N_0001) 25.0000 ($N_0002) 15.0000
($N_0003) 0.0000 ($N_0004) 4.2857
($N_0005) 10.0000 ($N_0006) 10.0000
($N_0007) 0.0000 ($N_0008) 10.0000
VOLTAGE SOURCE CURRENTS
NAME CURRENT
V_V4 -5.000E+00
V_V3 1.429E+00
V_V1 3.571E+00
TOTAL POWER DISSIPATION -3.57E+01 WATTS
Initial conditions (use steady-state techniques):
vC1(0) = Node Voltage 1 - Node Voltage 3 = 25 V
vC2(0) = Node Voltage 4 = 4.2857 V
iL1(0) = Current thru Vamp1 = -5 A
iL2(0) = Current thru Vamp2 = 1.429 A
Therefore: 
For t > 0, we have:
We can use Pspice to fine all of these values by setting each source (i.e., vc1, vc2, iL1, iL2, w1, and w2) to 1 in turne while killing the remaining 5 sources. Thus, we replace the capacitors and inductors by voltage sources and current sources and obtain the following Pspice schematic:
By running this model six times (once for each source set to 1 with remaining sources set to 0), we can find all the unknown responses in our state variable. Here is the Pspice output for each of the six cases:
**** 10/01/98 11:34:05 ******** NT Evaluation PSpice (October 1996) **********
**** CIRCUIT DESCRIPTION
******************************************************************************
* Schematics Version 7.1 - October 1996
* Thu Oct 01 11:34:04 1998
** Analysis setup **
.OP
* From [SCHEMATICS NETLIST] section of msim.ini:
.lib nom.lib
.INC "hw13a.net"
**** INCLUDING hw13a.net ****
* Schematics Netlist *
R_R4 $N_0002 $N_0001 2
R_R5 0 $N_0003 4
R_R3 $N_0004 $N_0002 1
R_R2 $N_0005 $N_0003 3
R_R1 $N_0006 $N_0005 4
V_Vc2 $N_0001 $N_0003 DC 0
V_V1 $N_0006 0 DC 0
I_IL1 $N_0006 $N_0004 DC 0
V_Vc1 $N_0005 0 DC 1
I_IL2 $N_0003 0 DC 0
I_I1 0 $N_0001 DC 0
**** RESUMING hw13a.cir ****
.INC "hw13a.als"
**** INCLUDING hw13a.als ****
* Schematics Aliases *
.ALIASES
R_R4 R4(1=$N_0002 2=$N_0001 )
R_R5 R5(1=0 2=$N_0003 )
R_R3 R3(1=$N_0004 2=$N_0002 )
R_R2 R2(1=$N_0005 2=$N_0003 )
R_R1 R1(1=$N_0006 2=$N_0005 )
V_Vc2 Vc2(+=$N_0001 -=$N_0003 )
V_V1 V1(+=$N_0006 -=0 )
I_IL1 IL1(+=$N_0006 -=$N_0004 )
V_Vc1 Vc1(+=$N_0005 -=0 )
I_IL2 IL2(+=$N_0003 -=0 )
I_I1 I1(+=0 -=$N_0001 )
.ENDALIASES
**** RESUMING hw13a.cir ****
.probe
.END
**** 10/01/98 11:34:05 ******** NT Evaluation PSpice (October 1996) **********
**** SMALL SIGNAL BIAS SOLUTION TEMPERATURE = 27.000 DEG C
******************************************************************************
NODE VOLTAGE NODE VOLTAGE NODE VOLTAGE NODE VOLTAGE
($N_0001) .5714 ($N_0002) .5714
($N_0003) .5714 ($N_0004) .5714
($N_0005) 1.0000 ($N_0006) 0.0000
VOLTAGE SOURCE CURRENTS
NAME CURRENT
V_Vc2 -1.143E-12
V_V1 2.500E-01
V_Vc1 -3.929E-01
TOTAL POWER DISSIPATION 3.93E-01 WATTS
iC1|vc1 = Current thru V_Vc1 = -0.3929 A
iC2|vc1 = Current thru V_Vc2 = 0 A
vL1|vc1 = Node Voltage 6 - Node Voltage 4 = -0.5714V
vL2|vc1 = Node Voltage 3 - Node Voltage 0 = 0.5714V
y|vc1 = Node Voltage 5 - Node Voltage 6 = 1 V
Second run with Vc2 = 1 V and rest of sources killed:
NODE VOLTAGE NODE VOLTAGE NODE VOLTAGE NODE VOLTAGE
($N_0001) 1.0000 ($N_0002) 1.0000
($N_0003)-3.429E-12 ($N_0004) 1.0000
($N_0005) 0.0000 ($N_0006) 0.0000
VOLTAGE SOURCE CURRENTS
NAME CURRENT
V_V1 1.000E-12
V_Vc1 -1.143E-12
V_Vc2 -2.000E-12
TOTAL POWER DISSIPATION 2.00E-12 WATTS
iC1|vc2 = Current thru V_Vc1 = 0 A
iC2|vc2 = Current thru V_Vc2 = 0 A
vL1|vc2 = Node Voltage 6 - Node Voltage 4 = -1 V
vL2|vc2 = Node Voltage 3 - Node Voltage 0 = 0 V
y|vc2 = Node Voltage 5 - Node Voltage 6 = 0 V
Third run with IL1 = 1 A and rest of sources killed:
NODE VOLTAGE NODE VOLTAGE NODE VOLTAGE NODE VOLTAGE
($N_0001) 1.7143 ($N_0002) 3.7143
($N_0003) 1.7143 ($N_0004) 4.7143
($N_0005) 0.0000 ($N_0006) 0.0000
VOLTAGE SOURCE CURRENTS
NAME CURRENT
V_V1 -1.000E+00
V_Vc1 5.714E-01
V_Vc2 1.000E+00
TOTAL POWER DISSIPATION 0.00E+00 WATTS
iC1|iL1 = Current thru V_Vc1 = 0.5714 A
iC2|iL1 = Current thru V_Vc2 = 1 A
vL1|iL1 = Node Voltage 6 - Node Voltage 4 = -4.7143 V
vL2|iL1 = Node Voltage 3 - Node Voltage 0 = 1.7143 V
y|iL1 = Node Voltage 5 - Node Voltage 6 = 0 V
Fourth run with IL2 = 1 A and rest of sources killed:
NODE VOLTAGE NODE VOLTAGE NODE VOLTAGE NODE VOLTAGE
($N_0001) -1.7143 ($N_0002) -1.7143
($N_0003) -1.7143 ($N_0004) -1.7143
($N_0005) 0.0000 ($N_0006) 0.0000
VOLTAGE SOURCE CURRENTS
NAME CURRENT
V_V1 -1.714E-12
V_Vc1 -5.714E-01
V_Vc2 3.429E-12
TOTAL POWER DISSIPATION 0.00E+00 WATTS
iC1|iL2 = Current thru V_Vc1 = -0.5714 A
iC2|iL2 = Current thru V_Vc2 = 0 A
vL1|iL2 = Node Voltage 6 - Node Voltage 4 = 1.7143 V
vL2|iL2 = Node Voltage 3 - Node Voltage 0 = -1.7143 V
y|iL2 = Node Voltage 5 - Node Voltage 6 = 0 V
Fifth run with w1 = 1 V and rest of sources killed:
NODE VOLTAGE NODE VOLTAGE NODE VOLTAGE NODE VOLTAGE
($N_0001) 1.714E-12 ($N_0002) 3.714E-12
($N_0003) 1.714E-12 ($N_0004) 4.714E-12
($N_0005) 0.0000 ($N_0006) 1.0000
VOLTAGE SOURCE CURRENTS
NAME CURRENT
V_Vc1 2.500E-01
V_Vc2 1.000E-12
V_V1 -2.500E-01
TOTAL POWER DISSIPATION 2.50E-01 WATTS
iC1|w1 = Current thru V_Vc1 = 0.25 A
iC2|w1 = Current thru V_Vc2 = 0 A
vL1|w1 = Node Voltage 6 - Node Voltage 4 = 1 V
vL2|w1 = Node Voltage 3 - Node Voltage 0 = 0 V
y|w1 = Node Voltage 5 - Node Voltage 6 = -1 V
Sixth and final run with w2 = 1 V and rest of sources killed:
NODE VOLTAGE NODE VOLTAGE NODE VOLTAGE NODE VOLTAGE
($N_0001) 1.7143 ($N_0002) 1.7143
($N_0003) 1.7143 ($N_0004) 1.7143
($N_0005) 0.0000 ($N_0006) 0.0000
VOLTAGE SOURCE CURRENTS
NAME CURRENT
V_Vc1 5.714E-01
V_Vc2 1.000E+00
V_V1 1.714E-12
TOTAL POWER DISSIPATION 0.00E+00 WATTS
iC1|w2 = Current thru V_Vc1 = 0.5714 A
iC2|w2 = Current thru V_Vc2 = 1 A
vL1|w2 = Node Voltage 6 - Node Voltage 4 = -1.7143 V
vL2|w2 = Node Voltage 3 - Node Voltage 0 = 1.7143 V
y|w2 = Node Voltage 5 - Node Voltage 6 = 0 V
The output equation is : 
State equation: