## Numerical Circuit Analysis with Analog Insydes

 Download this example as a Mathematica notebook.

To execute this notebook you need the add-on package Analog Insydes
by ITWM (see copyright message at end of document).

### Transient Circuit Analysis

With Analog Insydes' behavioral modeling functionality and its numerical solver for systems of nonlinear differential algebraic equations (DAE), you can describe and simulate nonlinear dynamic circuits in Mathematica. Here, we use the DAE solver to simulate the transient response of a differential amplifier circuit.

#### Netlist and Models

Consider the following differential amplifier circuit consisting of four bipolar junction transistors. We shall compute the transient response V(t) to a sinusoidal voltage pulse applied at node 1.

Differential Amplifier Circuit

`In[1]:= <<AnalogInsydes``
In[2]:= DifferentialAmplifier=
Circuit[
Netlist[
{RS1, {1, 2}, Symbolic -> RS1,Value-> 1.*^3},
{RS2, {3, 0}, Symbolic -> RS2,Value-> 1.*^3},
{RC1, {4, 8}, Symbolic -> RC1,Value-> 1.*^4},
{RC2, {5, 8}, Symbolic -> RC2,Value-> 1.*^4},
{RBIAS, {7, 8}, Symbolic -> RBIAS,Value-> 2.*^4},
{CLOAD, {4, 5}, Symbolic -> CLOAD,Value -> 5.*^-12},
{Q1, {4 -> C, 2 -> B, 6 -> E}, Model -> NPN,
Selector->View, betaf -> 255.9, betar -> 6.092,
Is -> 14.34*^-15, Vt -> 0.026},
{Q2, {5 -> C, 3 -> B, 6 -> E}, Model -> NPN,
Selector -> View, betaf -> 255.9,
betar -> 6.092, Is -> 14.34*^-15, Vt -> 0.026},
{Q3, {6 -> C, 7 -> B, 9 -> E},
Model -> NPN, Selector -> View, betaf -> 255.9,
betar -> 6.092, Is -> 14.34*^-15,
Vt -> 0.026},{Q4, {7 -> C, 7 -> B, 9 -> E},
Model -> NPN, Selector -> View, betaf->255.9,
betar -> 6.092, Is -> 14.34*^-15, Vt -> 0.026},
{VCC, {8, 0}, Symbolic -> VCC, Type -> VoltageSource,
Value ->12.},
{VEE, {9, 0}, Symbolic -> VEE, Value -> -12.},
{V1, {1, 0}, Symbolic->V1,Value->V1}]];

The transistors are modeled by the nonlinear DC Ebers-Moll
equations which are loaded from the Analog Insydes model library.

`In[3]:= LoadModel[NPN, EbersMoll,Scope->Global];`
`GlobalSubcircuits[]`
`Out[3]={{NPN,EbersMoll}}`

#### Setting Up Circuit Equations

Now we set up a system of time domain circuit equations
from the netlist.

In[4]:= mna=CircuitEquations[DifferentialAmplifier/.
View->EbersMoll,
TimeDomain->True]

#### Transient Circuit Analysis

To simulate the transient behavior of the circuit, we apply a
sine wave stimulus to the input and call the transient solver.
`In[5]:= Vpulse[t_]:= Sin[10^6 t]`
`In[6]:= TransientPlot[{Vpulse[t]}, {t,0,10^-5}]`

`Out[6]= -Graphics-`
```In[7]:= transient =
NDAESolve[mna /.V1-> Vpulse[t],{t,0.,1.5*10^-5},```
`Protocol->True,MaxStepSize->5.*10^-7]`
`10% successfully computed.`
`20% successfully computed.`
`30% successfully computed.`
`40% successfully computed.`
`50% successfully computed.`
`60% successfully computed.`
`70% successfully computed.`
`80% successfully computed.`
`90% successfully computed.`
`100% successfully computed.`
Solutions are computed from tstart=0. to
tstop=0.0000150953 in 108 steps.
Out[7]= {{V\$4->InterpolatingFunction[{{0.,0.0000150953}},<>],
V\$5->InterpolatingFunction[{{0.,0.0000150953}},<>],
V\$1->InterpolatingFunction[{{0.,0.0000150953}},<>],
V\$2->InterpolatingFunction[{{0.,0.0000150953}},<>],
V\$3->InterpolatingFunction[{{0.,0.0000150953}},<>],
V\$6->InterpolatingFunction[{{0.,0.0000150953}},<>],
V\$7->InterpolatingFunction[{{0.,0.0000150953}},<>],
V\$8->InterpolatingFunction[{{0.,0.0000150953}},<>],
V\$9->InterpolatingFunction[{{0.,0.0000150953}},<>],
I\$BC\$Q1->InterpolatingFunction[{{0.,0.0000150953}},<>],
I\$BE\$Q1->InterpolatingFunction[{{0.,0.0000150953}},<>],
I\$BC\$Q2->InterpolatingFunction[{{0.,0.0000150953}},<>],
I\$BE\$Q2->InterpolatingFunction[{{0.,0.0000150953}},<>],
I\$BC\$Q3->InterpolatingFunction[{{0.,0.0000150953}},<>],
I\$BE\$Q3->InterpolatingFunction[{{0.,0.0000150953}},<>],
I\$BC\$Q4->InterpolatingFunction[{{0.,0.0000150953}},<>],
I\$BE\$Q4->InterpolatingFunction[{{0.,0.0000150953}},<>],
I\$VCC->InterpolatingFunction[{{0.,0.0000150953}},<>],
I\$VEE->InterpolatingFunction[{{0.,0.0000150953}},<>],
I\$V1->InterpolatingFunction[{{0.,0.0000150953}},<>]}}

#### Visualizing the Results

Then we plot the time domain response with Analog Insydes'
transient waveform viewer.
`In[8]:= TransientPlot[transient,{V\$1[t],V\$5[t],I\$BC\$Q1[t]},`
`{t,0,1.5*10^-5},`
`SingleDiagram->False,PlotRange->All]`

`Out[8]= -GraphicsArray-`
`In[9]:=  TransientPlot[transient,{V\$1[t], V\$5[t]},`
`{t,1.*10^-6,1.5*10^-5},Parametric->True]`

`Out[9]= -Graphics-`

### Numerical Parametric DC Circuit Analysis

Next, we will demonstrate a special application of Analog Insydes' transient solver NDAESolve. Since NDAESolve solves not only differential equations but also purely algebraic systems of equations, and because the function does not care about the physical meaning of the independent variable (usually time ), we can use NDAESolve for
parametric DC analysis. "Parametric DC analysis" means to calculate DC currents and voltages as functions of a circuit element as the element value is varied within a given interval. This method will be illustrated in
the following example.

#### Netlist and Models

Consider the above differential amplifier circuit with a modified netlist entry for the resistor RS2. The resistor value is now left symbolic (see the netlist entry highlighted in red color).
In[10]:= DifferentialAmplifier2=
Circuit[
Netlist[
{RS1, {1, 2}, Symbolic -> RS1,Value-> 1.*^3},
{RS2, {3, 0}, Symbolic -> RS2, Value -> RS2,
Pattern -> Impedance},
{RC1, {4, 8}, Symbolic -> RC1,Value-> 1.*^4},
{RC2, {5, 8}, Symbolic -> RC2,Value-> 1.*^4},
{RBIAS, {7, 8}, Symbolic -> RBIAS,Value-> 2.*^4},
{CLOAD, {4, 5}, Symbolic -> CLOAD, Value -> 5.*^-12},

{Q1, {4 -> C, 2 -> B, 6 -> E}, Model -> NPN,
Selector ->View, betaf -> 255.9, betar -> 6.092,
Is -> 14.34*^-15, Vt -> 0.026},
{Q2, {5 -> C, 3 -> B, 6 -> E}, Model -> NPN,
Selector -> View, betaf -> 255.9, betar -> 6.092,
Is -> 14.34*^-15, Vt -> 0.026},
{Q3, {6 -> C, 7 -> B, 9 -> E},Model -> NPN,
Selector -> View, betaf -> 255.9, betar -> 6.092,
Is -> 14.34*^-15, Vt -> 0.026},
{Q4, {7 -> C, 7 -> B, 9 -> E}, Model -> NPN,
Selector -> View,
betaf -> 255.9,    betar -> 6.092, Is -> 14.34*^-15,
Vt -> 0.026},
{VCC, {8, 0}, Symbolic -> VCC, Type -> VoltageSource,
Value ->12.},
{VEE, {9, 0}, Symbolic -> VEE,Value -> -12.},
{V1, {1, 0},Symbolic->V1,Value->V1}
]
];

#### Setting Up Circuit Equations

First, we set up a system of "time-domain" circuit equations.
However, we use RS2 instead of time as the independent variable by specifying the option TimeVariable -> RS2.

In[11]:= parDCeqs=
CircuitEquations[DifferentialAmplifier2/.View->EbersMoll,

`TimeDomain->True, TimeVariable->RS2]`

#### Parametric DC Analysis

Then we let NDAESolve compute the trajectories of the circuit's DC operating point as the value of the parameter RS2 is swept from  to
`In[12]:= parDC=NDAESolve[parDCeqs/.V1-> 0,{RS2,1.*^3,5.*^4},`
` AnalysisMethod->DC,Protocol->True]`
`10% successfully computed.`
`20% successfully computed.`
`30% successfully computed.`
`40% successfully computed.`
`50% successfully computed.`
`60% successfully computed.`
`70% successfully computed.`
`80% successfully computed.`
`90% successfully computed.`
`100% successfully computed.`
Solutions are computed from tstart=1000.
to tstop=50067.4 in 92 steps.

Out[12]= {{V\$1->InterpolatingFunction[{{1000.,50067.4}},<>],
V\$2->InterpolatingFunction[{{1000.,50067.4}},<>],
V\$3->InterpolatingFunction[{{1000.,50067.4}},<>],
V\$4->InterpolatingFunction[{{1000.,50067.4}},<>],
V\$5->InterpolatingFunction[{{1000.,50067.4}},<>],
V\$6->InterpolatingFunction[{{1000.,50067.4}},<>],
V\$7->InterpolatingFunction[{{1000.,50067.4}},<>],
V\$8->InterpolatingFunction[{{1000.,50067.4}},<>],
V\$9->InterpolatingFunction[{{1000.,50067.4}},<>],
I\$RS2->InterpolatingFunction[{{1000.,50067.4}},<>],
I\$BC\$Q1->InterpolatingFunction[{{1000.,50067.4}},<>],
I\$BE\$Q1->InterpolatingFunction[{{1000.,50067.4}},<>],
I\$BC\$Q2->InterpolatingFunction[{{1000.,50067.4}},<>],
I\$BE\$Q2->InterpolatingFunction[{{1000.,50067.4}},<>],
I\$BC\$Q3->InterpolatingFunction[{{1000.,50067.4}},<>],
I\$BE\$Q3->InterpolatingFunction[{{1000.,50067.4}},<>],
I\$BC\$Q4->InterpolatingFunction[{{1000.,50067.4}},<>],
I\$BE\$Q4->InterpolatingFunction[{{1000.,50067.4}},<>],
I\$VCC->InterpolatingFunction[{{1000.,50067.4}},<>],
I\$VEE->InterpolatingFunction[{{1000.,50067.4}},<>],
I\$V1->InterpolatingFunction[{{1000.,50067.4}},<>]}}

#### Visualizing the Results

The influence of the resistor RS2 on the circuit voltages and
currents can be visualized by Analog Insydes' transient waveform viewer Transient Plot
`In[13]:= TransientPlot[parDC,{V\$3[RS2],I\$BC\$Q2[RS2]},`
`{RS2,1.*^3,5.*^4},Frame->True,GridLines->Automatic,`
`Axes->False,PlotRange->All,SingleDiagram->False]`

`Out[13]= -GraphicsArray-`

### Copyright © 1998 by ITWM e.V.

Written by Thomas Halfmann

Analog Insydes
ITWM - Institut für Techno- und Wirtschaftsmathematik e.V.
Erwin-Schrödinger-Str.
D-67663 Kaiserslautern, Germany

Analog Insydes is a trademark of ITWM.
Mathematica is a registered trademark of Wolfram Research, Inc. All other product names are trademarks of their respective owners.

Copying and redistribution of this notebook is welcomed and encouraged, provided that the entire contents of the document, in particular the above copyright notice, are left unchanged and that the file is redistributed free of charge.