RF Analysis

RF Analysis

The Network Object

DataNetwork
EquationNetwork

Stability Analysis

Stability Circles

VSWR Circles
Gain Equations

Maximum Unilateral Gain
Maximum Stable Gain
Maximum Available Gain

Transmission Line Calculations

The Network Object

Marconi is structured around a base AbstractNetwork object. This object can be constructed with data, equations, and the combination of other networks.

All networks provide attributes ports and Z0 for characteristic impedance.

DataNetwork

To build a network from a Touchstone file see File IO, otherwise we can simply use the constructor for DataNetwork.

Besides ports and Z0, a DataNetwork must also have frequency, a vector of frequencies for which the network is characterized, and s_params, for the S-Parameters themselves.

As of this release, the frequency/s_params lists must be ordered. If they are not evenly-spaced, all interpolation operations will be Grid interpolations instead of splines.

# DataNetwork(ports,Z0,frequency,s_params)
myNetwork = DataNetwork(1,50,[100e6, 200e6],[0.3+0.5im, 0.4+0.6im])

1-Port Network
 Z0 = 50
 Frequency = 100.0 MHz to 200.0 MHz
 Points = 2

For more than 1 port, the S-Parameters have to be an n-square matrix for n ports.

myNetwork = DataNetwork(2,50,[100e6, 200e6],[[0.3+0.5im 0.1+0.2im; 0.4+0.6im 0.3+0.5im], [0.3+0.5im 0.1+0.2im; 0.4+0.6im 0.3+0.5im]])

2-Port Network
 Z0 = 50
 Frequency = 100.0 MHz to 200.0 MHz
 Points = 2

EquationNetwork

To build a network from an equation, we start with an equation that defines some S-Parameters.

Take a series R-L network for example. The S-Parameters of an RL network would look like:

function inductorAndResistor(;freq,Z0)
    L = 1e-9
    R = 30
    z = R + im*2*pi*freq*L
    return (z-Z0)/(z+Z0)
end

inductorAndResistor (generic function with 1 method)

To use an equation-driven network in Marconi, the function must accept the kwags freq and Z0.

To build the EquationNetwork:

RL = EquationNetwork(1,50,inductorAndResistor)

1-Port Network
 Z0 = 50
 Equation-driven Network

Once again, for n-port networks, the function must provide an n-square matrix

function idealTransmissionLine(;freq,Z0)
    l = 1e-2 # 2cm
    λ = (3e8/freq)
    β = (2*pi)/λ
    return [0 exp(-β*l);exp(-β*l) 0]
end

tline = EquationNetwork(2,50,idealTransmissionLine)

2-Port Network
 Z0 = 50
 Equation-driven Network

Stability Analysis

One of the most important aspects in active microwave design is that of stability.

When designing an amplifier, we do not want oscillations on either of the ports. An oscillation would imply that $|\Gamma_{In}| > 1$ or $|\Gamma_{Out}| > 1$.

As both of these gammas depend on the input and output matching networks, the stability of an amplifier depends on $\Gamma_{S}$ and $\Gamma_{L}$.

There are two different kinds of stability:

Unconditional Stability - $|\Gamma_{In}| < 1$ and $|\Gamma_{Out}| < 1$ for all source and load impedances
Conditional Stability - $|\Gamma_{In}| < 1$ and $|\Gamma_{Out}| < 1$ for certain impedances.

To test for unconditional stability, we can use the Rollet Stability Criterion

\[K = \frac{1-|S_{11}|^2-|S_{22}|^2+|\Delta|^2}{2|S_{12}S_{21}|} > 1\]

along with the auxiliary condition that

\[|\Delta| = |S_{11}S_{22}-S_{12}S_{21}| < 1\]

For a device to be unconditionally stable, both of these conditions must be satisfied.

Alternatively, the μ stability parameter is a similar test for unconditional stability that integrates all conditions into a single parameter.

\[μ = \frac{1-|S_{11}|^2}{|S_{22}-\Delta S_{11}^{*}| + |S_{12}S_{21}|} > 1\]

To test these conditions (or plot them), use the following functions:

Example 12.2 from Microwave Engineering by David M. Pozar

gan_hemt = [0.869*exp(deg2rad(-159)im) 4.250*exp(deg2rad(61)im);0.031*exp(deg2rad(-9)im) 0.507*exp(deg2rad(-117)im)]

network = DataNetwork(2,50,[1.9e9],[gan_hemt])

2-Port Network
 Z0 = 50
 Frequency = 1.9 GHz to 1.9 GHz
 Points = 1

We now compute |Δ| with

testMagDelta(network)[1] # Only 1 point in this data

0.33618219546559625

and K with

testK(network)[1]

0.3825748331995004

So, now we can conclude that this device is NOT unconditionally stable at 1.9 GHz as $|\Delta| < 1 $,but $K<1$.

By calculating μ and testing the $\mu > 1$ stability condition a similar conclusion can be reached.

testμ(network)[1]

0.3549366514276895

We can also plot the stability factors as a function of frequency to find stable regions.

Take the data from this CE3520K3 low noise JFET:

jfet = readTouchstone("CE3520K3.s2p")

# Plot our criteria
ax = plotRectangular(jfet,testK,label="K")
plotRectangular!(ax,jfet,testMagDelta,label=raw"$|\Delta|$")

# Add a horizontal line to show stability regions
push!(ax,@pgf(HLine(@pgf({"dashed"}),1)))

# Adjust legend location
ax["legend pos"] = "outer north east"

# And a title, why not
ax["title"] = "Stability Tests"

From this plot, we can conclude that under 50Ω matching on both ports, this device is unconditionally stable from 14-18 GHz and above 25 GHz as those two regions satisfy the criterion.

Stability Circles

To observe the behavior of potentially unstable devices, we can plot the region that would push the device into oscillation. The circle that bounds these impedances are Stability Circles.

To plot these stability circles we can start with a network at one frequency:

LNA = DataNetwork(2,50,[800e6],[[∠(0.65,-95) ∠(0.035,40); ∠(5,115) ∠(0.8,-35)]])

2-Port Network
 Z0 = 50
 Frequency = 800.0 MHz to 800.0 MHz
 Points = 1

Now we can plot the the source and load stability circles on a smith chart.

sc = SmithChart()
plotSStabCircle!(sc,LNA,800e6,label="Source Stability")
plotLStabCircle!(sc,LNA,800e6,label="Load Stability")

VSWR Circles

Some times it is helpful to draw VSWR circles to know the bandwidth of a network. This is done with

# 10 dB RL == 1.92 VSWR
circleStyle = @pgf {"thick", color = "black"}
bpf = readTouchstone("BPF.s2p")
sc = plotSmithData(bpf,(1,1),label="S(1,1)")
plotVSWR!(sc,1.92,opts = circleStyle,label="VSWR = 1.92")

Gain Equations

These functions provide calculations to asses the gain performance of active microwave circuits

Maximum Unilateral Gain

Making the unilateral assumption, S12 = 0, we can calculate MUG as

\[MUG = \frac{ |S_{21}|^2 }{ (1 - |S_{11}|^2)(1 - |S_{22}|^2) }\]

jfet = readTouchstone("CE3520K3.s2p")
plotRectangular(jfet,testMUG,dB,label="MUG")

Maximum Stable Gain

The maximum gain out of a potentially unstable device

\[MSG = \frac{ |S_{21}| }{ |S_{12}| }\]

jfet = readTouchstone("CE3520K3.s2p")
plotRectangular(jfet,testMSG,dB,label="MSG")

Otherwise known as GMAX, MAG is the maximum gain from a stable network. This formula comes from Microwaves 101 - a modification from the textbook equation such that MAG behaves well for high K devices.

\[MAG = \frac{ |S_{21}| }{ |S_{12}| } * \frac{1}{K+\sqrt{K^2-1}}\]

jfet = readTouchstone("CE3520K3.s2p")
plotRectangular(jfet,testMAG,dB,label="MAG")

This makes sense as looking at the K plot from before, the device is only unconditionally stable around 14 GHz and above 25 GHz.

Just to compare this to the S21 parameter:

ax = plotRectangular(jfet,testMAG,dB,label="MAG")
plotRectangular!(ax,jfet,(2,1),label="S(2,1)")

Transmission Line Calculations

Marconi provides some basic calculations for transmission lines:

# Input impedance of lossless t-line terminated in Zr
Θ = 35
Zr = 100+im*50
Zin = inputZ(Zr,Θ)

72.66031465868264 - 55.85271589192081im

And to show the behavior we expect from a λ/4 line:

Θ = 89.99
Zr = 1e99+0im
Zin = inputZ(Zr,Θ)

2.500000076154357e-96 - 0.008726646348585729im