Matlab史密斯圆图与LNA设计

这几天需要用Matlab设计一个LNA，结果发现Matlab的史密斯圆图不太好用…

于是我决定自己整一个史密斯圆图。

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电阻圆族和电抗圆族公式如下：

$${(\Gamma_{r}-\frac{r}{1+r})}^2 + {\Gamma_{i}}^2 = {(\frac{1}{1+r})}^2$$

$${(\Gamma_{r}-1)}^2 + {(\Gamma_{i}-\frac{1}{x})}^2 = {(\frac{1}{x})}^2$$

那么，可以写出代码如下：

function [] = smith()
    for x = -4 : 0.2 : 4
        viscircles([1, 1 / x], 1 / abs(x), "LineWidth", 0.1, "Color", "blue");
        xlim([-1, 1]);
        ylim([-1, 1]);
    end
    line([-1, 1], [0, 0], "LineWidth", 0.1, "Color", "blue");
    
    for r = 0 : 0.2 : 5
        viscircles([r / (1 + r), 0], 1 / (1 + r), "LineWidth", 0.1, "Color", "blue");
        hold on; 
        axis equal;
        xlim([-1, 1]);
        ylim([-1, 1]);
    end     

    anglex={-1, -1, 1, 1};
    angley={1, -1, -1, 1};
    for i = 1 : 4 
        for theta = pi / 2 * i : pi / 1000 : pi / 2 * ( i + 1 )
            x = cos(theta); 
            y = sin(theta);
            x1 = cos(theta + pi / 1000); 
            y1 = sin(theta + pi / 1000);
            fill([x, x1, anglex{i}], [y, y1, angley{i}], "b");
        end
    end
end

效果是这样的

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LNA设计基于上文的史密斯圆图来表示。

具体公式比较复杂，可以去看《射频电路设计――理论与应用（第二版）》这本书📕，公式位于第九章。

代码如下：

clear;
close all;
clc;

% S11 A11 S21 A21 S12 A12 S22 A22 Topt_m Topt_a Fmin_db Rn/50 顺序 
Parameter_demo = [0.3 30 2.5 -80 0.2 -60 0.2 -15 0.5 45 1.5 0.08];
Parameter_2G = [];

% 选择S参数
Sel = Parameter_demo;

% 指定 增益
G_db = 8;
G = power(10, G_db / 10);

% 指定 VSWRimn
VSWRimn = 1.5;

Timn_abs = (VSWRimn - 1) / (VSWRimn + 1);

% 指定 噪声系数
F_db = 1.6; 
F = power(10, F_db / 10);

% Rn/50
Rn = Sel(12);

% 最小噪声系数
Fmin_db = Sel(11); 
Fmin = power(10, Fmin_db / 10);

% opt反射系数
Topt_a = ang2rad(Sel(10));
Topt = Sel(9) * exp (1i * Topt_a);

% 指定S参数
a11 = ang2rad(Sel(2));
s11 = Sel(1) * exp (1i * a11);

a21 = ang2rad(Sel(4));
s21 = Sel(3) * exp (1i * a21);

a12 = ang2rad(Sel(6));
s12 = Sel(5) * exp (1i * a12);

a22 = ang2rad(Sel(8));
s22 = Sel(7) * exp (1i * a22);

% 史密斯圆图
subplot(121);
smith();

% 稳定性计算
delta = s11 * s22 -s12 * s21;

k = (1 - power(abs(s11), 2) - power(abs(s22), 2) + power(abs(delta), 2)) / (2 * abs(s12) * abs(s21));

% 最大功率增益
Gmax = abs(s21) / abs(s12) * (k  - sqrt(power(k, 2) - 1));
Gmax_db = 10 * log10(Gmax);

% 比例系数
g0 = G / power(abs(s21), 2);

% 功率增益圆
dg0 = g0 * conj(s22 - delta * conj(s11)) / (1 + g0 * (power(abs(s22), 2) - power(abs(delta), 2)));
rg0 = sqrt(1 - 2 * k * g0 * abs(s12 * s21) + power(g0, 2) * power(abs(s12 * s21), 2)) / abs(1 + g0 * (power(abs(s22), 2) - power(abs(delta), 2)));

% 转换
dgs = ((1 - s22 * dg0) * conj((s11 - delta * dg0)) - power(rg0, 2) * conj(delta) * s22) / (power(abs(1 - s22 * dg0), 2) - power(rg0, 2) * power(abs(s22), 2));
rgs = rg0 * abs(s12 * s21) / abs(power(abs(1 - s22 * dg0), 2) - power(rg0, 2) * power(abs(s22), 2));

viscircles([real(dgs), imag(dgs)], rgs, "LineWidth", 2, "Color", "red");

% 噪声功率圆
Qk = power(abs(1 + Topt), 2) * (F - Fmin) / (4 * Rn);
df = Topt / (1 + Qk);
rf = sqrt((1 - power(abs(Topt), 2)) * Qk + power(Qk, 2)) / (1 + Qk);

viscircles([real(df), imag(df)], rf, "LineWidth", 2, "Color", "green");

% 噪声opt
scatter(real(Topt), imag(Topt), "red");

% 扫描取优
min_F = inf;
VSWRomn_target = inf;
for degree = 1:360
    Ts = dgs + rgs * exp (1i * ang2rad(degree));
    Tl = (s11 - conj(Ts)) / (delta - s22 * conj(Ts));

    Tout = s22 + s12 * s21 * Ts / (1 - s11 * Ts);
    Tomn = (conj(Tout) - Tl) / (1 - Tl * Tout);

    VSWRomn = (1 + abs(Tomn)) / (1 - abs(Tomn));

    if(abs(Ts - df) < rf && VSWRomn < VSWRomn_target)
        scan_F = Fmin + 4 * Rn * power(abs(Ts - Topt), 2) / ((1 - power(abs(Ts), 2)) * power(abs(1 + Topt), 2));

        if(scan_F < min_F)
            min_F = scan_F;
            Ts_final = Ts;
            Tl_final = Tl;
        end
        scatter(real(Ts), imag(Ts), "yellow");
    end
end

scatter(real(Ts_final), imag(Ts_final), "red");

% 下面进行VSWR的优化

Tin = s11 + (s21 * s12 * Tl_final) / (1 - s22 * Tl_final);

dv_imn = (1 - power(Timn_abs, 2)) * conj(Tin) / (1 - power(Timn_abs * abs(Ts), 2));
rv_imn = (1 - power(abs(Tin), 2)) * Timn_abs / (1 - power(Timn_abs * abs(Ts), 2));

viscircles([real(dv_imn), imag(dv_imn)], rv_imn, "LineWidth", 2, "Color", "cyan");
VSWRomn_list = zeros(360, 1);

for degree = 1:360
    Ts = dv_imn + rv_imn * exp (1i * ang2rad(degree));

    Tout = s22 + s12 * s21 * Ts / (1 - s11 * Ts);
    Tomn = (conj(Tout) - Tl_final) / (1 - Tl_final * Tout);

    VSWRomn_list(degree) = (1 + abs(Tomn)) / (1 - abs(Tomn));
end

subplot(122);
grid on;
xlim([1, 360]);
ylim([1, inf]);
hold on;

% 画出不同角度的VSWR，然后选择一个角度
plot(1:360, VSWRomn_list, "Color", "blue");
line([1, 360], [VSWRimn, VSWRimn], "Color", "red", "LineStyle", "--");

prompt = "选择角度：";
degree = input(prompt);

Ts_final = dv_imn + rv_imn * exp (1i * ang2rad(degree));
Tout = s22 + s12 * s21 * Ts_final / (1 - s11 * Ts_final);

% Gt值计算
Gt = (1 - power(abs(Tl_final), 2)) * power(abs(s21), 2) * (1 - power(abs(Ts_final), 2)) / (power(abs(1 - Ts_final * Tin), 2) * power(abs(1 - s22 * Tl_final), 2));
Gt_db = 10 * log10(Gt);

% 噪声系数
F_final = Fmin + 4 * Rn * power(abs(Ts_final - Topt), 2) / ((1 - power(abs(Ts_final), 2)) * power(abs(1 + Topt), 2));
F_final_db = 10 * log10(F_final);

% 输出
fprintf("Ts: %f∠%f°\nTout: %f∠%f°\n", abs(Ts_final), getAngle(Ts_final), abs(Tout), getAngle(Tout));
fprintf("Gt(dB): %f\nF(dB): %f\nVSWRomn: %f\n", Gt_db, F_final_db, VSWRomn_list(degree));

该代码应配合下面的函数使用。

function [rad] = ang2rad(ang)
    rad = ang / 360 * 2 * pi;
end

function [ang] = getAngle(z)
    ang = angle(z) / 2 / pi * 360;
end

运行结果如下：

左图中红圈就是对于$\Gamma_{S}$的等增益圆，绿圈则是等噪声圆。

绿圈中的红点是$\Gamma_{opt}$。

左图下方一大片黄点是满足参数要求（增益、噪声）的点，红色的则是噪声系数最小的点。

获得噪声系数最小的点后，进行驻波比优化。通过给定IMN的驻波比，画出图中蓝色的等驻波比圆。扫描该圆，计算驻波比，画出右图。

右图中实线是OMN的驻波比，需要找到一个比较小的点；虚线是IMN的驻波比。

将找到的OMN对应的角度输入Matlab，即可得到输出。