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Homework Statement
The attempt at a solution
Constructing the total impedance of the circuit as follows,
$$\frac{1}{Z_T}=\frac{1}{Z_R}+\frac{1}{Z_C}+\frac{1}{Z_L}$$
where $Z_R=R$, $Z_C=j\frac{1}{\omega C}$ and $Z_L=j\omega L$.
$$\frac{1}{Z_T}=\frac{1}{R}+j\omega C+\frac{1}{j\omega L}$$
solving for $Z_T$ gives us,
$$Z_T=\frac{Rj\omega L}{j\omega LRCL\omega^2+R}$$
where $\omega=2\pi f$
$$Z_T=\frac{2\pi fRLj }{2\pi fLjRCL(2\pi f)^2+R}$$
Plotting this function over the frequencies, whilst $L=10\ \mu H$, $C=5\ pF$ and $R=10^6\ \Omega
From here, I created the following MATLAB code to find:
 The magnitude of the impedance (Z_Mag) for a range of frequencies.
 Finding the maximum of Z_mag and it's corresponding frequency.
 finding the 3db points and the frequencies that correspond to these points.
 taking the difference of said frequencies in order to calculate the bandwidth.
Now when I vary the N (number of frequency samples), the bandwidth I find seems to change quite drastically. Ideally, I would expect the bandwidth to asymptotically approach some value with a greater accuracy for a greater N.
I have been trying to found out what I did wrong for some time now, does anyone know where I went wrong? Thanks!
The attempt at a solution
Constructing the total impedance of the circuit as follows,
$$\frac{1}{Z_T}=\frac{1}{Z_R}+\frac{1}{Z_C}+\frac{1}{Z_L}$$
where $Z_R=R$, $Z_C=j\frac{1}{\omega C}$ and $Z_L=j\omega L$.
$$\frac{1}{Z_T}=\frac{1}{R}+j\omega C+\frac{1}{j\omega L}$$
solving for $Z_T$ gives us,
$$Z_T=\frac{Rj\omega L}{j\omega LRCL\omega^2+R}$$
where $\omega=2\pi f$
$$Z_T=\frac{2\pi fRLj }{2\pi fLjRCL(2\pi f)^2+R}$$
Plotting this function over the frequencies, whilst $L=10\ \mu H$, $C=5\ pF$ and $R=10^6\ \Omega
From here, I created the following MATLAB code to find:
 The magnitude of the impedance (Z_Mag) for a range of frequencies.
 Finding the maximum of Z_mag and it's corresponding frequency.
 finding the 3db points and the frequencies that correspond to these points.
 taking the difference of said frequencies in order to calculate the bandwidth.
Code:
clear all
close all
N=100000; %number of frequency samples
L=10*10^(6); %inductance
C=5*10^(12); %capacitance
R=10^6; %resistance
f=linspace(1,10^8,N); %frequency of 1 Hz to 100 MHz
df=f(2)f(1);
Z_T=(2*pi*f*R*L*1i)./(2*pi*f*L*1iR*C*L*(2*pi*f).^2+R); %impedance
Z_mag=abs(Z_T); %magnitude of the complex impedance
plot(f,Z_mag); %plotting the frequency against the total impedance
title('Total impedance per frequency')
xlabel('frequency')
ylabel('total impedance')
[max_Z, max_index]=max(Z_mag); %maximum value of impedance
threedb=max_Z*sqrt(2)/2; %the 3db point
[Z_db,index_db] = min(abs(Z_magthreedb)); %closest value and index to the 3db point
f1=index_db*df; %the first frequency at the 3db point
%the second frequency at the 3db point (first frequency mirrored around the
%frequency at max magnitude
f2=(max_index+(max_indexindex_db))*df;
BW=abs(f2f1); %the bandwidth
Now when I vary the N (number of frequency samples), the bandwidth I find seems to change quite drastically. Ideally, I would expect the bandwidth to asymptotically approach some value with a greater accuracy for a greater N.
I have been trying to found out what I did wrong for some time now, does anyone know where I went wrong? Thanks!
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