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simulation to model new device types and for black-box modeling of complex systems. In this document, examples are used to show how the Analog Behavioral Modeling feature of PSpice can be used to: Calculate square roots Use ideal non-linearity from look-up tables Design small systems Pass parameters to sub-circuits

Introduction
Behavioral Modeling is the process of developing a model for a device or a system component representing the
behavior rather than from a microscopic description You can use Behavioral Modeling in the domain of analog
simulation to model new device types and for black box modeling of complex systems
In this document examples are used to show how the Analog Behavioral Modeling feature of PSpice can be used
Calculate square roots
Use ideal non linearity from look up tables
Design small systems
Pass parameters to sub circuits
Calculating Square Roots
Assume that you need to create a signal whose voltage is the square root of another signal s voltage A simple
solution is to use a feedback circuit to calculate square roots But this technique fails if the reference signal goes
negative The solution then is to use the functional form of Analog Behavioral Modeling
Esqrt out hi out lo value sqrt abs v input
Figure 1 Square Roots Sub circuit
This model takes the absolute value of the ground referenced signal input before evaluating the square root
function The absolute value function is a nonlinear function
Note You can also use a floating signal pair in the model for example replace v input with v in hi
v in lo or v in hi in lo
Using Ideal Non Linearity from Look up Tables
You can introduce ideal non linearity using the table look up form of Analog Behavioral Modeling For example
the following one line ideal OpAmp model has high gain but its output is clamped between 15 volts
Eamp out 0 table 200K v in hi v in lo
15 15 15 15
APPLICATION NOTE
Figure 2 Look up Tables Sub circuit
The input to the table is the differential gain formula but the look up table has only two entries The output of the
table is interpolated between these two endpoints and clamped when the input exceeds the table s range This is
a convenient use of the table look up form available in PSpice
Designing Small Systems
Small systems of behavioral models are easy to design using PSpice For example a true RMS circuit can be
built by decomposing the RMS function
1 Square the signal
2 Integrate over time
3 Take the square root of the time average
These three operations can be bundled in a tiny sub circuit for use as a module
Figure 3 RMS Sub circuit
subckt RMS in out G1 0 1 VALUE V IN V IN
C1 1 0 1 R1 1 0 1G E1 out 0 VALUE IF TIME 0
0 SQRT V 1 TIME ends
The current source G1 squares the signal which is then integrated in the capacitor C1 The voltage on the
capacitor is time averaged and the square root is taken The resistor is a dummy load that satisfies the algorithm
The voltage source E1 shows that the value of simulated time is available in Analog Behavioral Modeling and
may be used as a variable in a formula Note the if than else function If time is less than or equal to zero
APPLICATION NOTE
then the output of E1 is sqrt v 1 time This prevents convergence problems when sqrt v 1 time is
evaluated at time 0
Passing Parameters to Sub Circuits
Parameter passing into sub circuits also works with Analog Behavioral Modeling making your models more
flexible Here is a small system that is a voltage follower with hysteresis useful in simulating say a mechanical
system with gear backlash
Figure 4 Hysteresis Sub circuit
subckt HYS in out params H 1 G1 0 1 TABLE
V IN 1 H 2 2 1G 1 0 1 0 2 1G C1 1 0 1
R1 1 0 1G E1 out 0 1 0 1 ends
In the model the parameter H defines the size of the hysteresis and is used in the formula input to the table The
combination of the formula and table defines a dead band outside of which the output follows the input with an
offset of H 2 The capacitor serves as memory for the circuit and is nearly ideal except for the DC bias resistor
which provides a droop time constant of one billion seconds The voltage follower E1 prevents output loading
problems E1 could also have gain representing the gear ratio of a mechanical system then voltage would
represent the total turn angle of each gear and H the amount of angular backlash
Figure 5 Circuit using RMS and Hysteresis Sub circuit
APPLICATION NOTE
param H 1 V1 in 0 SIN 0 1 1 Xrms 1 rms RMSXhys 1 hys HYS param H 1 tran
A 1Hz sine wave is used for the stimulus to the RMS and HYS circuits
Figure 6 Output from RMS and HYS Circuit
Figure 6 shows a Probe plot of the input and the outputs from each circuit Note that the RMS circuit outputs the
well known result of 0 707 volts after one input cycle while the HYS circuit lags the input by a half volt in
each direction for a total hysteresis of one volt
Amplitude and Balanced Modulation
Multipliers
Multiplier is a behavioral block that performs the mathematical task of multiplying the two inputs and returns the
product as the output Multipliers are often used for signal processing applications In this note two examples are
presented to illustrate the use of a multiplier to make an amplitude modulator and to make a frequency doubler
Amplitude Modulation
Amplitude modulation is a technique which uses a low frequency signal to control the amplitude of a high
frequency signal A simple modulator can be constructed using a multiplier as shown in Figure 7
APPLICATION NOTE
Figure 7 A simple amplitude modulator circuit
One input is the high frequency or carrier signal and the other input is the modulating signal
A sinusoidal source at a frequency of 10 kHz is used to represent the carrier signal and a second source at a
frequency of 1 kHz is used to represent the modulating signal Notice that the peak amplitude for the carrier is set
to 1 volt using the parameter VcarrierPK The modulating index is the ratio of the peak of the modulating signal to
the peak of the carrier Here the index is set to 0 8 or 80 modulation A typical broadcast AM signal includes the
carrier as well as the sidebands in the transmission To get such a double sideband transmitted carrier signal
DSBTC we must bias or offset the modulating signal by a value equal to the carrier s peak voltage
The amplitude modulated signal and the modulating signal from this simulation are shown in Figure 8
Figure 8 An amplitude modulator output signal
APPLICATION NOTE
Balanced Modulation
A balanced modulator produces a double sideband suppressed carrier signal DSB SC By setting the offset of
the modulating signal to be zero in the above circuit we will suppress the carrier Notice the output of this
modulator shown in Figure 9 the shape of its upper A balanced modulator output signal envelope resembles a
full wave rectified AC source
Figure 9 A balanced modulator output signal
Frequency Multiplication
Another application for a multiplier is as a frequency doubler see Figure 10 Connecting a sinusoidal source
simultaneously to both inputs of a multiplier will yield a signal with double the input frequency The first multiplier
produces a waveform that has one half the amplitude of the original input signal with a DC offset of one half the
input waveform s peak value The DC offset is removed with a voltage source called Voffset The amplitude of the
original signal is restored with a second multiplier which doubles the signal
APPLICATION NOTE
Figure 10 A simple frequency doubler circuit
Figure 11 shows the original input signal as well as the frequency doubled output signal These examples have
illustrated how a multiplier implemented using the E device in PSpice can be used in signal processing
applications such as amplitude modulation and frequency doubling
Figure 11 Output results for frequency doubler
APPLICATION NOTE
Solving Simple Differential Equations
PSpice is well known for its ability to solve the equations which arise in circuit analysis What is less well known is
that PSpice can also be used to solve problems in other domains which can be expressed as differential
equations This article presents some examples of using PSpice as an analog computer to solve sets of
differential equations describing the kinetics of a chemical reaction
Consider a familiar example the voltage across a parallel capacitor resistor combination as a function of time
The circuit equation for this example is
rearranging a little
Because V is a function of the single variable t this is an Ordinary Differential Equation ODE Its solution is the
equation of exponential decay which is
where V0 is the initial voltage on the capacitor
To see how PSpice can be used to solve the equation above consider an ideal integrator Suppose its output is
the voltage we want V Now the input to the ideal integrator is evidently dV dt
A circuit which represents the equation is shown in the following schematic The schematic was drawn using
OrCAD Capture The parts shown come from the Analog Behavioral Modeling ABM symbol library abm olb
supplied with the program The symbols INTEG integrator and MULT multiplier are used
APPLICATION NOTE
Figure 12 RC circuit
The symbol containing 1 0 is an integrator with gain 1 Its output is obtained by integrating the voltage at its
input This voltage is constrained to be 1 RC times the output voltage V by closing the feedback loop
This way of setting up differential equations for solution is the way that analog computers were used In this
example an integrator block would be used the constant 1 RC would be supplied by a gain block and using the
inverting input to the integrator would provide the 1
The initial condition for this problem is that the initial voltage is V0 On an analog computer this voltage would be
derived from a reference and patched to the initial condition input of the integrator Using the ABM integrator
symbol INTEG the initial voltage is specified by setting the value of the IC attribute on the symbol
Running a Transient Analysis on the ABM representation of the problem shows the expected exponential decay
of voltage with time
Solving Coupled Differential Equations
Systems of interest usually contain more than one variable There may be several interacting voltages in a circuit
In a chemical reaction the rate of production of a component may depend on the concentrations of several other
components
For example if we have three components x1 x2 and x3 the equations controlling their rates of decay and
production might be
APPLICATION NOTE
Sets of equations like this can be solved using similar techniques to the first problem In this example we would
use three integrators with three feedback loops and three node voltages to solve for x1 x2 and x3
Let s look at a real example This is a chemical system which contains four components x1 x2 x3 and x4 They
are related by four equations containing rate constants the Ks and physical constants R and Q
A schematic containing ABM components for integration summing and taking square roots etc is shown in
Figure 13 It also contains definitions for Q and R the physical constants and the Ks
APPLICATION NOTE
Figure 13 Coupled Ordinary Differential Equations ODE implemented using ABM components for integration summing
square roots etc The physical constants Q and R are shown as well as the rate constant K to demonstrate a chemical
Design Systems Inc All others are properties of their respective holders
APPLICATION NOTE

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