# Design Broadband Matching Networks for Amplifier

This example shows how to design broadband matching networks for a low noise amplifier (LNA).

In an RF receiver front end, the LNA is commonly found immediately after the antenna or after the first bandpass filter that follows the antenna. Its position in the receiver chain ensures that it deals with weak signals that have significant noise content. As a result the LNA has to not only provide amplification to such signals but also minimize its own noise footprint on the amplified signal.

In this example you will design an LNA to achieve the target gain and noise figure specifications over a specified bandwidth, using lumped LC elements. A direct-search based approach is used to arrive at the optimum element values in the input and output matching network.

Figure 1: Impedance matching of an amplifier

### Set Design Parameters

The design specifications are as follows.

• Amplifier is an LNA amplifier

• Center Frequency = 250 MHz

• Bandwidth = 100 MHz

• Transducer Gain greater than or equal to 10 dB

• Noise Figure less than or equal to 2.0 dB

• Operating between 50-Ohm terminations

### Specify Design Parameters

You are building the matching network for an LNA with a bandpass response, so specify the bandwidth of match, center frequency, gain, and noise figure targets.

BW = 100e6;            % Bandwidth of matching network (Hz)
fc = 250e6;            % Center frequency (Hz)
Gt_target = 10;        % Transducer gain target (dB)
NFtarget = 2;          % Max noise figure target (dB)

Specify the source impedance, reference impedance, and the load impedance.

Zs = 50;               % Source impedance (Ohm)
Z0 = 50;               % Reference impedance (Ohm)
Zl = 50;               % Load impedance (Ohm)

### Create Amplifier Object and Perform Analysis

Use the read method to create an amplifier object using data from the file lnadata.s2p.

Define the number of frequency points to use for analysis and set up the frequency vector.

Npts = 32;                           % No. of analysis frequency points
fLower = fc - (BW/2);                % Lower band edge
fUpper = fc + (BW/2);                % Upper band edge
freq = linspace(fLower,fUpper,Npts); % Frequency array for analysis
w = 2*pi*freq;                       % Frequency (radians/sec)

Use the analyze method to perform frequency-domain analysis at the frequency points in the vector freq.

analyze(Unmatched_Amp,freq,Zl,Zs,Z0);   % Analyze unmatched amplifier

### Examine Stability, Power Gain, and Noise Figure

The LNA must operate in a stable region, to examine stability, plot Delta and K for the transistor. Use the plot method of the rfckt object to plot Delta and K as a function of frequency to see if the transistor is stable.

figure
plot(Unmatched_Amp,'Delta','mag')
hold all
plot(Unmatched_Amp,'K')
title('Device stability parameters')
hold off
grid on

As the plot shows, $K>1$ and $\Delta <1$ for all frequencies in the bandwidth of interest. This means that the device is unconditionally stable. It is also important to view the power gain and noise figure behavior across the same bandwidth. Together with the stability information this data allows you to determine if the gain and noise figure targets can be met.

plot(Unmatched_Amp,'Ga','Gt','dB')

This plot, shows the power gain across the 100-MHz bandwidth. It indicates that the transducer gain varies linearly between 5.5 dB to about 3.1 dB and achieves only 4.3 dB at band center. It also suggests there is sufficient headroom between the transducer gain Gt and the available gain Ga to achieve our target Gt of 10 dB.

plot(Unmatched_Amp,'Fmin','NF','dB')
axis([200 300 0 2])
legend('Location','NorthEast')

This plot shows the variation of the noise figure with frequency. The unmatched amplifier clearly meets the target noise figure requirement. However this would change once the input and output matching networks are included. Most likely, the noise figure of the LNA would exceed the requirement.

The region of operation is between 200 300 MHz. Therefore, choose a bandpass topology for the matching networks which is shown here.

Figure 2: Matching network topology

The topology chosen, as seen in Figure 2, is a direct-coupled prototype bandpass network of parallel resonator type with top coupling [2], that is initially tuned to the geometric mean frequency with respect to the bandwidth of operation.

N_input = 3;           % Order of input matching network
N_output = 3;          % Order of output matching network
wU = 2*pi*fUpper;      % Upper band edge
wL = 2*pi*fLower;      % Lower band edge
w0 = sqrt(wL*wU);      % Geometric mean

For the initial design all the inductors are assigned the same value on the basis of the first series inductor. As mentioned in [3], choose the prototype value to be unity and use standard impedance and frequency transformations to obtain denormalized values [1]. The value for the capacitor in the parallel trap is set using this inductor value to make it resonate at the geometric mean frequency. Note that there are many ways of designing the initial matching network. This example shows one possible approach.

LvaluesIn = (Zs/(wU-wL))*ones(N_input,1);    % Series and shunt L's [H]
CvaluesIn = 1 / ( (w0^2)*LvaluesIn(2));      % Shunt C [F]

### Form Complete Circuit with Matching Networks and Amplifier

Use either the rfckt.seriesrlc or rfckt.shuntrlc constructor to build each branch of the matching network. Then, form the matching network from these individual branches by creating an rfckt.cascade object. The output matching network for this example is the same as the input matching network.

LC_InitialIn = [LvaluesIn;CvaluesIn];
LvaluesOut = LvaluesIn;
CvaluesOut = CvaluesIn;
LC_InitialOut = [LvaluesOut;CvaluesOut];

{rfckt.seriesrlc('L',LvaluesIn(1)), ...
rfckt.shuntrlc('C',CvaluesIn,'L',LvaluesIn(2)), ...
rfckt.seriesrlc('L',LvaluesIn(3))});

{rfckt.seriesrlc('L',LvaluesOut(1)), ...
rfckt.shuntrlc('C',CvaluesOut,'L',LvaluesOut(2)), ...
rfckt.seriesrlc('L',LvaluesOut(3))});

Put together the LNA network consisting of matching networks and amplifier by creating an rfckt.cascade object as shown in previous section.

{InputMatchingNW,Unmatched_Amp,OutputMatchingNW});

### Optimize Input & Output Matching Network

There are several points to consider prior to the optimization.

• Objective function: The objective function can be built in different ways depending on the problem at hand. For this example, the objective function is shown in the file below.

• Choice of cost function: The cost function is the function you would like to minimize (maximize) to achieve near optimal performance. There could be several ways to choose the cost function. For this example you have two requirements to satisfy simultaneously, i.e. gain and noise figure. To create the cost function you first, find the difference, between the most current optimized network and the target value for each requirement at each frequency. The cost function is the L2-norm of the vector of gain and noise figure error values.

• Optimization variables: In this case it is a vector of values, for the specific elements to optimize in the matching network.

• Optimization method: A direct search based technique, the MATLAB® function fminsearch, is used in this example to perform the optimization.

• Number of iterations/function evaluations: Set the maximum no. of iterations and function evaluations to perform, so as to tradeoff between speed and quality of match.

• Tolerance value: Specify the variation in objective function value at which the optimization process should terminate.

The objective function used during the optimization process by fminsearch is shown here.

% returns the current value of the objective function stored in OUTPUT
% evaluated after updating the element values in the object, AMP. The
% inductor and capacitor values are stored in the variable LC_OPTIM.
%
% BROADBAND_MATCH_AMPLIFIER_OBJECTIVE_FUNCTION is an objective function of RF Toolbox demo:
% Designing Broadband Matching Networks (Part II: Amplifier)

%   Copyright 2008 The MathWorks, Inc.

% Ensure positive element values
if any(LC_Optim<=0)
output = inf;
return;
end
% Update matching network elements - The object AMP has several properties
% among which the cell array 'ckts' consists of all circuit objects from
% source to load.  Since RFCKT.CASCADE was used twice, first to form the
% matching network itself and a second time to form the LNA, we have to
% step through two sets of cell arrays to access the elements
for loop1 = 1:3
AMP.ckts{1}.ckts{loop1}.L  = LC_Optim(loop1);
AMP.ckts{3}.ckts{loop1}.L  = LC_Optim(loop1+4);
end
AMP.ckts{1}.ckts{2}.C    = LC_Optim(4);
AMP.ckts{3}.ckts{2}.C    = LC_Optim(8);

% Perform analysis on tuned matching network
Npts           = length(freq);
analyze(AMP,freq,Zl,Zs,Z0);

% Calculate target parameters of the Amplifier
target_param   = calculate(AMP,'Gt','NF','dB');
Gt             = target_param{1}(1:Npts,1);
NF_amp         = target_param{2}(1:Npts,1);

% Calculate Target Gain and noise figure error
errGt          = (Gt - Gt_target);
errNF          = (NF_amp - NF);

% Check to see if gain and noise figure target are achieved by specifying
% bounds for variation.
deltaG         = 0.40;
deltaNF        = -0.05;
errGt(abs(errGt)<=deltaG) = 0;
errNF(errNF<deltaNF) = 0;

% Cost function
err_vec        = [errGt;errNF];
output         = norm((err_vec),2);

% Animate
Gmax           = (Gt_target + deltaG).*ones(1,Npts);
Gmin           = (Gt_target - deltaG).*ones(1,Npts);
plot(AMP,'Gt','NF','dB');
hold on
plot(freq.*1e-6,Gmax,'r-*')
plot(freq.*1e-6,Gmin,'r-*')
legend('G_t','NF','Gain bounds','Location','East');
axis([freq(1)*1e-6 freq(end)*1e-6 0 Gt_target+2]);
hold off
drawnow;

The optimization variables are all the elements (inductors and capacitors) of the input and output matching networks.

nIter = 125; % Max No of Iterations
options = optimset('Display','iter','TolFun',1e-2,'MaxIter',nIter);   % Set options structure
LC_Optimized = [LvaluesIn;CvaluesIn;LvaluesOut;CvaluesOut];
LC_Optimized,freq,Gt_target,NFtarget,Zl,Zs,Z0),LC_Optimized,options);

Iteration   Func-count     min f(x)         Procedure
0            1          30.4869
1            9          28.3549         initial simplex
2           11          25.5302         expand
3           12          25.5302         reflect
4           13          25.5302         reflect
5           14          25.5302         reflect
6           16          22.8228         expand
7           17          22.8228         reflect
8           19          19.0289         expand
9           20          19.0289         reflect
10           21          19.0289         reflect
11           22          19.0289         reflect
12           24          14.8785         expand
13           25          14.8785         reflect
14           27           10.721         expand
15           28           10.721         reflect
16           29           10.721         reflect
17           31          9.84796         expand
18           32          9.84796         reflect
19           33          9.84796         reflect
20           34          9.84796         reflect
21           35          9.84796         reflect
22           37          9.84796         contract outside
23           39          9.84796         contract outside
24           41          9.84796         contract inside
25           43          9.64666         reflect
26           45          9.64666         contract inside
27           46          9.64666         reflect
28           48          9.64666         contract inside
29           49          9.64666         reflect
30           51          9.64666         contract inside
31           53           7.9372         expand
32           55           7.9372         contract outside
33           56           7.9372         reflect
34           57           7.9372         reflect
35           58           7.9372         reflect
36           59           7.9372         reflect
37           60           7.9372         reflect
38           62          5.98211         expand
39           63          5.98211         reflect
40           64          5.98211         reflect
41           65          5.98211         reflect
42           66          5.98211         reflect
43           68          4.31973         expand
44           70          4.31973         contract inside
45           71          4.31973         reflect
46           72          4.31973         reflect
47           73          4.31973         reflect
48           74          4.31973         reflect
49           75          4.31973         reflect
50           77          2.83135         expand
51           79          1.17624         expand
52           80          1.17624         reflect
53           81          1.17624         reflect
54           82          1.17624         reflect
55           84         0.691645         reflect
56           85         0.691645         reflect
57           86         0.691645         reflect
58           88         0.691645         contract inside
59           90         0.691645         contract outside
60           91         0.691645         reflect
61           93         0.691645         contract inside
62           95         0.691645         contract inside
63           96         0.691645         reflect
64           97         0.691645         reflect
65           98         0.691645         reflect
66          100         0.691645         contract inside
67          102         0.691645         contract outside
68          103         0.691645         reflect
69          105         0.691645         contract inside
70          107         0.497434         reflect
71          109         0.497434         contract inside
72          111         0.497434         contract inside
73          112         0.497434         reflect
74          114         0.497434         contract inside
75          116         0.497434         contract inside
76          118         0.444957         reflect
77          120         0.402851         expand
78          122                0         reflect
79          123                0         reflect
80          125                0         contract inside
81          127                0         contract inside
82          128                0         reflect
83          129                0         reflect
84          130                0         reflect
85          131                0         reflect
86          132                0         reflect
87          133                0         reflect
88          134                0         reflect
89          135                0         reflect
90          137                0         contract inside

91          139                0         contract outside

Optimization terminated:
the current x satisfies the termination criteria using OPTIONS.TolX of 1.000000e-04
and F(X) satisfies the convergence criteria using OPTIONS.TolFun of 1.000000e-02

### Update Matching Network and Re-analyze LNA

When the optimization routine stops, the optimized element values are stored in LC_Optimized. The following code updates the input and output matching network with these values.

for loop1 = 1:3
Matched_Amp.ckts{1}.ckts{loop1}.L = LC_Optimized(loop1);
Matched_Amp.ckts{3}.ckts{loop1}.L = LC_Optimized(loop1 + 4);
end
Matched_Amp.ckts{1}.ckts{2}.C = LC_Optimized(4);
Matched_Amp.ckts{3}.ckts{2}.C = LC_Optimized(8);
analyze(Matched_Amp,freq,Zl,Zs,Z0); % Analyze LNA

### Verify Design

The results of optimization can be viewed by plotting the transducer gain and the noise figure across the bandwidth, and comparing it with the unmatched amplifier.

plot(Matched_Amp,'Gt')
hold all
plot(Unmatched_Amp,'Gt')
plot(Matched_Amp,'NF')
plot(Unmatched_Amp,'NF')
legend('G_t  - Matched','G_t  - Unmatched','NF - Matched',...
'NF - Unmatched','Location','East')
axis([freq(1)*1e-6 freq(end)*1e-6 0 12])
hold off

The plot shows, the target requirement for both gain and noise figure have been met. To understand the effect of optimizing with respect to only the transducer gain, use the first choice for the cost function (which involves only the gain term) within the objective function shown above.

### Display Optimized Element Values

The optimized inductor and capacitor values for the input matching network are shown below.

Lin_Optimized = LC_Optimized(1:3)
Lin_Optimized = 3×1
10-7 ×

0.5722
0.9272
0.3546

Cin_Optimized = LC_Optimized(4)
Cin_Optimized = 6.8526e-12

Similarly, here are the optimized inductor and capacitor values for the output matching network

Lout_Optimized = LC_Optimized(5:7)
Lout_Optimized = 3×1
10-6 ×

0.0517
0.1275
0.0581

Cout_Optimized = LC_Optimized(8)
Cout_Optimized = 5.4408e-12

### References

[1] Ludwig, Reinhold, and Gene Bogdanov. RF Circuit Design: Theory and Applications. Upper Saddle River, NJ: Prentice-Hall, 2009.

[2] Cuthbert, Thomas R. Broadband Direct-Coupled and Matching RF Networks. Greenwood, Ark.: T.R. Cuthbert, 1999.

[3] Cuthbert, T.R. “A Real Frequency Technique Optimizing Broadband Equalizer Elements.” In 2000 IEEE International Symposium on Circuits and Systems. Emerging Technologies for the 21st Century. Proceedings (IEEE Cat No.00CH36353), 5:401–4. Geneva, Switzerland: Presses Polytech. Univ. Romandes, 2000. https://doi.org/10.1109/ISCAS.2000.857453.

[4] Pozar, David M. Microwave Engineering. 4th ed. Hoboken, NJ: Wiley, 2012.