<content> Numerical Inverse Laplace Transform </content>

c1a2a71c-d990-4f06-9a0d-8d7195bbce88 1269 4 5 false false false 0 3 decimal

&[DATE] &[TIME] - &[FILENAME]

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

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

Fs# T# α# β# ℐℒ s F# Fs# : f# 0 : 12 mat 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 31 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 </content> <content> STAMPS AC </content>

AC simulation

s v MNA Clear 1

Subckt

M X MNA 12 el MNA 13 el MNA 14 el MNA 15 el MNA 16 el 16 mat :

v

Resistor

M M R 211 v / 1 v / - 1 v / - 1 v / 22 mat 0021 mat 0021 mat 16 mat stack :

v R ≡

Capacitor

M M C 21 s v * s v * - s v * - s v * 22 mat 0021 mat 0021 mat 16 mat stack :

v C ≡

Inductor

M M 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

M M I 21000022 mat v - v 21 mat 0021 mat 16 mat stack :

v I ≡

Voltage
source

M M V 21001001 - 11 - 033 mat 00 v 31 mat 00 I 31 mat 16 mat stack :

v V ≡

Power
source

M M 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 ≡

v 2 el P ≡

OpAmp

M M O 30000000010000101 - 044 mat 000041 mat 000 I 41 mat 16 mat stack :

VCCS

M M G 410000100001 - 00000000000011 - 1 v / - 55 mat 0000051 mat 0000 I 51 mat 16 mat stack :

v G ≡

CCCS

M M F 410000100001 - 00001 v / 00001 v / - 0011 - 055 mat 0000051 mat 0000 I 51 mat 16 mat stack :

v G ≡

VCVS

M M E 410000100001 - 000000000011 - v - v 055 mat 0000051 mat 0000 I 51 mat 16 mat stack :

v G ≡

CCVS

M M H 41000001000001 - 00001000001 - 00011 - 0011 - 00 v - 066 mat 00000061 mat 0000 Ix I 61 mat 16 mat stack :

v G ≡

Ideal
Transf.

M M K 4100001 - 0000 v 0000 v - 000011 v - v 1 - 055 mat 0000051 mat 0000 I 51 mat 16 mat stack :

v T ≡

End

M M num2str str2num :

M rows 13

s v MNA M :

M Clear 1

A# B# MLinSol A# 1 -^B# * 11 line :

Carriage Return and Line Feed Unicode chars CR LF 12 mat \ 000D 000A 12 mat \ concat str2num num2str vectorize :

ENTER CR :

χ# P# O# MNA So# f p n μ m K M G T u k MEG strsplit : Vo# -15 -12 -9 -6 -3 3 6 9 12 -6 3 6 strsplit : Vo# *(10^Vo#) concat vectorize : U# V A Ω ohm S F s Hz C T Hy J strsplit : ξ# V# X# ξ# strrep : -.0123456789 X# 11 substr findstr 1 - ≡ X# k# 1 So# rows range X# X# So# k# el Vo# k# el strrep : for k# 1 U# rows range X# X# U# k# el *' U# k# el concat strrep : for X# 31 line if 21 line : O# s ≡ S# χ# : k# 0 : Y# 0 : 13 mat S# IsString num2str 1 ≡ (S# strlen 0 > (& S# findstr 1 - ≡ (S# 11 substr . ≠ (& S# description(@) @ S# strrep str2num : if S# S# LF CR strrep CR strsplit : 21 line S# χ# stack eval : if s# S# s# strlen 1 > s# findstr 1 - ≠ s# s# strrep : while s# 11 substr ≡ s# s# 2 s# strlen 1 - substr : if s# s# strlen 1 substr ≡ s# s# 1 s# strlen 1 - substr : if s# 11 substr * ≠ s# 11 substr + ≡ Y# k# el Y# k# el s# 3 substr concat : s# strlen P# ≥ Y# k# k# 1 + : el s# : if if 41 line if for Y# 41 line s# P# 8 s MNA s# strlen 8 > (s# 16 substr .PARAM ≡ (& s# s# 8 substr strsplit : σ# s# σ# σ# strrep = strsplit : σ# 1 el : σ# 2 el V# concat str2num 21 line for 21 line if 11 line for 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 k# 1 E# rows range : No# k# E# k# 1 el augment : 12 mat s# χ# 5 s MNA σ# s# strsplit : n# No# σ# 1 el 2 findrows : (0 ≡ n# No# σ# 1 el 11 substr 2 findrows : if n# 0 ≠ n# n# 1 el : ι# ι# 1 + : 12 mat n# 1 ≡ MNA σ# σ# rows el str2num : if C# ι# 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 V# : t# 0 : j# 1 np# range : 12 mat t# 1 j# el σ# E# n# 2 el 1 + j# + el V# str2num : C# ι# 4 el t# num2str : 31 line if t# 0 : j# 1 E# n# 2 el range : 12 mat C# ι# 3 el t# j# el σ# j# 1 + el str2num : (: N# N# t# stack max eval : 71 line if 31 line for M# N# 1 + : k# 1 M# range : 12 mat X# k# el'V(k# 1 - num2str) concat str2num : k# 1 ι# range r# C# k# 1 el : n# E# r# 2 el : 12 mat u# E# v F# E# r# u# el : C# k# 4 el str2num F# 21 line : u# N# u# n# ≤ C# k# 3 el u# el 1 + M# u# + n# - if 11 line : Σ# 4 E# : ν# Σ# rows : 12 mat Λ# 1 ν# range λ# Λ# N# : B# λ# el 5 E# Λ# el B# λ# el + : 12 mat Λ# n# > s#' 6 E# Λ# el num2str (' C# k# 2 el) concat : X# λ# el s# str2num X# λ# el + : 21 line if sc# 1 ν# range c# sc# N# : A# λ# c# el Σ# Λ# sc# el A# λ# c# el + : 12 mat for 31 line for M# M# ν# + n# - : 61 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# B# MLinSol : Y# A# B# X# VI# 1 N# range el VI# N# 1 + (M# 1 - (range el 15 mat : 21 line O# num2str ABX ≡ Y# A# B# X# 13 mat : Y# 0 : if Y# num2str str2num 111 line if 61 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# dot# MNA cir# dot# MNA 11 line :

cir# MNA cir# MNA 11 line :

<content> Spice </content>

Wraper for spice#

χ# A# X# Spice C# spice#("* circuit CR concat : Y# 02 matrix : 12 mat c# χ# 3 s MNA C# C# CR c# concat : for a# A# α# a# 3 s MNA C# C# CR α# concat : for for C# C# CR .SAVE @ CR .END","@") concat : x# X# num2str vectorize' strrep vectorize Y# Y# C# @ x# strrep str2num x# - 12 mat try stack eval : for Y# 61 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 ≡

~X n m + el N# V M# I 21 mat ≡

~B n m + el I.X Σ V m el 21 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.

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

X have the node voltages and currents trhough voltage sources.

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 χs :

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

G1 1 R1 / :

G2 1 R2 / :

G3 1 R3 / :

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

G4 1 R4 / :

G5 1 R5 / :

G G 00000 G4 000000000044 mat + 0000000000 G5 0000044 mat + :

D 011 mat ≡

~B 000 Is - Vb 51 mat ≡

C 100041 mat ≡

V.b

at node 4

First V source

A B X V I 15 mat χs MNA :

X 1 V 2 V 3 V 4 V Vb I 51 mat

B 000 Is Vb 51 mat

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

D G G1 G2 G3 G4 G5 Clear 1

Numerically:

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

A B X V I 15 mat χ MNA :

V 4030122041 mat

A 1 -^B * 403012205 - 51 mat

I 5 - 11 mat

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

X 1 V 2 V 3 V 4 V Vb I 51 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 χ :

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

B 00 1.5 141 mat

X 1 V 2 V 3 V V1 I 41 mat

V 1 5.67 16.17 31 mat

χ .OP X 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 χs :

V1 1 0 5R1 1 0 100R2 1 0 1k χ χ :

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 V1 I Spice 0 0.06 - 12 mat

V1 1 2R1 1 0R2 2 0 χs χs :

V1 1 2 5R1 1 0 100R2 2 0 1k χ χ :

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

I 011 mat

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

A B X V I 15 mat χ MNA :

VI χ .OP X Spice :

V 8 8.41 16.07 - 8 - 41 mat

VI 080 8.41 0 16.07 - 08 - 0 0.1 0 0.9 62 mat

X 1 V 2 V 3 V 4 V V1 I V2 I 61 mat

I 0.1 0.9 21 mat

<content> RL Circuit </content>

RL Circuit

Symbolic and Numerical analysis

V1 1 0 V.1R1 1 2 R.1L1 2 0 L.1 χs χs :

V1 1 0 5R1 1 2 300L1 2 0 100 χ χ :

O .IC V(2) = 0 .TRAN 0 3 0.01 21 mat :

.IC I(L1) = 0

A B X V I 15 mat χs MNA :

inverting laplace transforms:

V V.1 V.1 s * L.1 * R.1 s L.1 * + / 21 mat

I V.1 R.1 s L.1 * + / - 11 mat

V 2 el s / ℐℒ t V.2 ≡

I 1 el s / - ℐℒ t I.L ≡

t V.2 V.1 e R.1 t * L.1 / -^* :

t I.L V.1 R.1 / 1 e R.1 t * L.1 / -^- (* :

V.1 5 :

R.1 300 :

L.1 100 :

T 03 0.1 range :

χ O 2 V Spice T T V.2 vectorize augment 2 1 sys 1 1 line

V 2 el s / T ℐℒ

χ O L1 I Spice T T I.L vectorize augment 2 1 sys 1 1 line

I 1 el s / - T ℐℒ

Pulse

V1 1 0 PULSE(0 4 0 1-1n 0 1n 1)R1 1 2 300L1 2 0 100 χ χ :

V.1 Clear 1

t V.1 4 t t 1 - u t * - (* : t u 0.5 1 t sign + (* : 21 line

x 1 mod V.1 x 0 > ( * 1 1 sys

Using

t a - u t * ℒ e a s * -^t a + ℒ * ≡

2nd shift prop

t a - u ℒ e a s * -^s / ≡

t V.1 ℒ 4 t ℒ * 4 e a s * -^t 1 + ℒ * * - ≡

then

ℒV.1 41 - e s -^s * + e s -^+ (* s 2^/ - :

in the freq domain

Finally, using that a periodic function f with period p and "first period" expression f* have

t f ℒ t f\002A\ ℒ 1 e p s * -^- / ≡

ℒV.2 V 2 el V.1 ℒV.1 equrep 1 e 1 s * -^- / : 4 s e s^- 1 + (* s 3 s + (* e s^1 - (* / -

ℒI.1 I 1 el - V.1 ℒV.1 equrep 1 e 1 s * -^- / : s e s^- 1 + 25 s 2^* 3 s + (* e s^1 - (* / -

χ O 1 V Spice χ O 2 V Spice T T 1 mod V.1 vectorize . augment ℒV.2 T ℐℒ . augment 4 1 sys 1 1 line χ O L1 I Spice ℒI.1 T ℐℒ . augment 2 1 sys 1 1 line <content> RC Circuit </content>

RC Circuit

Symbolic and Numerical analysis

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

V1 1 0R1 1 2C1 2 0 χs χs :

V1 1 0 12R1 1 2 22kC1 2 0 0.9m χ χ :

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 :

t V.2 V 2 el s / s t invlaplace maple : 1215 t * 99 / - exp - (*

t I.C I 1 el s / - s t invlaplace maple : 35 t * 99 / - exp * 5500 /

RL Circuit

Symbolic and Numerical analysis

V1 1 0 12R1 1 2 22kC1 2 0 0.9m χ χ :

O .IC V(2) = 0 .TRAN 0 140 5 21 mat :

A B X V I 15 mat χs MNA :

inverting laplace transforms:

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

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

V 2 el s / ℐℒ t V.2 ≡

I 1 el s / - ℐℒ t I.L ≡

t V.2 121 e 5 t * 99 / -^- (* :

t I.L 35500 / e 5 t * 99 / -^* :

t V.2 χ O 2 V Spice 2 1 sys 1 1 line

V 2 el s / 01405 range ℐℒ

t I.C χ O C1 I Spice 2 1 sys 1 1 line

I 1 el s / - 01405 range ℐℒ

<content> RLC Circuit </content>

RLC Circuit

* RLC SerieV1 1 0 PULSE(0 5 0 1ns 1ns 5ms 10ms)R1 1 2 10L1 2 3 1mC1 3 0 100u.TRAN 0.1ms 10ms 0 0.05ms.SAVE V(1) V(2) V(3) I(L1).END χs χs :

V1 1 0 V.1R1 1 2 10L1 2 3 1mC1 3 0 100u χo χo :

S χs description 1 V 2 V 3 V L1 I 41 sys spice# :

t V1 t V2 t V3 t IL1 4 1 sys

Using MNA

V.1 Vo to Clear 1

t V.1 Vo Vo t to - Heaviside * - :

A B X V I 15 mat χo MNA :

Vo 5 :

to 5103 -^* :

T 02 0.02 range to * :

V.1 1 e s to * -^- s / Vo * :

V.2 V 2 el T ℐℒ :

I I 1 el - T ℐℒ :

V.3 V 3 el T ℐℒ :

t V.1 t 0 > ( * V.2 V.3 I 4 1 sys <content> Controlled sources </content> <content> E: VCVS </content>

Voltage Controlled Voltage Source (VCVS)

Example

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

A B X V I 15 mat χ MNA :

VI χ .OP X Spice :

V 12655641 mat

VI 01206505060 0.006 - 0 0.65 - 0 0.65 - 72 mat

X 1 V 2 V 3 V 4 V V1 I V2 I E1 I 71 mat

I 0.01 - 0.65 - 0.65 - 31 mat

Example

R1 2 1 4R2 4 0 6V1 1 0 12V2 4 3 4E1 2 3 0 4 2 χ χ :

A B X V I 15 mat χ MNA :

VI χ .OP X Spice :

V 124452 - 48 - 41 mat

VI 012044052 - 048 - 080808 - 72 mat

X 1 V 2 V 3 V 4 V V1 I V2 I E1 I 71 mat

I 888 - 31 mat

Current Controlled Current Source (CCCS)

Example

R1 1 0 200R2 2 1 100I1 0 2 4G1 0 2 1 2 3 χ χ :

A B X V I 15 mat χ MNA :

VI χ .OP X Spice :

V 2.66 3.99 21 mat

VI 0 2.66 0 3.99 0 3.99 - 32 mat

X 1 V 2 V G1 I 31 mat

I 3.99 - 11 mat

Current Controlled Current Source (CCCS)

Example

Adding a node for the MNA method:

5

I1 1 2 0.5V1 5 0 10R1 2 1 2R2 0 1 60R3 2 3 40R4 3 0 80R5 4 3 20F1 2 0 4 5 -3 χ χ :

I1 1 2 0.5V1 4 0 10R1 2 1 2R2 0 1 60R3 2 3 40R4 3 0 80R5 4 3 20F1 2 0 V1 -3 χ.S χ.S :

A B X V I 15 mat χ MNA :

VI χ.S .OP X Spice :

V 51.82 54.55 21.3 101051 mat

VI 0 51.82 0 54.55 0 21.3 010 V(5) - 0 0.56 0 1.69 - 72 mat

X 1 V 2 V 3 V 4 V 5 V V1 I F1 I 71 mat

I 0.56 1.69 - 21 mat

Example

Adding 0V source:

I1 4 2 2V1 4 0 10R1 4 0 2R2 0 1 3R3 3 2 2R4 1 2 1F1 1 2 3 4 -2 χ χ :

I1 4 2 2V1 4 0 10R1 4 0 2R2 0 1 3R3 3 2 2R4 1 2 1Vx 3 4 0F1 1 2 Vx -2 χ.S χ.S :

A B X V I 15 mat χ MNA :

VI χ.S .OP X Spice :

V 6.75 9.5 101041 mat

VI 0 6.75 0 9.5 0100100 7.25 - 0 0.5 62 mat

X 1 V 2 V 3 V 4 V V1 I F1 I 61 mat

I 7.25 - 0.5 21 mat

Current Controlled Voltage Source (CCVS)

Example

For MNA we need to add N3 for specify the current-sensing branch.

3

V1 2 3 {V1}R1 1 2 {R1}H1 1 0 3 0 {G} χ χ :

V1 2 0 {V1}R1 1 2 {R1}H1 1 0 V1 {G} χ2 χ2 :

.PARAM V1=12 R1=100 G=-5.OP Π Π :

A B X V I 15 mat χ Π MNA :

VI χ2 Π X Spice :

V 0.57 12031 mat

VI 0 0.5714 012 V(3) - 0 0.1143 - Ix(H1) - 0 0.1143 62 mat

X 1 V 2 V 3 V V1 I H1 Ix H1 I 61 mat

I 0.11 - 0.11 - 0.11 31 mat

Another way for spice is adding a 0 V source in series with the current-sensing branch.

V1 2 3 {V1}R1 1 2 {R1}Vx 3 0 0H1 1 0 Vx {G} χ3 χ3 :

Renaming:

X 5 el H1 Ix

X 5 el Vx I :

χ3 Π X Spice 0 0.57 012000 0.11 - 0 0.11 - 0 0.11 62 mat

X 1 V 2 V 3 V V1 I Vx I H1 I 61 mat

That is optional when we have a V source, but we need it if the current control is not across a V, like in the next example.

Example

Adding 0V source:

V 1 el V 3 el ≡

I1 0 2 1I2 0 1 2R1 0 2 4R2 0 1 3R3 2 3 5 H1 2 0 3 1 -6 χ χ :

I1 0 2 1I2 0 1 2R1 0 2 4R2 0 1 3R3 2 3 5Vx 3 1 0H1 2 0 Vx -6 χ.S χ.S :

A B X V I 15 mat χ MNA :

X 4 el Vx I :

VI χ.S .OP X Spice :

V 4.71 2.57 4.71 31 mat

VI 0 4.71 0 2.57 0 4.71 0 0.43 - 0 0.79 52 mat

X 1 V 2 V 3 V Vx I H1 I 51 mat

I 0.43 - 0.79 21 mat

<content> Controlled sources Example </content>

Example

---

I1 0 4 0.6V1 3 0 5R1 1 4 10R2 3 4 20R3 2 3 10R4 0 5 20H1 2 1 2 5 3G1 0 1 3 4 2 χ χ :

I1 0 4 0.6V1 3 0 5R1 1 4 10R2 3 4 20R3 2 3 10R4 0 5 20Vx 2 5 0H1 2 1 Vx 3G1 0 1 3 4 2 χ.S χ.S :

A B X V I 15 mat χ MNA :

Renaming

X 7 el H1 Ix

X 7 el Vx I :

VI χ.S .OP X Spice :

V 0.17 - 0.2 - 5 5.55 0.2 - 51 mat

VI 0 0.17 - 0 0.2 - 050 5.55 0 0.2 - 0 0.49 - 0 0.01 - 0 0.53 0 1.1 - 92 mat

X 1 V 2 V 3 V 4 V 5 V V1 I Vx I H1 I G1 I 91 mat

I 0.49 - 0.01 - 0.53 1.1 - 41 mat

<content> opamps </content>

Opamp for spice#

.subckt OA 1 2 3G1 0 2 1 3 100kR3 2 0 1C3 2 0 0.001591549430919.ends OA OA O 30000000010000101 - 044 mat 000041 mat 000 I 41 mat 16 mat :

spice# subcircuit

_Aol 100103^* :

_GBW 10106^* :

nonivin

_Aol 2 π *_GBW * / 0.001591549430919

invin

OA# OA description :

Ideal opamp stamp

Summary of basic
op amp circuits

Shorthands:

n Vs χ MNA 4 el n el 11 line :

Symbolic MNA

n Vn χ Π MNA 4 el n el 11 line :

Numric MNA

n V# χ .OP Π OA# 31 mat n V Spice 11 line :

Numeric spice#

.PARAM Vi=5 R1=3K R2=2K R3=6K.PARAM V1=8 V2=5 V3=12 Rf=2K Π 1 :

Numeric values

Inverter amplifier

Vi 1 0 {Vi}R1 1 2 {R1}R2 2 3 {R2}X1 0 3 2 OA χ 1 :

3 Vs Vi R2 * R1 / -

3 Vn 3.33 -

3 V# 0 3.33 - 12 mat

Non-inverter amplifier

Vi 1 0 {Vi}R1 0 2 {R1}R2 2 3 {R2}X1 1 3 2 OA χ 1 :

3 Vs R1 R2 + (Vi * R1 /

3 Vn 8.3333

3 V# 0 8.3332 12 mat

Voltage Follower

Vi 1 0 {Vi}X1 1 2 2 OA χ 1 :

2 Vs Vi

2 Vn 5

2 V# 0512 mat

Summer

V1 1 0 {V1}V2 2 0 {V2}V3 3 0 {V3}R1 1 4 {R1}R2 2 4 {R2}R3 3 4 {R3}Rf 4 5 {Rf}X1 0 5 4 OA χ 1 :

5 Vn 14.3333 -

5 V# 0 14.3329 - 12 mat

5 Vs Rf V1 R2 * V2 R1 * + (R3 * V3 R2 * R1 * + (* R3 R2 * R1 * / -

Difference Amplifier

V1 1 0 {V1}V2 2 0 {V2}R1 1 3 {R1}R2 2 4 {R2}R3 3 4 {R3}Rf 4 5 {Rf}X1 0 5 4 OA χ 1 :

5 Vn 6.7778 -

5 V# 0 6.7776 - 12 mat

5 Vs Rf V1 R2 * R1 R3 + (V2 * + (* R1 R3 + (R2 * / -

n Vs n Vn n V# Clear 1

<content> opamp: Capacitance multiplier </content>

Capacitance multiplier

Vb 1 0 V.bR1 1 3 1k R2 1 2 100kC1 2 0 100nX1 2 3 3 OA χ χ :

Vb 1 0 PULSE(0 12 0 0 0 5m 10m)R1 1 3 1k R2 1 2 100kC1 2 0 100nX1 2 3 3 OA χ.S χ.S :

O OA# .TRAN 0 60m 0.1m 21 mat :

T 0601 range 103 -^* :

T rows 61

A B X V I 15 mat χ MNA :

Laplace transform of the pulse:

To 5103 -^* :

x u 0.5 1 x sign + (* :

t V.b 12 t u t To - u - (* :

Time domain

V.b 12 s 1 e To s * -^+ (* / :

Freq domain

χ.S O 1 V Spice χ.S O 2 V Spice T T 2 To * T 2 To * / trunc * - V.b vectorize augment V 2 el T ℐℒ 4 1 sys 1 1 line χ.S O Vb I Spice χ.S O R1 I Spice I 1 el T ℐℒ I 2 el T ℐℒ 4 1 sys 1 1 line

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 5X1 0 2 3 OAX2 0 4 5 OA χs χs :

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 OAX2 0 4 5 OA χ χ :

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

A B X 13 mat χs ABX 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 1 V 2 V 3 V 4 V 5 V Vb I X1 I X2 I 81 mat

B 00000 Vb 0081 mat

A B X 13 mat χ ABX MNA :

s A A :

s B B :

0 A 0 B augment rref

ToDo: rref!

χ .OP OA# 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

s Vb R1 R2 R3 R4 C1 C2 Clear 1

Alvaro

4 appVersion 1.73.9126.0