使用MATLAB迭代间接方程(Iterations of an indirect equation using MATLAB)

我被困在绘制表面电位（shy_s）Vs栅极电压（vgb）的图表上。我只需要解决下面的这个等式，找到每次迭代的根

vgb=vfb+shy_s+gama.*sqrt(shy_s+shy_t.*exp((shy_s-2.*shy_f)/shy_t))

哪里

shy_f=0.347; %shy_f=shy_t*ln(Na/ni) shy_t=0.0259; %Thermal voltage = KT/e ; where k = 1.3806*10^-23 @ 300 K es=11.7*8.85*10^-12; Na=10^10; %[unit]=[m^-3) cox=6.93*10^-12; %[unit]=[F/m^2] and t_ox=550 A q=1.6*10^-19; vfb=0; gama=(sqrt(2*q*es*Na)/cox);

在这里，我必须找到不同的shy_s(surface potential)值vgb(gate voltage)的shy_s(surface potential)的值。

所以我尝试了不同的方法来解决它，比如

a=zeros(1,100); b=zeros(1,100); for vgb=0:0.1:10 shy_s=0; % Say p=shy_s; % And j=vgb-vfb-((sqrt(2*q*es*10^10))/cox).*sqrt(shy_s+shy_t.*exp((shy_s-2.*shy_f)/shy_t)); D=p-j; if D>0 for shy_s=0:0.1:30; D=p-j; if D<0 a=shy_s; break end end elseif D<0 for shy_s=0:0.1:30 D=p-j; if D>0 a=shy_s; break end end end b(1,vgb)=a; end plot(vgb,b)

以这种方式出现以下错误：

??? 下标索引必须是实数正整数或逻辑。

错误= =>> shy_s_vs_vgb_latest2 at 78 b（1，vgb）= a;

我再次尝试使用相当简单的技术 -

vgb=fzero(@(shy_s)vfb+shy_s+gama.*sqrt(shy_s+shy_t.*exp((shy_s-(2.*shy_f))/shy_t)),2)

但它说 -

退出fzero：中止搜索包含符号更改的间隔，因为搜索期间遇到复杂的函数值。（-0.56处的功能值为-0.56 + 62.1585i。）检查功能或使用不同的起始值重试。

vgb =

为NaN

另一种关系可用于相同的目的

(vgb-vfb-shy_s)/gama)^2 = shy_s+shy_t.*(exp((shy_s-2*shy_f)/shy_y))+shy_t.*(exp(-shy_s/shy_t)-1)

I am stuck on plotting a graph of surface potential(shy_s) Vs Gate voltage(vgb). I just have to solve this equation below and find the root for every iterations

vgb=vfb+shy_s+gama.*sqrt(shy_s+shy_t.*exp((shy_s-2.*shy_f)/shy_t))

where

shy_f=0.347; %shy_f=shy_t*ln(Na/ni) shy_t=0.0259; %Thermal voltage = KT/e ; where k = 1.3806*10^-23 @ 300 K es=11.7*8.85*10^-12; Na=10^10; %[unit]=[m^-3) cox=6.93*10^-12; %[unit]=[F/m^2] and t_ox=550 A q=1.6*10^-19; vfb=0; gama=(sqrt(2*q*es*Na)/cox);

Here I have to find the value of shy_s(surface potential) for different values of vgb(gate voltage).

So I tried different methods to solve it, such as

a=zeros(1,100); b=zeros(1,100); for vgb=0:0.1:10 shy_s=0; % Say p=shy_s; % And j=vgb-vfb-((sqrt(2*q*es*10^10))/cox).*sqrt(shy_s+shy_t.*exp((shy_s-2.*shy_f)/shy_t)); D=p-j; if D>0 for shy_s=0:0.1:30; D=p-j; if D<0 a=shy_s; break end end elseif D<0 for shy_s=0:0.1:30 D=p-j; if D>0 a=shy_s; break end end end b(1,vgb)=a; end plot(vgb,b)

At this manner the following error shows up:

??? Subscript indices must either be real positive integers or logicals.

Error in ==> shy_s_vs_vgb_latest2 at 78 b(1,vgb)=a;

Again I tried to use rather a simpler technique-

vgb=fzero(@(shy_s)vfb+shy_s+gama.*sqrt(shy_s+shy_t.*exp((shy_s-(2.*shy_f))/shy_t)),2)

but it says-

Exiting fzero: aborting search for an interval containing a sign change because complex function value encountered during search. (Function value at -0.56 is -0.56+62.1585i.) Check function or try again with a different starting value.

vgb =

NaN

Another relation can be used for the same purpose

(vgb-vfb-shy_s)/gama)^2 = shy_s+shy_t.*(exp((shy_s-2*shy_f)/shy_y))+shy_t.*(exp(-shy_s/shy_t)-1)

最满意答案

虽然不太可能是最好的解决方案，但以下是一个快速而肮脏的技巧：

opt = optimset('TolFun',1e-8); vgb=@(shy_s) vfb+shy_s+gama.*sqrt(shy_s+shy_t.*exp((shy_s-2.*shy_f)/shy_t)); b = fminsearch(@(shy_s) abs(vgb(shy_s)-VAL),10,opt);

VAL是您希望找到的逆数。

Here how to use the function fzero for a simple iteration with out doing much -

for i=1:length(vgb)

c=@(shy_s)((vgb(i)-vfb-shy_s)/gama)-sqrt(abs(shy_s+shy_t.*exp((shy_s-2.*shy_f)/shy_t)));

shy_s=fzero(c,[-3 10])

a(i)=shy_s

end

'a' gives the correct iterated value!

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