LINC2 Software Enhances the Design and Analysis of VCOs and Other Tunable
RF and Microwave Circuits

By Dale D. Henkes,
ACS

The most commonly used simulation programs and methods have
become popular for good reasons; they have become essential tools for enabling
the circuit designer to be more productive and to practice the trade
successfully.  SPICE based programs
are appropriately used to analyze transient and circuit start-up behavior as
current or voltage waveforms in the time domain.  RF and microwave circuit simulation
programs commonly display the circuit’s s-parameters (or quantities
related to these s-parameters) in the frequency domain.  More advanced circuit simulation
software will include additional methods of analyzing the circuit or viewing the
resulting simulation data.

In addition to the customary frequency response analysis,
this article will demonstrate a few other methods and simulation tools for
enhancing the design and analysis of RF and microwave circuits.  The LINC2 RF CAE (Computer Aided
Engineering) software suite from ACS (Applied Computational Sciences, Escondido, CA)
will be used to demonstrate the following:

·
Swept
Variables and Circuit Component Parameter Sweeps

A circuit or component parameter
can be swept through a range of values by assigning a variable to the parameter
and performing a variable sweep.
The circuit response can be viewed against the variable at any fixed
frequency point.

·
User
Defined Equations

New component models can be created
or existing component models modified by user defined equations that formulate
new relationships between variables and circuit parameters.

·
Special
Output Functions for Post Processing Simulation Data

These special intrinsic functions provide
new ways of processing and viewing simulation results.  For example, finding and tracking the frequency
point at which the maximum value of a circuit response occurs or tracking the
frequency point for zero transmission phase during a variable parameter sweep.

In the following example, the design of a voltage-controlled
oscillator (VCO) will be analyzed using these special simulation and analysis
tools.

Voltage-Controlled
Oscillator (VCO) Example

The crystal oscillator circuit of Figure 1 has been
reconfigured in Figure 2 so as to enable the oscillator to be analyzed by
linear Bode (gain and phase response) techniques.  The modifications include breaking the
feedback path through the crystal, removing the crystal and connecting input
and output ports at the circuit nodes where the crystal was attached.  Breaking the feedback path and attaching
source and load measurement ports allows for analyzing the oscillator as a
phase shift network and amplifier cascade.

The oscillator will oscillate at the frequency where the
total open loop phase shift equals zero degrees as long as there is sufficient
positive gain (> 0 dB) in the vicinity around this zero-phase crossover
point.  Since the circuit can potentially
oscillate at frequencies where the phase shift is multiples of 360 degrees,
care should be taken to ensure that there is no positive gain (< 0 dB) at
any undesired frequency where the loop phase shift is 0 or multiples of
360°.  The frequency selective
tank circuit formed by L1, C1, C2 and C3 ensures that the circuit meets the
oscillation criteria at only the desired frequency.

Figure 1, 100 MHz Butler Crystal
Oscillator

Figure 2, LINC2
Schematic Setup for Oscillator Analysis

In Figure 2, the crystal has been removed to allow for
tuning the oscillator over a relatively wide frequency range.  The tuning range for this example, using
a varactor tuning diode, is around 30% (+/- 15% of the center frequency).  The tuning range using other tuning
methods may be larger.

Circuit Simulation
Using Variables for Tuning Control

In LINC2 a named variable can be placed on the schematic
page and assigned to one or more component parameters as needed.  In this example, the variable CVar is
given an initial value of 9.44 pF and assigned to the VCO tuning capacitor C3.  This sets the oscillator’s center
frequency to 100 MHz.  With the
oscillator configured as in Figure 2, the linear gain/phase swept frequency
response is displayed in Figure 3 for several settings of the variable tuning
capacitor C3.

Interactive real-time tuning is convenient and easy to perform
in a LINC2 graph window.  Simply
selecting the variable from the Tune menu and tapping the up or down arrow key increases
or decreases the value of the variable while the plotted circuit response is immediately
updated.  With the variable CVar assigned
to the capacitance value for C3, tuning CVar varies the frequency of
oscillation (zero phase point) as shown in Figure 3.

Figure 3, LINC2 Oscillator
Gain-Phase Frequency Response

As is the case for a well designed oscillator of this type,
the gain peaks at nearly the same frequency point at which the open loop
transmission phase becomes zero.  In
Figure 3, the three gain peaks and phase zero pairs occur at 80 MHz, 100 MHz
and 120 MHz for CVar (tuning capacitor C3) values of 21.4 pF, 9.44 pF and 3 pF
respectively.  This suggests the
following way to determine the VCO tuning range for a given range of tuning
capacitance:

Set the variable CVar to its
largest value and note the resulting frequency corresponding to zero open loop
transmission phase (P21 = Phase Angle[S21] = 0).  Then, set CVar to its smallest value and
again note the frequency at which P21 = 0.
The difference between these two frequency points is the tuning range.

For CVar = 21.4 (pF), P21 = 0 occurs at 80 MHz.  When CVar is tuned to 3 (pF), the P21 =
0 point shifts upward to 120 MHz.
The result is a 40 MHz frequency range (centered at 100 MHz) for tuning
capacitance ranging from 3 pF to 21.4 pF.
This amounts to a 40% tuning range over frequency for a 7 to 1 ratio in
tuning capacitance.  A more direct
and automatic method for determining the VCO tuning range will be demonstrated
next, after introducing the use of swept variables.

Using Swept Variables
in Circuit Simulations

Instead of manually tuning a component parameter using a
variable as in the previous section, the variable can be set up to
automatically sweep over its entire range of values while the circuit response
is plotted as a function of the swept variable.  As shown in Figure 4, the LINC2 program
provides a checkbox for enabling a variable parameter sweep, resulting in an
ordinary variable being converted to a swept variable.  The parameters of a swept variable are
its nominal value, the starting value, the stop value and number of sweep
points.

Figure 4, LINC2
Swept Variable Setup

Variable Parameter
Sweeps and LINC2 Special Output Functions

There are a number of special built-in functions in the
LINC2 program for post processing of the simulation data.  The Zero[Data]
function finds the frequency at which the selected data has a value of exactly
zero.  The function continues to
find all frequency points corresponding to zero data values in the simulation
data for each value of a swept variable, thus generating an array of frequency
points.  The result is a plot of
frequency (vertical axis) versus a selected variable (horizontal axis).  The meaning of “frequency”
in this example is the frequency of oscillation (Frequency[P21 = 0]) for each
value of the tuning capacitance.

As just described, the Zero[P21]
function will be employed in this example to plot the VCO oscillation frequency
as a function of the tuning capacitor’s swept capacitance value (CVar).  Since the P21 = 0 value (zero
transmission phase point) shifts in frequency as the swept variable (CVar) is automatically
stepped through its values, plotting the frequency of zero P21 against this
variable will plot out the VCO’s tuning response.  This unique LINC2 function produces the oscillator’s
tuning response to the tuning capacitance CVar as shown in Figure 5.

Figure 5, The
VCO’s Oscillation Frequency as a Function of the Tuning Capacitance

The oscillation frequency (Freq[P21=0]) is plotted on the
right vertical axis while the frequency at which the gain peaks is plotted on
the left vertical axis as a function of the tuning capacitance (CVar).  These two plots almost completely
overlap because the gain peaks (Max[S21(dB)]) occur at nearly the same
frequency as the zero phase point (see Figure 3).

Comparing Figure 5 to Figure 3, Figure 5 shows the
VCO’s tuning response to the tuning capacitance in a much more direct and
clearer way.  In Figure 5 the
oscillator’s tuning characteristics are captured and displayed in a
simple easy to visualize graphical format.

Creating New Circuit
Models with User Defined Equations

The tunable oscillator schematic in Figure 2 uses a variable
(CVar) to control the tuning capacitance (C3).  The variable capacitor C3 might be
implemented in a number of different ways.
This example will use a varactor tuning diode to accomplish voltage-controlled
tuning of the oscillator, making it a VCO (or Voltage-Controlled Oscillator).  The diode’s capacitance as a
function of its tuning voltage can be simulated by using a user defined
equation in LINC2.  A useful
approximation for the varactor diode’s voltage to capacitance
transformation is given by [1]:

Equation 1)       C(VR)
= CJ0/(1 + VR/VJ)^M + CP, where CJ0
is the diode’s zero-bias junction capacitance, VR is the
applied reverse DC bias voltage, VJ is the junction potential, M is
a device dependant constant called the grading coefficient, and CP
is the device package capacitance.

Using an equation to model the diode may be preferred over
using a built-in schematic model because it defines the function explicitly and
there is almost no limit to the model details and complexity that can be
embodied in the equation.  Moreover,
it is easy to edit the equation model to accommodate the characteristics of a
different device.  Figure 6 shows a
LINC2 schematic of the VCO with the varactor (represented by C3) modeled by
equation CVarEqn.

Figure 6, LINC2 VCO
Schematic with Varactor Equation Model

In the schematic (Figure 6), the variable CVar (from Figure 2)
has been replace with the equation CVarEqn that models the varactor’s
capacitance as a function of its DC bias voltage.  The equation uses the model parameters
given in [1] for the SMV1248 diode.
The diode model parameters were extracted from measured CV(VR)
data.  More complete models may
include at least some series resistance and package inductance and these
parasitic components could also be included explicitly on the schematic.

A simulation run on the circuit in Figure 6 will produce a conventional
frequency response plot as in Figure 3 with the exception that, instead of
varying the capacitance directly, the VCO is tuned by varying the varactor
voltage (via the Varactor_V variable in the schematic).  However, the LINC2 simulator also
produces the characteristic tuning plot shown in Figure 7.  This graph window simultaneously plots
the VCO’s tuning response (Zero[P21]) and the equation (CVarEqn) that
describes the tuning diode’s capacitance, both as a function of the
varactor DC bias voltage (Varactor_V).

Figure 7, The
VCO’s Tuning Characteristics as a Function of Varactor Voltage

Comparing the tuning frequency response in Figure 7 to that
in Figure 5, the following observations can be made.  The tuning frequency slope in Figure 5
is negative because the tuning capacitance is increasing to the right, lowering
the frequency of oscillation.  However,
in Figure 7 the tuning frequency slope is positive because it is plotted as a
function of the varactor voltage.
As the varactor voltage increases to the right, the varactor capacitance
decreases (and the frequency of oscillation moves higher).

Figure 7 indicates that the VCO’s tuning frequency is a slightly upward
curving function of the tuning voltage (relative to a straight line) whereas
the tuning diode’s capacitance curves downward in a non-linear curve
characteristic of the exponential nature of the diode’s voltage to
capacitance relation (Equation 1).
With this plot, it can see at a glance that the VCO can be tuned between
84.75 MHz and 115.75 MHz with a tuning voltage ranging between 0.30 volts and 2.30
volts.  The corresponding varactor
tuning capacitance will range between 4.07 pF and 17.7 pF respectively.  Markers placed on the plots have their
numerical values displayed at the top of the graph for various values of
varactor voltage.

Figure 8, LINC2
Automatically Calculates the Loaded Q of the Oscillator

Summary of LINC2 VCO
Analysis

The LINC2 program provides new ways to analyze and
characterize the VCO’s response to tuning control.  For example, in Figure 7 the linearity
of the VCO’s tuning control can be seen at a glance.  The required control voltage range and
tuning capacitance range can also be immediately determined from the graph by
inspection.

When a user defined equation is part of a LINC2 schematic
(such as equation CVarEqn in Figure 6), the equation can be plotted simultaneously
on the same graph along with the simulation data.  For example, in Figure 7 the plot of
equation CVarEqn shows the tuning diode’s CV(VR)
characteristics and how they relate to the overall tuning characteristics of
the VCO.

In addition to these new LINC2 output functions and analysis
techniques, the conventional frequency sweeps (as in Figure 3 and Figure 8) of
S21 (Gain) and P21 (Phase) yield important additional information about the VCO
and its quality of performance.  For
example, in Figure 8 the Find menu
can be used to locate the frequency of oscillation and calculate the loaded Q
for the oscillator.  Find > Loaded Q > At P21 Phase Zero
places a small circle on the phase (P21) curve at the frequency point where the
VCO will oscillate for the given value of tuning voltage/capacitance.  This is also the point where the loaded
Q is calculated.  The loaded Q is
one of the most dominate factors in determining the phase noise of the
oscillator.  The LINC2 GRAPH window
(Figure 8) reports a loaded Q of 26.23 for the VCO at 100 MHz.

This article has repeatedly cited the open loop phase zero
point as the indicator for the frequency of oscillation.  This concept can be found in reference
[2] under the subject Oscillation Conditions (Chapter 1).  Reference [2] refers to the closed loop phase
of 0° (or a multiple of 360°) as one of the requirements for
oscillation.  Indeed, the loop is
closed in the actual oscillator circuit.
This article refers to the open loop phase zero requirement because the
(amplifier-feedback) loop is broken and the cascade is simulated open-loop for
analysis purposes.  Reference [2]
also refers to the gain condition for oscillation, which was mentioned earlier
as an additional requirement for oscillation, that there should be sufficient
positive gain at the zero-phase point.

The circuit theory, such as presented in reference [2], is
an excellent source to acquire an understanding of the circuit operation and
design requirements.  Moreover, it
is gratifying to verify the theory through circuit simulation and actual
hardware built.  The author has used
the LINC2 program to design a crystal controlled version of this
oscillator.  The oscillator was then
built and employed in an up-converter in communication equipment for a
satellite uplink.

When different simulation programs are available and time
permits, it is a good idea to verify the circuit using a different kind of
simulator.  There are many versions
of the SPICE program available (and some are available free of charge).  The inner workings of the way SPICE analyzes
a circuit are very different from the way RF EDA programs (such as LINC2)
perform circuit simulation.  And
yet, as one would hope and expect, the results are the same- though presented
from a different viewpoint (i.e. in the time domain for SPICE and the frequency
domain for LINC2).  The same circuit
analyzed by the LINC2 program (in Figures 2 and 6) was entered into Linear
Technology’s LTSpice program [3] and a SPICE simulation was run.  The results are shown in Figure 9.

The SPICE simulation in Figure 9 shows the startup transient
after 300 ns and continuing on until the oscillations eventually stabilize
another 200 ns later.  There are two
complete cycles for every 20 ns time period, for a frequency of 100 MHz.  An oscillation frequency of 100 MHz was
exactly as predicted by the zero phase (P21=0) point on the LINC2 simulation in
Figure 8.  This completes the
analysis of the voltage-controlled oscillator.

The LINC2 Software
Suite

LINC2 is a high performance RF and microwave design and
simulation program from ACS.  In
addition to schematic based circuit simulation, optimization and statistical
yield analysis, LINC2 Pro includes many value-added features for automating
design tasks, including circuit synthesis.

LINC2 directly interfaces to leading RF and microwave design
suites, allowing it to be used stand-alone or by leveraging its capabilities
with those of other major packages.
LINC2 offers exact circuit synthesis, schematic capture, circuit
simulation, circuit optimization and yield analysis in a single affordable
design environment.  More
information about LINC2 can be found on the ACS web site at www.appliedmicrowave.com.

Figure 9, LTSpice
Version of the Oscillator Simulation

References

Application
note APN1004, Alpha Industries.

Foundations
of Oscillator Circuit Design, Guillermo Gonzalez, Artech House 2007, Chapter
1, Section 1.2, Oscillation Conditions.

The
LTSpice/SwitcherCAD III program was provided courtesy of Linear Technoloy,
www.linear.com.