<content> Numerical Inverse Laplace Transform </content>

c1a2a71c-d990-4f06-9a0d-8d7195bbce88 1077 4 5 false false false 0 0 decimal

spice# and MNA

<content> Numerical Inverse Laplace Transform </content> <content> Gamma </content>

Γ.κ 0.99999999999999709182 57.156235665862923517 59.597960355475491248 - 14.136097974741747174 0.49191381609762019978 - 0.33994649984811888699 104 -^* 0.46523628927048575665 104 -^* 0.98374475304879564677 104 -^* - 0.15808870322491248884 103 -^* 0.21026444172410488319 103 -^* - 0.21743961811521264320 103 -^* 0.16431810653676389022 103 -^* - 0.84418223983852743293 104 -^* 0.26190838401581408670 104 -^* - 0.36899182659531622704 105 -^* 151 mat :

ζ Γ z ζ 1 - : zh 1 - 2 ζ * + 2 / : zg zh 607128 / + : 13 mat 2 π * sqrt zg 0.5 zh *^(2^* e zg^/ Γ.κ 1 el Γ.κ κ 1 + el z κ + / κ 114 sum + (* eval 21 line :

Valid for Re(ζ)gt;0

Heaviside x Φ 1 x sign + 2 / :

Based on algorithm of Oscar:
https://en.smath.info/forum/yaf_postst1341_Signals-and-Systems.aspx

Fs# T# α# β# ℐℒ Fs# IsString num2str 1 ≡ F#(s#):@(s#) @ Fs# num2str strrep str2num s F# Fs# : if f# 0 : k# 2 T# length range st# α# T# k# el / : βt# β# T# k# el / : 12 mat f# k# el βt# st# F# * Re vectorize eval sum : 21 line for T# f# 1 el f# 2 el T# 1 el T# 2 el - T# 3 el T# 2 el - / f# 3 el f# 2 el - (* + : augment eval 41 line :

Fs# T# ℐℒ Fs# T# ℐℒ.α ℐℒ.β ℐℒ 11 line :

a ns nd ℐℒ.αβ a ns nd 13 mat 6201913 mat : α 0 : β 0 : γ 0 : m ns nd + 1 + : 14 mat ν 1 m range : β ν el 1 - (ν 1 +^eval : 12 mat n h nd 1 + Γ 2 nd^nd 2 + n - Γ * n Γ * / eval : n 2 nd range : γ 1 el 1 h : γ n el n h γ n 1 - el + eval : 13 mat n ns 3 + (m range : β 1 el 0.5 β 1 el * : β n el β n el γ m n - 1 + el * eval : 13 mat α ν el a ν 1 - (π * i * + : e a^β * 12 mat 71 line :

ℐℒ.α ℐℒ.β 12 mat 62019 ℐℒ.αβ eval :

<content> MNA WITH STAMPS </content> <content> STAMPS AC </content>

AC simulation

s v Clear 1

Impedance

MNA.AC Z 211 v / 1 v / - 1 v / - 1 v / 22 mat 0021 mat 0021 mat 16 mat :

v Z ≡

Resistor

MNA.AC MNA.AC R 211 v / 1 v / - 1 v / - 1 v / 22 mat 0021 mat 0021 mat 16 mat stack :

v R ≡

Capacitor

MNA.AC MNA.AC C 21 s v * s v * - s v * - s v * 22 mat 0021 mat 0021 mat 16 mat stack :

v C ≡

Inductor

MNA.AC MNA.AC L 211 s v * / 1 s v * - / 1 s v * - / 1 s v * / 22 mat 0021 mat 0021 mat 16 mat stack :

v L ≡

Current
source

MNA.AC MNA.AC I 21000022 mat v - v 21 mat 0021 mat 16 mat stack :

v I ≡

Voltage
source

MNA.AC MNA.AC V 21001001 - 11 - 033 mat 00 v 31 mat 00 I 31 mat 16 mat stack :

v V ≡

Power
source

MNA.AC MNA.AC P 221 v 1 el / 1 v 1 el / - 1 v 1 el / - 1 v 1 el / 22 mat 8 v 2 el * v 1 el / sqrt 8 v 2 el * v 1 el / sqrt - 21 mat 0021 mat 16 mat stack :

v 1 el r ≡

Figures frome QUCS online help

v 2 el P ≡

OpAmp

MNA.AC MNA.AC O 30000000010000101 - 044 mat 000041 mat 000 I 41 mat 16 mat stack :

VCCS

MNA.AC MNA.AC G 41000000000100001 - 000001001 - 1 v / - 55 mat 0000051 mat 0000 I 51 mat 16 mat stack :

v G ≡

CCCS

MNA.AC MNA.AC F 4100001 v / 0000100001 - 00001 v / - 1001 - 055 mat 0000051 mat 0000 I 51 mat 16 mat stack :

v G ≡

VCVS

MNA.AC MNA.AC E 410000000001 - 0000100000 v 1 - 1 v - 055 mat 0000051 mat 0000 I 51 mat 16 mat stack :

v G ≡

CCVS

MNA.AC MNA.AC H 41000010000001 - 00000100001 - 0011 - 0 v - 01001 - 0066 mat 00000061 mat 0000 Iin Iout 61 mat 16 mat stack :

v G ≡

Ideal
Transf.

MNA.AC MNA.AC K 4100001 - 0000 v 0000 v - 000011 v - v 1 - 055 mat 0000051 mat 0000 I 51 mat 16 mat stack :

v T ≡

End

MNA.AC MNA.AC num2str str2num :

MNA.AC rows 13

s v MNA MNA.AC :

Carriage return and Line feed Unicode char CR LF 12 mat \ 000D 000A 12 mat \ concat str2num num2str vectorize : 12 mat

ENTER CR :

A## MInverse A## 1 -^11 line :

str# MNA.Val x# str# strrep : -.0123456789 x# 11 substr findstr 1 - ≡ x# v# V A Ω ohm S F H C T s Hz J K 113 mat : k# 1 v# cols range x# x# v# k# el *' v# k# el concat strrep : for v# f p n μ m k M G T u 15 - 12 - 9 - 6 - 3 - 369126 - 210 mat : k# 1 v# cols range x# x# v# 1 k# el *(10^v# 2 k# el num2str) concat strrep : for x# 51 line if 21 line :

str# n# MNA.Description s# mat(description(str#),1,1) concat str2num 1 el : s# findstr 1 - ≠ s# s# strrep : while s# s# ENTER strsplit : k# 0 : out# 0 : 13 mat j# 1 s# length range s# j# el strlen 0 > x# l x# j# el strlen 1 - : s# j# el 11 substr ≡ s# j# el s# j# el 2 s# l substr : if s# j# el s# l 1 + 1 substr ≡ s# j# el s# j# el 1 s# l substr : if s# j# el strlen n# ≥ out# k# k# 1 + : el s# j# el : if 41 line if for out# 51 line :

χ# O# Σ# MNA A# 9999 matrix eval : B# A# 1 col : X# B# : 13 mat C# 0 : N# 0 : ι# 0 : s v Clear E# s v MNA : 15 mat str# χ# 5 MNA.Description : Σ# line(Clear(12 mat : 12 mat k# 1 E# rows range : POS# E# k# 1 el k# augment : 12 mat k# 1 str# length range σ# str# k# el strsplit : n# POS# σ# 1 el 1 findrows 2 el POS# σ# 1 el 11 substr 1 findrows 2 el 0 try try : (0 ≠ C# ι# ι# 1 + : 1 el n# : C# ι# 2 el σ# 1 el : np# E# n# 3 el : 13 mat np# 0 ≡ (E# n# 2 el 2 + σ# length > (np# 1 ≡ (& (| C# ι# 4 el σ# 1 el : np# 1 ≡ C# ι# 4 el σ# E# n# 2 el 2 + el MNA.Val : t# 0 : j# 1 np# range : 12 mat t# 1 j# el σ# E# n# 2 el 1 + j# + el MNA.Val str2num : C# ι# 4 el t# num2str : 31 line if if t# 0 : j# 1 E# n# 2 el range : C# ι# 3 el t# j# el σ# j# 1 + el str2num : (: 13 mat N# N# t# stack max eval : Σ# Σ# 1 el σ# 1 el : C# ι# 4 el, concat Σ# 2 el σ# 1 el, concat 12 mat : 51 line if 21 line for M# N# 1 + : k# 1 M# range : X# k# el'V_k# 1 - num2str concat str2num : 13 mat Σ# Σ# 1 el 1, ι# 1 + num2str,1) concat Σ# 2 el v) concat 12 mat : k# 1 ι# range r# C# k# 1 el : n# E# r# 2 el : 12 mat u# EV# v F# E# r# u# el : C# k# 4 el str2num F# 21 line : u# NOD# u# n# ≤ C# k# 3 el u# el 1 + M# u# + n# - if 11 line : 12 mat stamp# 4 EV# : ν# stamp# rows : 12 mat sr# 1 ν# range nr# sr# NOD# : B# nr# el 5 EV# sr# el B# nr# el + : 12 mat sr# n# > X# nr# el' 6 EV# sr# el num2str_C# k# 2 el concat str2num X# nr# el + : if sc# 1 ν# range nc# sc# NOD# : A# nr# nc# el stamp# sr# sc# el A# nr# nc# el + : 12 mat for 31 line for M# M# ν# + n# - : 51 line for r# 2 M# range : A# A# r# r# el : B# B# r# el : X# X# r# el : 14 mat O# num2str ≡ VI# A# MInverse B# * : A# B# X# VI# 1 N# range el VI# N# 1 + (M# 1 - (range el 15 mat num2str str2num 21 line O# num2str ABX ≡ A# B# X# 13 mat num2str str2num if 101 line :

N Number_of_nodes ≡

MC STAMP_POS NAMES NODES_VEC NUMBER_OF_VALUES VALUES 15 mat ≡

ι elements_in_cir ≡

np number_of_params ≡

cir# MNA cir# MNA.VARS MNA 11 line :

<content> Spice </content>

Wraper for handle some issues with spice#, like taking strings from the description, and a way to add string lines, just for rehuse the code withing MNA.

χ# A# X# Spice C# // circuit CR description(#1) #1 χ# strrep str2num concat : a# A# C# C# CR a# concat : for x# X# C# C# CR .SAVE x# concat : for C# spice#("#1","#2") #1 C# CR .END concat strrep : ans# 02 matrix eval : x# X# ans# ans# C# #2 x# strrep str2num stack eval : for ans# 71 line :

<content> MNA Method </content>

MNA Method

Given a linear circuit with n Nodes and m sources, solve the system

~A ~X * ~B ≡

~A n m + n m + el G n n * el C n m * el C m n * el D m m * el 22 mat ≡

G have the passive elements: the diagonal elements G(n,n) are the sum of the conductance of each element connected to the node n, and the others G(i,j) are the negative conductance of the element connected to the nodes i,j.

C have the voltage sources, where if a voltage source is connected to nodes m, n, then C(m,n) = ±1, or zero otherwise.

~X n m + el V_#N I_V#M 21 mat ≡

D is the same of B, but with dependent sorces.

X have the node voltages and currents trhough voltage sources.

~B n m + el I.X Σ V m el 21 mat ≡

For b, b(n) = sum of the currents through the passive elements into the node n, and b(m) = value of the independent voltage sources m.

Example MNA Method

Vb 1 0R1 1 2R2 2 3R3 3 4R4 2 0R5 3 0Is 0 4 χs 1 :

Vb 1 0 40R1 1 2 2R2 2 3 9R3 3 4 8R4 2 0 10R5 3 0 4Is 0 4 1 χ 1 :

G1 1 R1 / :

G2 1 R2 / :

G3 1 R3 / :

G4 1 R4 / :

G5 1 R5 / :

G G1 G1 - 00 G1 - G1 000000000044 mat 00000 G2 G2 - 00 G2 - G2 0000044 mat + 0000000000 G3 G3 - 00 G3 - G3 44 mat + 00000 G4 000000000044 mat + 0000000000 G5 0000044 mat + :

D 011 mat :

~B 000 Is - Vb 51 mat ≡

C 100041 mat ≡

V.b

A B X V I 15 mat χs MNA :

X V_1 V_2 V_3 V_4 I_Vb 51 mat

at node 4

First V source

A 1 R1 / 1 R1 / - 0011 R1 / - R2 R4 + (R1 * R2 R4 * + R1 R2 * R4 * / 1 R2 / - 0001 R2 / - R3 R5 + (R2 * R3 R5 * + R2 R3 * R5 * / 1 R3 / - 0001 R3 / - 1 R3 / 01000055 mat

B 000 Is Vb 51 mat

D G G1 G2 G3 G4 G5 Clear 1

Numerically:

A B X V I 15 mat χ MNA :

V 4030122041 mat

A 1 -^B * 403012205 - 51 mat

χ .OP V(1) V(2) V(3) V(4) I(Vb) 51 mat Spice 04003001202005 - 52 mat

I 5 - 11 mat

Example spice# MNA Method

spice sharp also uses MNA. From:

https://spicesharp.github.io/SpiceSharp/articles/tutorials/writing_behaviors/modified_nodal_analysis.html

V1 1 0 1R1 1 2 5R2 2 3 7R3 2 0 10I1 0 3 1.5 χ 1 :

A B X V I 15 mat χ MNA :

A 0.2 0.2 - 01 0.2 - 0.44 0.14 - 00 0.14 - 0.14 0100044 mat

X V_1 V_2 V_3 I_V1 41 mat

B 00 1.5 141 mat

V 1 5.67 16.17 31 mat

χ .OP V(1) V(2) V(3) I(V1) 41 mat Spice 010 5.67 0 16.17 0 0.93 42 mat

I 0.93 11 mat

Ohm's Law

V1 1 0R1 1 0R2 1 0 χs 1 :

V1 1 0 5R1 1 0 100R2 1 0 1k χ 1 :

1

A B X V I 15 mat χs MNA :

V V1 11 mat

I R1 R2 + (V1 * R1 R2 * / - 11 mat

A B X V I 15 mat χ MNA :

V 511 mat

I 0.06 - 11 mat

χ .OP I(V1) Spice 0 0.06 - 12 mat

V1 1 2R1 1 0R2 2 0 χs 1 :

V1 1 2 5R1 1 0 100R2 2 0 1k χ 1 :

1

2

A B X V I 15 mat χs MNA :

V R1 V1 * R1 R2 + / R2 V1 * R1 R2 + / - 21 mat

I V1 R1 R2 + / - 11 mat

A B X V I 15 mat χ MNA :

V 0.45 4.55 - 21 mat

χ .OP V(1) V(2) I(V1) 31 mat Spice 0 0.45 0 4.55 - 0032 mat

I 011 mat

<content> Resistive networks </content>

Resistive networks

V1 1 0 8R1 1 2 4I1 0 2 5R2 2 0 6R3 2 3 7R4 3 4 9V2 0 4 8R5 3 0 10I2 3 0 6 χ 1 :

A B X V I 15 mat χ MNA :

V 8 8.41 16.07 - 8 - 41 mat

χ .OP V(1) V(2) V(3) V(4) I(V1) I(V2) 61 mat Spice 080 8.41 0 16.07 - 08 - 0 0.1 0 0.9 62 mat

I 0.1 0.9 21 mat

<content> RL Circuit </content>

RL Circuit

Symbolic and Numerical analysis

V1 1 0R1 1 2L1 2 0 χs 1 :

V1 1 0 5R1 1 2 300L1 2 0 100 χ 1 :

A B X V I 15 mat χs MNA :

I V1 R1 s L1 * + / - 11 mat

V V1 V1 s * L1 * R1 s L1 * + / 21 mat

A B X V I 15 mat χ MNA :

O .IC V(2) = 0 .TRAN 0 2 0.1 21 mat :

.IC I(L1) = 0

χ O V(2) Spice V 2 el s / 0 2 0.1 range ℐℒ 2 1 sys 1 1 line χ O I(L1) Spice I 1 el s / - 0 2 0.1 range ℐℒ 2 1 sys 1 1 line <content> RC Circuit </content>

X 0.9m is wrong, it correct expression is 900u or 900μ

RL Circuit

Symbolic and Numerical analysis

V1 1 0R1 1 2C1 2 0 χs 1 :

V1 1 0 12R1 1 2 22kC1 2 0 0.9m χ 1 :

A B X V I 15 mat χs MNA :

I s C1 * V1 * 1 s C1 * R1 * + / - 11 mat

V V1 V1 1 s C1 * R1 * + / 21 mat

A B X V I 15 mat χ MNA :

O .IC V(2) = 0 .TRAN 0 140 0.5 21 mat :

χ O V(2) Spice V 2 el s / 0 140 5 range ℐℒ 2 1 sys 1 1 line χ O I(C1) Spice I 1 el s / - 0 140 5 range ℐℒ 2 1 sys 1 1 line <content> opamps </content>

Summary of basic
op amp circuits

Shorthand:

n V χ MNA 4 el n el 11 line :

Inverter
amplifier

V.i 1 0R.1 1 2R.2 2 3O 0 3 2 χ 1 :

V.o 3 V : V.i R.2 * R.1 / -

Non-
inverter
amplifier

V.i 1 0R.1 0 2R.2 2 3O 1 3 2 χ 1 :

V.o 3 V : R.1 R.2 + (V.i * R.1 /

Voltage
Follower

V 1 0 V.iO 1 2 2 χ 1 :

V.o 2 V : V.i

Summer

V 1 0 V.1V 2 0 V.2V 3 0 V.3R 1 4 R.1R 2 4 R.2R 3 4 R.3R 4 5 R.fO 0 5 4 χ 1 :

V.o 5 V : R.f V.1 R.2 * V.2 R.1 * + (R.3 * V.3 R.2 * R.1 * + (* R.3 R.2 * R.1 * / -

V.o R.f V.1 / R.f V.2 / + R.f V.3 / + (- ≡

Difference
Amplifier

V 1 0 V.1V 2 0 V.2R 1 3 R.1R 2 4 R.2R 3 4 R.3R 4 5 R.fO 0 5 4 χ 1 :

V.o 5 V : R.f V.1 R.2 * R.1 R.3 + (V.2 * + (* R.1 R.3 + (R.2 * / -

V.o R.2 R.1 / V.2 V.1 - (* ≡

Opamp for spice#

x E 10 x^:

.subckt opamp 1 2 3G1 0 2 1 3 100kR3 2 0 1C3 2 0 0.001591549430919.ends opamp opamp 1 :

_Aol 1003 E * :

_GBW 106 E * :

_Aol 2 π *_GBW * / 0.001591549430919

opamp opamp description :

nonivin

invin

Example with spice#: Inverter
amplifier

V.i 1 0 5R.1 1 2 10R.2 2 3 20X1 0 3 2 opamp χ 1 :

V.3 V.i R.2 * R.1 / - ≡

χ .OP opamp 21 mat V(1) V(2) V(3) 31 mat Spice 0500010 - 32 mat

Sweep analysis

V 1 0 5VV 2 0 3VV 3 0 4VR 1 4 2ΩR 2 4 8ΩR 3 4 7ΩR 4 5 R.fO 0 5 4 χ 1 :

R.f V.o 5 V :

V.o 10 V * - :

R.f R.f Ω * V.o V.o ≡ R.f roots Ω * : 2.9 Ω *

x Ω * V.o x 0 > ( * R.f V.o o augment 2 1 sys

Sanity check

V 53431 mat V * :

R 28731 mat Ω * :

R.f V.o V 1 el R 1 el / V 2 el R 2 el / + V 3 el R 3 el / + / - : 2.9 Ω *

TODO: Sweep analysis with spice#

<content> opamp: Capacitance multiplier </content>

Capacitance multiplier

Vb 1 0R1 1 3 1k R2 1 2 100kC1 2 0 100nO1 2 3 3 χ 1 :

A B X V I 15 mat χ MNA :

Laplace transform of a pulse:

To 5103 -^* :

Vb 5 s 1 e To s * -^+ (* / :

Vb 1 0 PULSE(0 5 0 0 0 5m 10m)R1 1 3 1k R2 1 2 100kC1 2 0 100nX1 2 3 3 opamp χ 1 :

O opamp .TRAN 0 0.05 0.1m 21 mat :

T 0 0.05 0.001 range :

T rows 51

χ O V(1) Spice χ O V(2) Spice V 1 el T ℐℒ V 2 el T ℐℒ 4 1 sys 1 1 line χ O I(Vb) Spice χ O I(R1) Spice I 1 el T ℐℒ I 2 el T ℐℒ 4 1 sys

Vb Clear 1

<content> opamp: DC OP Point </content>

DC OP Point

Vb 1 0R1 2 3 R2 3 4 R3 1 2 R4 5 0 C1 2 3C2 4 5O1 0 2 3O2 0 4 5 χs 1 :

Vb 1 0 12R1 2 3 1kR2 3 4 100kR3 1 2 100kR4 5 0 100kC1 2 3 10nC2 4 5 10nX1 0 2 3 opampX2 0 4 5 opamp χ 1 :

R1 R2 R3 R4 14 mat 1 G1 / 1 G2 / 1 G3 / 1 G4 / 14 mat :

A B X 13 mat χs ABX Σ MNA :

A X * B ≡

A G3 G3 - 000100 G3 - G3 G1 + s C1 * + G1 s C1 * + (- 000100 G1 s C1 * + (- G2 G1 + s C1 * + G2 - 000000 G2 - G2 s C2 * + s C2 * - 001000 s C2 * - G4 s C2 * + 00010000000001 - 0000000001 - 00088 mat

X V_1 V_2 V_3 V_4 V_5 I_Vb I_O1 I_O2 81 mat

B 00000 Vb 0081 mat

DC Operating point (C as open circuits):

From ltspice:

χ .OP opamp 21 mat V(1) V(2) V(3) I(R4) I(Vb) 51 mat Spice 0120 1.21 109 -^* 0 1.2 109 -^* 000052 mat

--- Operating Point ---

V(1): 12 voltage
V(3): -0.119999 voltage
V(2): 1.20119e-006 voltage
V(n001): 11999.8 voltage
V(4): -0.119999 voltage
I(C2): 1.19999e-016 device_current
I(C1): -1.2e-021 device_current
I(R4): -0.119998 device_current
I(R3): -0.00012 device_current
I(R2): 1.19999e-016 device_current
I(R1): -0.00012 device_current
I(V1): -0.00012 device_current
Ix(u1:3): 0.00012 subckt_current
Ix(u2:3): -0.119998 subckt_current

s Vb R1 R2 R3 R4 C1 C2 Clear 1

<content> E Source </content>

Example

E1 1 0 4 5 0.5R1 1 2 8R2 2 3 6V1 3 0 120R3 2 4 2R4 4 5 4V2 5 0 60 χ 1 :

O .DC V1 120 120 0.1 :

O .op :

A B X V I 15 mat χ MNA :

Different node convention?

X V_1 V_2 V_3 V_4 V_5 I_E1 I_V1 I_V2 81 mat

V 38.18 38.18 120 10.91 6051 mat

χ O V(1) Spice 0212 mat

χ O V(2) Spice 06612 mat

χ O V(4) Spice 06412 mat

I 25.91 13.64 - 12.27 - 31 mat

χ O I(R1) Spice 08 - 12 mat

χ O I(R4) Spice 0112 mat

SMath determinant can't handle units:

A## MInverse A## inverse maple 11 line :

<content> Controled Sources with Feedback </content>

Controled Sources with Feedback

https://ultimateelectronicsbook.com/dependent-source-feedback/

R 0 1 5ΩR 1 2 10ΩI 0 1 10AE 1 2 0 0 0.5 χ 1 :

R1 0 1 5R2 1 2 10I1 0 1 10E2 2 1 0 0 0.5 χ' 1 :

2

1

A B X V I 15 mat χ MNA :

0

3

V V 402021 mat

χ' .OP V(1) V(2) I(R1) 31 mat Spice 050050010 - 32 mat

I A 2 - 11 mat

Voltage Controled Voltage Source (VCVS)

https://ultimateelectronicsbook.com/dependent-sources/

https://www.circuitlab.com/editor/#?id=77x8xhr2v36v

1

2

V1 1 0 12VV2 3 0 5VR  1 4 1kΩR  0 4 1kΩR  0 2 100ΩE  1 2 3 4 10 χ 1 :

V1 1 0 12V2 3 0 5R1  1 4 1kR2  0 4 1kR3  0 2 100E1  1 2 3 4 10 χ' 1 :

4

3

0

A B X V I 15 mat χ MNA :

V V 12655641 mat

χ' .OP V(1) V(2) V(3) V(4) I(R1) I(R2) I(R3) 71 mat Spice 01202205060 0.01 0 0.01 - 0 0.22 - 72 mat

I mA 6 - 650 - 65031 mat

X 57 range el I_V1 I_V2 I_E 31 mat

A det 0 S *

Current Controled Current Source (CCCS)

http://www.solved-problems.com/circuits/electrical-circuits-problems/561/nodal-analysis-problem-dependent-voltage-current-sources/

I 4 0 9AV 3 0 2VR 1 4 1ΩR 3 4 2ΩR 2 3 1ΩR 0 5 2ΩHx 5 2 1 2 3ΩGy 3 0 1 4 2S χ 1 :

A B X V I 15 mat χ MNA :

2

3

V V 5222 - 251 mat

I A 2 - 1 - 1841 mat

5 I.x -

1

4

X 69 range el I_V Iin_Hx Iout_Hx I_Gy 41 mat

0

A det 4.25 S *

http://www.solved-problems.com/circuits/electrical-circuits-problems/641/nodal-analysis-dependent-current-source/

Current Controled Current Source (CCCS)

I 3 2 2AV 3 0 10VR 0 3 2ΩR 0 1 3ΩR 2 4 2ΩR 1 2 1ΩFx 4 1 2 3 -2 χ 1 :

A B X V I 15 mat χ MNA :

V V 6.75 9.5 101041 mat

I A 7.25 - 0.5 21 mat

2

1

X V_1 V_2 V_3 V_4 I_V I_Fx 61 mat

LOS VOLTAJES ESTÁN BIEN, CHEKAR LAS CORRIENTES.

3

4 I.x -

A det S 2^/ 0.67 -

0

Current Controled Voltage Source (CCVS)

http://www.solved-problems.com/circuits/electrical-circuits-problems/698/nodal-analysis-dependent-voltage-source/

I 0 2 1AI 0 1 2AR 0 2 4ΩR 0 1 3ΩR 2 3 5ΩH 3 2 0 1 -6Ω χ 1 :

A B X V I 15 mat χ MNA :

V V 4.71 2.57 4.71 31 mat

I A 0.43 - 0.79 - 21 mat

2

X V_1 V_2 V_3 Iin_H Iout_H 51 mat

0

1

3 I.x -

Alvaro

4 appVersion 1.2.9018.0