גרסה מתאריך 21:28, 26 בדצמבר 2018

תוכן עניינים

1 מטלאב
2 ג'אווה-סקריפט
3 מאקסימה
- 3.1 שליפת תדר בסריקה ליניארית
- 3.2 תדר תהודה

1 מטלאב

1.1 גרף הדגמת FFT ו-THD

איור 1: הדגמה של THD

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%% FFT with THD example                                        %%%

%%% based on https://www.mathworks.com/help/matlab/ref/fft.html %%%

%%% with a few modifications                                    %%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Fs = 1000; % Sampling frequency

```
T = 1/Fs; % Sampling period       
```
```
L = 300; % Length of signal
```
```
t = (0:L-1)*T; % Time vector
```
```
freq = 50; % frequency
```

S1 = 5*sin(2*pi*freq*t); % 1st (base) harmony

S2 = 2*sin(2*pi*3*freq*t); % 2nd harmony distortion

```
S = S1 + S2;
```
```
f = Fs*(0:(L/2))/L;
```
```
Y = fft(S);
```
```
P2 = abs(Y/L);
```
```
P1 = P2(1:L/2+1);
```
```
P1(2:end-1) = 2*P1(2:end-1);
```
```
 
```
```
figure('Name','THD example');
```
```
subplot(311);
```
```
plot(t,S1,t,S2);
```
```
title('Two pure sine waves');
```
```
ylabel('Amplitude [V]');
```
```
xlabel('Time [S]');
```

legend(['1st harmony: ',num2str(freq),' Hz'],['2nd harmony: ',num2str(3*freq),' Hz']);

```
subplot(312);
```
```
plot(t,S);
```

title(['Addition of the above two sine waves with frequencies of ',num2str(freq),' and ',num2str(3*freq),' Hz']);

```
ylabel('Amplitude [V]');
```
```
xlabel('Time [S]');
```
```
subplot(313);
```
```
plot(f,P1);
```

title('Single-Sided Amplitude Spectrum of 1st (base) harmony + 2nd harmony');

```
xlabel('f (Hz)');
```
```
ylabel('Amplitude [V]');
```
```
 
```

set(gcf, 'Position', [600, 80, 800, 900]);

1.2 ייצוג ספקטראלי של גלים יסודיים

יש לשנות את המשתנה wave לערכים של בין 1 ל-6 כדי לקבל את ששת הגלים השונים:

סינוסי
ריבועי
שן-מסור
משולש
חצי-גל סינוסי מיושר
סדרת-דפקים

איור 2: גל שן-מסור והייצוג הספקטראלי שלו

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%% FFT based on                                       %%%

%%% https://www.mathworks.com/help/matlab/ref/fft.html %%%

%%% with a few modifications                           %%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

```
freq = 1000; % frequency
```

Fs = freq*20; % Sampling frequency

```
T = 1/Fs; % Sampling period       
```
```
L = 200; % Length of signal
```
```
t = (0:L-1)*T; % Time vector
```
```
 
```
```
% three different waves
```
```
RMS_Amplitude = 1;
```
```
x = 2*pi*freq*t;
```
```
d=20/100;  % Duty cycle for pulse wave
```

Names = {'Sine', 'Square', 'Sawtooth', 'Triangle','Half rectified sine','Pulse'};

Amplitudes = [sqrt(2), 1, sqrt(3), sqrt(3), 2, 1/sqrt(d)] * RMS_Amplitude;

half_wave = sin(x); half_wave(find(half_wave<0))=0;

pulse_wave = pulstran(x,(0:9)*max(x)/10,@rectpuls,d*max(x)/10);

Waves = {sin(x), square(x), sawtooth(x), sawtooth(x,0.5), half_wave, pulse_wave};

```
wave = 6; % actual wave to display
```
```
 
```
```
S=Amplitudes(wave)*Waves{wave};
```
```
f = Fs*(0:(L/2))/L;
```
```
Y = fft(S);
```
```
P2 = abs(Y/L);
```
```
P1 = P2(1:L/2+1);
```
```
P1(2:end-1) = 2*P1(2:end-1);
```
```
 
```
```
figure('Name','FFT example');
```

set(gcf, 'Position', [600, 80, 800, 900]);

```
ax=subplot(211);
```
```
plot(t,S); % plot wave
```

title(['Original ',Names{wave},' wave']);

```
ylabel('Amplitude [V]');
```
```
xlabel('Time [S]');
```
```
if max(ax.YLim)==max(S)
```
```
    ax.YLim=ax.YLim*1.1;
```
```
end
```
```
ax=subplot(212);
```
```
P1_RMS=P1;
```

non_DC=find(f>freq/2); % find non DC frequencies

P1_RMS(non_DC)=P1_RMS(non_DC)/sqrt(2); % calculate RMS

```
plot(f,P1_RMS); % plot FFT of wave
```

title(['FFT of the ',Names{wave},' wave']);

```
xlabel('f (Hz)');
```
```
ylabel('Amplitude [RMS V]');
```
```
if max(ax.YLim)==max(P1_RMS)
```
```
    ax.YLim=ax.YLim*1.1;
```
```
end
```

1.3 בניית גלים יסודיים באמצעות הרמוניות

זוהי תוכנה הופכית ממה שמופיע בייצוג ספקטראלי של גלים יסודיים, הפעם באמצעות נוסחא של ההרמוניות - בונים את הגל הסופי.

יש לשנות את המשתנה wave_num לערכים של בין 1 ל-6 כדי לקבל את ששת הגלים השונים:

גל ריבועי
סדרת-דפקים
מיישר חצי-גל
מיישר גל-מלא
גל שן-מסור
גל משולש

איור 3: הדגמה של בניית סדרת-פולסים באמצעות הרמוניות

איור 4: סדרת-פולסים אשר מורכבת מ-1000 הרמוניות

```
% structure for fourer waves
```

% 1: Name, 2: RMS Coefficient, 3: DC Average, 4: Series of harmonies, 5: Stand alone harmony if any

```
fourier_waves = {;
```

    {'Square', '1', '0', '4*A/pi*sin((2*n-1)*w0.*t)/(2*n-1)'};

    {'Pulse', '1/sqrt(d)', 'A*d', '2*A/(n*pi)*sin(n*pi*d)*cos(n*w0.*t)'};

    {'Half wave rectified sine', '2', 'A/pi', '-2*A/pi*cos(2*n*w0.*t)/(4*n^2-1)', 'A/2*sin(w0.*t)'};

    {'Full wave rectified sine wave', 'sqrt(2)', '2*A/pi', '-4*A/pi*cos(2*n*w0.*t)/(4*n^2-1)'};

    {'Saw tooth wave', 'sqrt(3)', '0', '-2*A/pi*sin(n*w0.*t)/n'};

    {'Triangle wave', 'sqrt(3)', '0', '8*A/pi^2*cos((2*n-1)*w0.*t)/(2*n-1)^2'}

```
};
```

wave_num = 2;                   % which wave to depict

wave = fourier_waves{wave_num}; % get data for selected wave

```
 
```
```
A_RMS = 1; % Amplitude in volts RMS
```
```
d=20/100;  % Duty cycle for pulse wave
```
```
RMS_coef=eval(wave{2});
```
```
A=A_RMS * RMS_coef;
```
```
f = 1000;  % frequency in Hz
```

harmonies = 1000; % How many harmonies to sum up

```
T=1/f;     % period
```

t=linspace(-T,T,1001); % resolution of two full periods

w0=2*pi*f;             % fundamental harmony

y=zeros(size(t));      % allocate memory for the harmonics addition

figure('Name', [wave{1},' wave']);         % create a plot window

set(gcf, 'Position', [600, 80, 800, 900]); % change window size

fourier_wave_prefix=wave{3}; % prefix of Fourier series

fourier_wave=wave{4};        % addends of the Fourier series

```
if length(wave)==5
```

    fourier_wave1=wave{5};   % out of serise harmony

```
else
```
```
    fourier_wave1=0;
```
```
end
```

subplots=9;                  % how many subplot steps to show

```
rows=sqrt(subplots); cols=rows;
```

axs=cell(1,subplots);        % save all axes for normalization

```
y_max=0; y_min=0;
```

for harmony=0:harmonies            % sums up all the harmonis

```
    switch harmony
```
```
        case 0
```
```
            n=0;
```
```
            % calculate DC (harmony 0)
```

            f_n=eval(fourier_wave_prefix);

```
        case 1
```
```
            % calculate another harmony
```
```
            if fourier_wave1==0
```
```
                f_n=eval(fourier_wave);
```
```
            else
```

                f_n=eval(fourier_wave1);

```
                n=n-1;
```
```
                fourier_wave1=0;
```
```
            end
```
```
        otherwise
```
```
            f_n=eval(fourier_wave);
```
```
    end
```
```
    y=y+f_n;
```

    % plot the intermediary series summation

```
    if harmony<subplots-1
```

        axs{harmony+1}=subplot(rows,cols,harmony+1);

```
        plot(t,y);
```
```
        switch harmony
```
```
            case 0
```

                title('DC (harmony 0)');

```
            case 1
```

                title('First harmony only');

```
            otherwise
```

                title(sprintf('Summation of %d harmonies',harmony));

```
        end
```

        y_min=min(min(y), y_min); y_max=max(max(y), y_max);

```
    end
```
```
    n=n+1;
```
```
end
```

axs{subplots}=subplot(rows,cols,subplots);

```
plot(t,y); % plot the final wave
```

title(['Summation of ',num2str(harmonies),' harmonies']);

```
% normalize all axes
```
```
dy=y_max-y_min;
```
```
ylim=[y_min-dy/10, y_max+dy/10];
```
```
for i=1:subplots
```
```
    axs{i}.YLim=ylim;
```
```
end
```
```
 
```

% plots actual wave (with 1k harmonies) in a new window

```
figure('Name','Actual wave');
```
```
plot(t,y);
```
```
ax=gca;
```
```
ax.YLim=ylim;
```
```
xlabel('Time [S]');
```
```
ylabel('Amplitude [V]');
```
```
title([wave{1},' wave']);
```

1.4 גרף הפרש-מופע

איור 5: הפרש-מופע בציר הזמן

```
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
```

%%% Plots two sine waves with       %%%

%%% phase and amplitude differences %%%

```
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
```
```
f = figure; % create a plot window
```

cycles = 2+1/16;    % display about 2 cycles

```
frequency = 1000; % 1 kHz frequency
```

total_x = 2*pi*cycles; % total horizontal length

x = linspace(0,total_x,1000); % horizontal axis - time scale

```
A1 = 1; % 1st amplitude
```

y1 = A1*sin(x); % vertical axis - voltage/current scale

phase = 2*pi/3; % create a phase difference

```
A2 = 0.7; % 2nd amplitude
```

y2 = A2*sin(x-phase); % another wave with phase difference

plot(x,y1,'k',x,y2,'b','LineWidth',3); % plot the graphs

h1 = gca; % Get handle to Current Axis (1)

h1.YLim = [-1.25, 1.25]; % set scale of figure 1

left1 = pi/2; % left cursor at maximum point

```
right1 = left1 + 2*pi; % next peak
```
```
dx = (right1-left1)/20;
```

annotation('doublearrow',x_to_norm_v2(left1+dx, right1-dx),y_to_norm_v2(A1, A1),'Color','k','LineWidth',2);

text(left1+(right1-left1)/2,A1+0.1,'T','Color','k','HorizontalAlignment','Center');

refline(0,0); % add horizontal reference line

```
left2 = pi/2 + phase;
```
```
right2 = left2 + 2*pi; % next peak
```

annotation('doublearrow',x_to_norm_v2(left2+dx, right2-dx),y_to_norm_v2(A2, A2),'Color','b','LineWidth',2);

text(left2+(right2-left2)/2,A2+0.1,'T','Color','b','HorizontalAlignment','Center');

% plot vertical lines (cursors) for peaks

```
hold on;
```

cursor1x=[left1, left1]; cursor1y=[-0.75, A1]; % left cursor

cursor2x=[left2, left2]; cursor2y=[-0.75, A2]; % right cursor

plot(cursor1x, cursor1y, 'k--', cursor2x, cursor2y, 'b--','LineWidth',2); % plot cursors

annotation('doublearrow',x_to_norm_v2(left1+dx, left2-dx), y_to_norm_v2(-0.75, -0.75), 'Color','r','LineWidth',2); % add ruler

text(left1+(left2-left1)/2,-0.75+0.1,'\Delta T','Color','r','HorizontalAlignment','Center');

% plot vertical lines (cursors) for zero crossings

```
hold on;
```
```
left1z=2*pi;
```
```
right1z=2*pi+phase;
```

cursor1xz=[left1z, left1z]; cursor1yz=[-1, 0]; % left cursor

cursor2xz=[right1z, right1z]; cursor2yz=[-1, 0]; % right cursor

plot(cursor1xz, cursor1yz, 'k--', cursor2xz, cursor2yz, 'b--','LineWidth',2); % plot cursors

annotation('doublearrow',x_to_norm_v2(left1z+dx, right1z-dx), y_to_norm_v2(-1, -1), 'Color','m','LineWidth',2); % add ruler

text(left1z+(right1z-left1z)/2,-1+0.1,'\Delta T','Color','m','HorizontalAlignment','Center');

1.5 גרף עקומות ליסאז'ו

איור 6: עקומות ליסאז'ו בקפיצות של 30°

```
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
```
```
%%% Plots Lissajous curve shapes %%%
```
```
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
```
```
figure('Name','Liss');
```
```
% draw a unit circle in the middle
```
```
subplot(5,5,[7:9,12:14,17:19]);
```
```
phi_ind=0:11;
```

phi_rad=phi_ind*pi/6; % phase shifts in radians

phi_deg=phi_ind*30;   % phase shifts in degrees

```
z=exp(j*phi_rad);
```
```
compass(z);
```
```
numbers=findall(gcf,'Type','text');
```
```
for i=1:numel(numbers)
```
```
    txt = numbers(i).String;
```
```
    if isempty(regexp(txt,'0$'))
```
```
        % erase non degrees numbers
```
```
        numbers(i).Visible='off';
```
```
    else
```
```
        % add degrees symbol
```

        numbers(i).String=[txt,'^\circ'];

```
    end
```
```
end
```
```
xt=linspace(0,2*pi,1000);
```
```
x=sin(xt);
```

plot2angle=[15, 10, 4, 3, 2, 6, 11, 16, 22, 23, 24, 20];

```
for i=phi_ind
```
```
    ax = subplot(5,5,plot2angle(i+1));
```

    y=sin(xt + phi_rad(i+1)); % add phase to y

```
    plot(x,y); % plot the curve
```

    % increase gap between plot and frame

```
    ax.XLim=ax.XLim*1.1;
```
```
    ax.YLim=ax.YLim*1.1;
```
```
    % remove labels
```
```
    ax.YTickLabel={};
```
```
    ax.XTickLabel={};
```
```
    % create title for each subplot
```

    phis = [phi_deg(i+1), phi_deg(i+1)-360];

```
    if (phis(1) > 180)
```
```
        phis = fliplr(phis);
```
```
    else
```
```
        if phis(1) == 0
```

            phis(2) = 360; % instead of -360

```
        end
```
```
    end
```

    title(sprintf('%d^\\circ = %d^\\circ', phis));

```
    grid on;
```
```
end
```
```
% equalize the scale of y and x axes
```

set(gcf, 'Position', [100, 100, 900, 1000]);

1.6 תגובה לתדר של מעגל RC

איור 7: תגובה לתדר של מעגל RC

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%% Create a frequency response for RC circuit %%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

```
f = 1000;         % 1kHz frequency
```
```
R = 1000;         % 1kΩ Resistance
```

C = 1/(2*pi*f*R); % 0.159µF Capacitance

numer = [1]; denom = [R*C*1 1]; % Hc(s)=1/(1+RCs)

w = logspace(0,6,1000); % 10^0 to 10^6 angular-frequency range

freqs(numer,denom,w); % create the graph

set(gcf, 'Position', [600, 80, 800, 900]); % enhance resolution

1.7 אופיין חשל מגנטי

איור 8: עקומת חשל מגנטי

```
%%%%%%%%%%%%%%%%%%%%%%
```
```
%%% Plot B-H curve %%%
```
```
%%%%%%%%%%%%%%%%%%%%%%
```
```
try
```
```
  % worked on matlab version R2017a
```

  bhsim = sim('elec_inductor_hysteresis','MaxStep','1e-5');

  simlog = get(bhsim,'simlog_elec_inductor_hysteresis');

```
catch
```
```
  % worked on matlab version R2016b
```

  bhsim = sim('elec_inductor_with_hysteresis','MaxStep','1e-5');

```
  simlog = get(bhsim,'simlog');
```
```
end
```

inductor=simlog.Nonlinear_Inductor.inductor;

```
H=inductor.H.series.values;
```
```
B=inductor.B.series.values;
```
```
figure('Name','B-H Hysteresis');
```
```
plot(H,B);
```

set(gcf, 'Position', [100, 100, 900, 1000]);

```
grid on;
```
```
xlabel('H [A/m]');
```
```
ylabel('B [T]');
```

% to disable 'Error due to multiple causes' for next runs

```
close_system();
```

1.8 גרף תופעת מעבר במעגל RL

איור 9: תופעת מעבר הזרם של מעגל RL טורי

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%% Draw transient reponse of RL circuit %%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

```
V=10;     % power-supply voltage [V]
```
```
R=10000;  % resistor value [Ω]
```
```
L=10;     % inductor value [H]
```
```
tau=L/R;  % time constant tau [s]
```
```
t=linspace(0,10*tau,1001); % time axis
```
```
il=V/R+(0-V/R)*exp(-t/tau);
```
```
figure('Name','RL circuit');
```
```
plot(t,il,'LineWidth',2);
```
```
% signage
```
```
xlabel('time [s]');
```
```
ylabel('i_L [A]');
```
```
title('RL circuit transient response');
```
```
hold on;
```
```
% extract special tau points
```
```
x = (1:5)*tau;
```
```
[~,il_tau]=intersect(t,x);
```
```
y = il(il_tau);
```
```
% calculate axis limits
```
```
max_y = max(y) * 1.2;
```
```
max_x = max(t) * 1.1;
```
```
min_y = -max_y / 12;
```
```
min_x = -max_x / 11;
```

% draw cross lines to emphasize special tau points

```
for i = 1:length(x)
```

    plot([min_x,x(i),x(i)],[y(i),y(i),min_y]);

```
end
```

legend('i_L','1\tau','2\tau','3\tau','4\tau','5\tau');

```
% change axis
```
```
ax=gca;
```
```
ax.YLim=[min_y max_y];
```
```
ax.YAxis.Exponent = -3;
```
```
ax.XLim=[min_x max_x];
```

1.9 תגובה לתדר ותדר-זוויתי של מעגל RL טורי

איור 10: תגובה לתדר-זוויתי של מעגל RL טורי

איור 11: תגובה לתדר של מעגל RL טורי

```
R=10000;   % Resistor value [Ω]
```
```
L=10;      % Inductor value [H]
```
```
tau=L/R;   % time constant
```

cutoff=1/(2*pi*tau);         % Cutoff frequency [Hz]

f=linspace(0,cutoff*5,1000); % Frequency sweep [Hz]

w=2*pi*f;      % Angular frequency sweep [rad/sec]

```
Vin=10;
```
```
ZL=j*w*L; % Inductor Impedance
```

Vr=Vin*R./(R+ZL);  % Complex Voltage of resistor

VL=Vin*ZL./(R+ZL); % Complex Voltage of inductor

mag_vr=abs(Vr);% get the magnitude of resistor's voltage

mag_vL=abs(VL);% get the magnitude of inductor's voltage

```
 
```
```
% Plot voltage vs. angular frequency
```
```
figure; hold on;
```
```
set(0, 'DefaultLineLineWidth', 2);
```

plot(w,mag_vr,w,mag_vL);            % plot graphs

plot([2*pi*cutoff 2*pi*cutoff],[0 Vin]);        % mark the cutoff vertically

plot([0 max(w)],[Vin/sqrt(2) Vin/sqrt(2)]); % mark the cutoff horizontally

```
% add signage
```
```
title('V_{OUT} vs. Angular Frequency');
```
```
xlabel('Angular Frequency [rad/s]');
```
```
ylabel('Voltage [V]');
```

legend('Resistor voltage','Inductor voltage','Cutoff X','Cutoff Y');

```
 
```
```
% Plot voltage vs. frequency
```
```
figure; hold on;
```

plot(f,mag_vr,f,mag_vL);            % plot graphs

plot([cutoff cutoff],[0 Vin]);        % mark the cutoff vertically

plot([0 max(f)],[Vin/sqrt(2) Vin/sqrt(2)]); % mark the cutoff horizontally

```
% add signage
```
```
title('V_{OUT} vs. Frequency');
```
```
xlabel('Frequency [1/s]=[Hz]');
```
```
ylabel('Voltage [V]');
```

legend('Resistor voltage','Inductor voltage','Cutoff X','Cutoff Y');

1.10 תופעת מעבר במעגל RC עם מקור סינוסי

איור 12: תופעת מעבר במעגל RC עם מקור סינוסי והעלמתה באמצעות שינוי זווית המופע של אות המקור

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%% Plot RC circuit transient response with Sine wave input %%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

```
theta=pi/4; % Phase angle of AC input
```
```
AC = calcRC(theta,'ac');
```
```
 
```
```
% RC Transient response equation
```
```
RCTrans=calcRC(theta,'trans');
```
```
RCSteadyState=calcRC(theta,'steady');
```
```
RCTransWithSine=RCTrans+RCSteadyState;
```
```
 
```
```
% calculate exponential decay
```
```
RCDecay=calcRC(theta,'decay');
```
```
 
```

% calculate theta which will cancel the transient response

```
corrected_theta=calcRC(0,'correction');
```

RCNoTrans=calcRC(corrected_theta,'both');

```
 
```

% plot input and output of original theta

```
figure('Name','RC with AC source');
```
```
subplot(2,1,1);
```
```
t=calcRC(0,'time');
```

plot(t,AC,t,RCTransWithSine,t,RCDecay,[min(t),max(t)],[0,0]);

legend('AC Input','Output (capacitor)','Decay envelope','Reference point');

xlabel('Time [s]'); ylabel('Voltage [v]');

```
title(calcRC(theta,'title'));
```

% plot input and output of corrected theta

```
subplot(2,1,2);
```

plot(t,calcRC(corrected_theta,'ac'),t,RCNoTrans,[min(t),max(t)],[0,0]);

legend('AC input shifted','Output (capacitor)','Reference point');

xlabel('Time [s]'); ylabel('Voltage [v]');

```
title(calcRC(corrected_theta,'title'));
```
```
% change axis
```
```
ax=gca;
```
```
top=max(AC);
```
```
ax.YLim=[-top-1, top+1];
```

set(gcf, 'Position', [100, 100, 600, 800]);

```
 
```

% AC input/Transient/SteadyState response of RC circuit

```
function response = calcRC(angle,role)
```
```
    Vm=10; % Amplitude of sine wave
```

    V0=2; % Initial voltage of capacitor

```
    f=700; % frequency of AC input
```
```
    w=2*pi*f; % angular frequency
```

    count=10; % how many time constants to plot

```
    R=10e3; % Resistor of 10kΩ
```
```
    C=0.1e-6; % Capacitor of 0.1µF
```
```
    tau=R*C; % Time constant
```

    t=linspace(0,count*tau,1000); % Time axis

```
 
```
```
    switch lower(role)
```
```
        case 'title'
```

            response = sprintf('V_S(t)=%dsin(2\\pi\\cdot%dt+%d^\\circ); V_0=%dV',Vm,f,round(angle/pi*180),V0);

```
        case 'time'
```
```
            %%% returns the time axis
```
```
            response = t;
```
```
        case 'correction'
```

            %%% returns angle to cancel out transient response

            response = asin(V0/Vm*sqrt(1+(w*tau)^2))-atan(-w*tau);

```
        case 'both'
```

            %%% return entire response (transient and steady state)

            response = calcRC(angle,'trans') + calcRC(angle,'steady');

```
        case 'trans'
```

            %%% return transient response

            response = (V0-Vm*sin(angle+atan(-w*tau))/sqrt(1+(w*tau)^2))*exp(-t/tau);

```
        case 'steady'
```

            %%% return steady state response

            response = Vm/sqrt(1+(w*tau)^2)*sin(w*t+angle+atan(-w*tau));

```
        case 'ac'
```
```
            %%% AC input of circuit;
```

            response = Vm*sin(w*t+angle);

```
        case 'decay'
```
```
            %%% returns decay envelope
```

            % max voltage of full response

            [peakVolt, peakInd]=max(calcRC(angle,'both'));

```
            peakTime=t(peakInd);
```

            % voltage diff of peak and steady state

            MAXSteady=max(calcRC(angle,'steady'));

```
            dV=peakVolt-MAXSteady;
```

            response = dV*exp(-(t-peakTime)/tau)+MAXSteady;

```
    end
```
```
end
```

1.11 גרף מצבי ריסון

איור 13: ארבעת סוגי הריסון

```
%%%%%%%%%%%%%%%%%%%%%%
```
```
%%% Damping ratios %%%
```
```
%%%%%%%%%%%%%%%%%%%%%%
```
```
figure('Name','Damping');
```
```
hold on;
```

% iterate thru a few different resistor values

```
Rs=[1000,5000,20e3,100e3,1e6];
```
```
Rs_len = length(Rs);
```
```
h = zeros(1,Rs_len);
```
```
legend_text = cell(1,Rs_len);
```
```
t=linspace(0,0.005,10000);
```
```
for i=1:Rs_len
```
```
    R=Rs(i); % Current resitor in Ω
```

    vc=solve_ode(R); % solve current ODE

```
    vct=double(vc(t)); % fill in values
```

    h(i)=plot(t,vct,'LineWidth',2); % plot current solution

    % create description for each resistor type

```
    if R < 5e3
```
```
        info = '- overdamped';
```
```
    elseif R == 5e3
```
```
        info = '- critically damped';
```
```
    elseif R >= 1e6
```
```
        info = '~ undamped (lossless)';
```
```
    else
```
```
        info = '- underdamped';
```
```
    end
```

    legend_text{i}=['R=',num2str(R),' \Omega ',info];

    legend(h(1:i),legend_text{1:i}); % update legend

```
    drawnow; % view plots during run
```
```
end
```
```
% signage
```
```
xlabel('Time [S]');
```
```
ylabel('V_C [V]');
```
```
title('Different damping ratios');
```
```
grid on;
```
```
set(gcf,'Position',[100,100,800,600])
```
```
 
```
```
% solve second order ode of type
```
```
% y''+2α*y'+w^2*y=w^2*V
```
```
% y(0)=V0
```
```
% y'(0)=-1/C*(V0/R+I0);
```
```
function solution = solve_ode(R)
```
```
    C=0.01e-6; % Capacitor in F
```
```
    L=1; % Inductor in H
```
```
 
```
```
    tau=R*C; % Time constant
```
```
    alpha=1/(2*tau); % Damping factor
```
```
    w0=1/sqrt(L*C); % natural frequency
```
```
 
```
```
    V = 10; % Input voltage
```
```
    V0=5; % Initial capacitor voltage
```
```
    I0=0; % Initial inductor current
```
```
 
```

    dVc0=-1/C*(V0/R+I0); % Derivate of capacitor's voltage at t=0

```
 
```
```
    % Based on
```

    % https://www.mathworks.com/help/symbolic/solve-a-single-differential-equation.html

```
    syms y(t)
```
```
    Dy = diff(y);
```

    ode = diff(y,t,2)+2*alpha*Dy+w0^2*y == w0^2*V;

```
    cond1 = y(0) == V0;
```
```
    cond2 = Dy(0) == dVc0;
```
```
    conds = [cond1, cond2];
```
```
    ySol(t) = dsolve(ode, conds);
```
```
 
```
```
    solution = simplify(ySol);
```
```
end
```

1.12 ניתוח נתונים מאורקד

איור 14: חישוב ערכי סימולציית אורקד באמצעות מטלאב

```
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
```
```
% Analyze rectifier data from orcad %
```
```
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
```
```
csv = load('orcaddata.csv');
```
```
% get timestamps
```
```
time = csv(:,1);
```
```
% get voltage of resistor
```
```
vr = csv(:,2);
```
```
% maximum voltage
```
```
Vm = max(vr);
```
```
% calculate average
```
```
VDC = mean(vr);
```
```
% calculate rms
```
```
VDCRMS = rms(vr);
```
```
% calculate form factor
```
```
FF = VDCRMS/VDC;
```
```
% calculate ripple factor
```
```
RF = sqrt(FF^2-1);
```
```
% plot all data
```
```
figure('Name','Rectifier');
```
```
total_x=[0 max(time)];
```

plot(time,vr,total_x,[Vm Vm],total_x,[VDC VDC],0,0,0,0,0,0);

```
txtVm=['V_m=',num2str(Vm),'V'];
```
```
txtVDC=['V_{DC}=',num2str(VDC),'V'];
```

txtVDCRMS=['V_{DCRMS}=',num2str(VDCRMS),'V'];

```
txtFF=['Form Factor=',num2str(FF)];
```
```
txtRF=['Ripple Factor=',num2str(RF)];
```

legend('V_R',txtVm,txtVDC,txtVDCRMS,txtFF,txtRF);

```
xlabel('Time [s]');
```
```
ylabel('Amplitude [V]');
```
```
ax=gca;
```
```
ax.YLim=[min(vr)-1, max(vr)+1];
```

1.13 שליפת הפרש-מופע מעקומת ליסאז'ו

איור 15: סימוני עקומת ליסאז'ו המדגימים כיצד לשלוף את הפרמטרים המאפשרים את חישוב הפרש-המופע בין שני גלים

```
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
```
```
%%% Plot of Lissajous curve with  %%%
```
```
%%%   markes to extract the phase %%%
```
```
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
```

t = linspace(0,2*pi,1000); % time between 0 and 2*pi

```
f = 1; % frequency in Hz
```
```
V1 = 4; % amplitude of first wave
```
```
wave1 = V1*sin(2*pi*f*t); % a sine wave
```

phi = pi/4; % phase difference of 45 degrees

```
V2 = 30; % amplitude of second wave
```

wave2 = V2*sin(2*pi*f*t+phi); % same sine wave but with a phase

```
 
```
```
figure('Name','Lissajous');
```
```
hold on;
```

plot(wave1, wave2, 'k','LineWidth',3); % plot the graphs

ax = gca; % Get handle of Current Axis (1)

ax.YLim = ax.YLim*1.25; % increase Y scale

ax.XLim = ax.XLim*1.25; % increase X scale

```
 
```

% calculate the contact points on the Lissajous curve

```
a1 = [0, abs(V2*sin(phi))]; % top a
```
```
a2 = [0, -abs(V2*sin(phi))]; % bottom a
```
```
b1 = [V1*cos(phi), abs(V2)]; % top b
```

b2 = [-V1*cos(phi), -abs(V2)]; % bottom b

```
 
```
```
% write the names of the points
```
```
gap_x = V1*0.1;
```
```
gap_y = V2*0.1;
```

text(a1(1)+gap_x, a1(2), 'a1', 'FontSize', 14, 'Color', 'r');

text(a2(1)+gap_x, a2(2), 'a2', 'FontSize', 14, 'Color', 'r');

text(b1(1), b1(2)+gap_y, 'b1', 'FontSize', 14, 'Color', 'b');

text(b2(1), b2(2)-gap_y, 'b2', 'FontSize', 14, 'Color', 'b');

```
 
```

% plot horizontal lines for the contact points

a1x=[-V1/2, a1(1)]; a1y=[a1(2), a1(2)]; % top middle

plot(a1x, a1y, 'r--', 'LineWidth',2); % plot line

a2x=[-V1/2, a2(1)]; a2y=[a2(2), a2(2)]; % bottom middle

plot(a2x, a2y, 'r--', 'LineWidth',2); % plot line

b1x=[0, b1(1)+gap_x]; b1y=[b1(2), b1(2)]; % top middle

plot(b1x, b1y, 'b--', 'LineWidth',2); % plot line

b2x=[b2(1), b1(1)+gap_x]; b2y=[b2(2), b2(2)]; % bottom middle

plot(b2x, b2y, 'b--', 'LineWidth',2); % plot line

```
 
```

% plot a vertical line between a1 and a2

annotation('doublearrow',x_to_norm_v2(-V1/3, -V1/3),y_to_norm_v2(a1(2), a2(2)),'Color','r','LineWidth',2);

% plot a vertical line between b1 and b2

annotation('doublearrow',x_to_norm_v2(b1(1), b1(1)),y_to_norm_v2(b1(2), b2(2)),'Color','b','LineWidth',2);

```
 
```
```
% write the names of the vertical lines
```

text(-V1/3+gap_x, a1(2)/2, 'a', 'FontSize', 14, 'Color', 'r');

text(b1(1)+gap_x, -V2/2, 'b', 'FontSize', 14, 'Color', 'b');

```
 
```
```
% turn on grid and draw X, Y axes
```
```
grid on;
```
```
gray=0.4*ones(1,3);
```

annotation('arrow',x_to_norm_v2(ax.XLim(1), ax.XLim(2)),y_to_norm_v2(0, 0),'Color',gray,'LineWidth',2);

annotation('arrow',x_to_norm_v2(0, 0),y_to_norm_v2(ax.YLim(1), ax.YLim(2)),'Color',gray,'LineWidth',2);

text(0-gap_x,V2+gap_y,'y', 'FontSize', 14, 'Color', gray);

text(V1+gap_x,0-gap_y,'x', 'FontSize', 14, 'Color', gray);

```
 
```
```
% draw green circles as the points
```
```
for dots={a1,a2,b1,b2}
```
```
    dot=dots{1};
```

    plot([dot(1), dot(1)], [dot(2), dot(2)], 'o', 'Color', 'g', 'MarkerSize',10, 'MarkerFaceColor','g');

```
end
```

2 ג'אווה-סקריפט

2.1 מסנן מעביר גבוהים או נמוכים

איור 16: אופיין של מסנן מעביר נמוכים

```
var rand = {random};
```
```
 
```
```
// high pass = 1, low pass = 0
```
```
var high_pass = Math.round(rand);
```
```
// grid frequency (MUST BE ROUND)
```

var grid_frequency = Math.round(6*rand+4);

```
// maximum voltage
```

var volt_size = Math.round((10*rand+2)*100)/100;

// Maximum frequency for sweep is between 100 kHz and 10 MHz

var max_frequency = Math.pow(10,Math.round(3*rand+5));

```
 
```
```
// frequency of internal sine wave
```
```
var frequency = 30;
```

// the frequency in the middle of the sweep

var middle_frequency = max_frequency / 2;

```
// Resistor size in Ohm
```
```
var R = 160;
```
```
// Capcitor size in Farad
```
```
var C = 1e-6;
```
```
 
```

// define a canvas dimensions along with grid lines

```
var w=500;
```
```
var h=360;
```
```
var grid_y=grid_frequency;
```
```
var grid_x=grid_frequency*2-1;
```

var hs=h*(grid_y-1)/grid_y; // sub height

```
 
```
```
// create a canvas inside the table
```

var c = document.createElement("canvas");

```
c.setAttribute("width",w);
```
```
c.setAttribute("Height",h);
```

var element = document.getElementById("insertcanvashere_sweep");

```
element.appendChild(c);
```
```
 
```
```
var ctx = c.getContext("2d");
```
```
// set style of text
```
```
ctx.font="20px Georgia";
```
```
 
```
```
ctx.save();
```
```
// set the style of grid lines
```

ctx.setLineDash([1, 3]);/*dashes are 5px and spaces are 3px*/

```
ctx.strokeStyle="blue";
```
```
 
```
```
// dispersion of x grid lines
```
```
var skip_x = w/(grid_frequency*2);
```
```
 
```
```
var y;
```
```
// draw x grid
```
```
var half = Math.floor(grid_y/2);
```
```
// pixels amount of each y gridline
```
```
var skip_h = h/grid_frequency;
```
```
// position in which to right voltage
```

var mark_y = Math.round(rand*(half-2))+1;

```
for (var i=0; i<grid_y; i++) {
```
```
	y=(-half+i)*skip_h+h/2;
```
```
	ctx.beginPath();
```
```
	ctx.moveTo(0,y);
```
```
	ctx.lineTo(w,y);
```
```
	ctx.stroke();
```
```
	// write voltages
```
```
	if (i==half) {
```
```
		ctx.fillText("0 V",w/2,y - 5);
```

		// draw different color for the period

```
		ctx.save();
```
```
		ctx.setLineDash([1, 0]);
```
```
		ctx.strokeStyle="black";
```
```
		ctx.stroke();
```
```
		ctx.restore();
```
```
	}
```
```
	if (i==mark_y) {
```
```
		// calculate voltage at current point
```

		var voltage = (half - mark_y) * skip_h * 2 / hs * volt_size;

		round_volt = Math.round(voltage * 1000) / 1000;

		ctx.fillText(round_volt.toString() + 'V',w/2,y - 5);

		// draw different color for the period

```
		ctx.save();
```
```
		ctx.setLineDash([10, 6]);
```
```
		ctx.strokeStyle="black";
```
```
		ctx.stroke();
```
```
		ctx.restore();
```
```
	}
```
```
}
```
```
min_y=y;
```
```
 
```
```
// draw y grid
```

var period=middle_frequency / grid_frequency;

```
var total_period=0;
```
```
var x_counter=0;
```
```
for (var x=skip_x; x<w; x+=skip_x) {
```
```
	ctx.beginPath();
```
```
	ctx.moveTo(x,0);
```
```
	ctx.lineTo(x,h);
```
```
	ctx.stroke();
```
```
	// write time period
```
```
	total_period+=period;
```
```
	x_counter++;
```

	if ((x_counter==2) || (x_counter==grid_x-2)) {

		var round_period = Math.round(total_period * 10) / 10;

```
		var units = 'Hz';
```
```
		if (x_counter != 2) {
```
```
			round_period = total_period / 1000; 
```

			round_period = Math.round(round_period * 10) / 10;

```
			units = 'kHz';
```
```
		}
```

		ctx.fillText(round_period.toString() + units,x,h - 10);

		// draw different color for the period

```
		ctx.save();
```
```
		ctx.setLineDash([10, 6]);
```
```
		ctx.strokeStyle="black";
```
```
		ctx.stroke();
```
```
		ctx.restore();
```
```
	}
```
```
}
```
```
ctx.restore();
```
```
 
```
```
// choose LPF
```
```
var filter_function='1/(omega*R*C+1)';
```
```
if (high_pass == 1) {
```
```
	// choose HPF
```

	filter_function='omega*R*C/(omega*R*C+1)';

```
}
```
```
// draw a positive RC response graph
```
```
ctx.beginPath();
```
```
ctx.moveTo(0,h/2);
```

var max_frequency_log=Math.log10(max_frequency);

```
for (var x=1; x<w; x++) {
```
```
	// calculate the logarithmic frequency
```

	current_frequency=Math.pow(10,x/w*max_frequency_log);

	var omega = 2 * Math.PI * current_frequency;

```
	y=eval(filter_function);
```
```
	y=h/2-y*hs/2;
```
```
	ctx.lineTo(x,y);
```
```
}
```
```
ctx.strokeStyle="red";
```
```
ctx.stroke();
```
```
// draw a negative RC response graph
```
```
ctx.beginPath();
```
```
ctx.moveTo(0,h/2);
```
```
for (var x=1; x<w; x++) {
```
```
	// calculate the logarithmic frequency
```

	current_frequency=Math.pow(10,x/w*max_frequency_log);

	var omega = 2 * Math.PI * current_frequency;

```
	y=eval(filter_function);
```
```
	y=y*hs/2+h/2;
```
```
	ctx.lineTo(x,y);
```
```
}
```
```
ctx.strokeStyle="red";
```
```
ctx.stroke();
```

// draw a sine wave inside the RC response

```
ctx.beginPath();
```
```
ctx.moveTo(0,h/2);
```
```
for (var x=1; x<w; x++) {
```

	current_frequency=Math.pow(10,x/w*max_frequency_log);

	var omega = 2 * Math.PI * current_frequency;

```
	var amplitude=eval(filter_function);
```

	y=amplitude*hs/2*Math.sin(2*Math.PI*frequency*x/w)+h/2;

```
	ctx.lineTo(x,y);
```
```
}
```
```
ctx.strokeStyle="green";
```
```
ctx.stroke();
```

2.2 ניתוח גל סינוס - הוצאת תדר

איור 17: גל סינוסואידלי לשליפת התדר

```
var total_time = {total_time};
```
```
var volt_size = {volt_size};
```
```
var frequency = {frequency};
```
```
 
```
```
var w=500;
```
```
var h=360;
```
```
var grid_y=7;
```
```
var grid_x=frequency*2-1;
```

var hs=h*(grid_y-1)/grid_y; // sub height

```
 
```

var c = document.createElement("canvas");

```
c.setAttribute("width",w);
```
```
c.setAttribute("Height",h);
```
```
 
```

var element = document.getElementById("insertcanvashere_freq");

```
element.appendChild(c);
```
```
 
```
```
var ctx = c.getContext("2d");
```
```
ctx.font="20px Georgia";
```
```
 
```
```
ctx.save();
```

ctx.setLineDash([1, 3]);/*dashes are 5px and spaces are 3px*/

```
ctx.strokeStyle="blue";
```
```
 
```
```
var skip_x = w/(frequency*2);
```
```
 
```
```
var y, max_y;
```
```
// draw x grid
```
```
var half = Math.floor(grid_y/2);
```
```
var skip_y = volt_size / half;
```
```
var voltage = 1;
```
```
for (var i=0; i<grid_y; i++) {
```
```
	y=(-half+i)*(h/grid_y)+h/2;
```
```
	if (i==0) {
```
```
		max_y=y;
```
```
	}
```
```
	ctx.beginPath();
```
```
	ctx.moveTo(0,y);
```
```
	ctx.lineTo(w,y);
```
```
	ctx.stroke();
```
```
	// write voltages
```
```
	if (i==half-1) {
```

		//ctx.fillText(skip_y.toString() + 'V',skip_x * 2,y - 5);

```
	}
```
```
}
```
```
min_y=y;
```
```
 
```
```
// draw y grid
```
```
var period=total_time / frequency;
```
```
var total_period=0;
```
```
var half = Math.ceil(grid_x/2);
```
```
for (var x=skip_x; x<w; x+=skip_x) {
```
```
	ctx.beginPath();
```
```
	ctx.moveTo(x,0);
```
```
	ctx.lineTo(x,h);
```
```
	ctx.stroke();
```
```
	// write time period
```
```
	total_period+=period;
```
```
	half--;
```
```
	if (half==0) {
```

		round_period = Math.round(total_period * 1000) / 1000;

		ctx.fillText(round_period.toString() + 'ms',x,min_y + 10);

		// draw different color for the period

```
		ctx.save();
```
```
		ctx.setLineDash([10, 6]);
```
```
		ctx.strokeStyle="black";
```
```
		ctx.stroke();
```
```
		ctx.restore();
```
```
	}
```
```
}
```
```
ctx.restore();
```
```
 
```
```
// draw a sine wave
```
```
ctx.beginPath();
```
```
ctx.moveTo(0,h/2);
```
```
for (var x=1; x<w; x++) {
```

	y=hs/2*Math.sin(2*Math.PI*frequency*x/w)+h/2;

```
	ctx.lineTo(x,y);
```
```
}
```
```
ctx.strokeStyle="red";
```
```
ctx.stroke();
```

2.3 ניתוח גל סינוס - מתח אפקטיבי

איור 18: גל סינוסואידלי לשליפת המתח

```
var total_time = {total_time};
```
```
var frequency = {frequency};
```
```
var volt_size = {volt_size};
```
```
 
```
```
var w=500;
```
```
var h=360;
```
```
var grid_y=Math.round(frequency+4);
```
```
var grid_x=frequency*2-1;
```

var hs=h*(grid_y-1)/grid_y; // sub height

```
 
```

var c = document.createElement("canvas");

```
c.setAttribute("width",w);
```
```
c.setAttribute("Height",h);
```
```
 
```

var element = document.getElementById("insertcanvashere_volt");

```
element.appendChild(c);
```
```
 
```
```
var ctx = c.getContext("2d");
```
```
ctx.font="20px Georgia";
```
```
 
```
```
ctx.save();
```

ctx.setLineDash([1, 3]);/*dashes are 5px and spaces are 3px*/

```
ctx.strokeStyle="blue";
```
```
 
```
```
var skip_x = w/(frequency*2);
```
```
 
```
```
var y, max_y;
```
```
// draw x grid
```
```
var half = Math.floor(grid_y/2);
```

var skip_y = volt_size / ((grid_y-1)/2);

```
var voltage = 1;
```
```
for (var i=0; i<grid_y; i++) {
```
```
	y=(-half+i)*(h/grid_y)+h/2;
```
```
	if (i==0) {
```
```
		max_y=y;
```
```
	}
```
```
	ctx.beginPath();
```
```
	ctx.moveTo(0,y);
```
```
	ctx.lineTo(w,y);
```
```
	ctx.stroke();
```
```
	// write voltages
```
```
	if (i==half) {
```
```
		ctx.fillText('0',w/2,y - 5);
```

		// draw different color for the period

```
		ctx.save();
```
```
		ctx.setLineDash([1, 0]);
```
```
		ctx.strokeStyle="black";
```
```
		ctx.stroke();
```
```
		ctx.restore();
```
```
	}
```
```
	if (i==half-1) {
```

		round_volt = Math.round(skip_y * 1000) / 1000;

		ctx.fillText(round_volt.toString() + 'V',w/2,y - 5);

		// draw different color for the period

```
		ctx.save();
```
```
		ctx.setLineDash([10, 6]);
```
```
		ctx.strokeStyle="black";
```
```
		ctx.stroke();
```
```
		ctx.restore();
```
```
	}
```
```
}
```
```
min_y=y;
```
```
 
```
```
// draw y grid
```
```
var period=total_time / frequency;
```
```
var total_period=0;
```
```
var half = Math.ceil(grid_x/2);
```
```
for (var x=skip_x; x<w; x+=skip_x) {
```
```
	ctx.beginPath();
```
```
	ctx.moveTo(x,0);
```
```
	ctx.lineTo(x,h);
```
```
	ctx.stroke();
```
```
	// write time period
```
```
	total_period+=period;
```
```
	half--;
```
```
	if (half==1000) {
```

		round_period = Math.round(total_period * 1000) / 1000;

		ctx.fillText(round_period.toString() + 'ms',x,min_y + 10);

		// draw different color for the period

```
		ctx.save();
```
```
		ctx.setLineDash([10, 6]);
```
```
		ctx.strokeStyle="black";
```
```
		ctx.stroke();
```
```
		ctx.restore();
```
```
	}
```
```
}
```
```
ctx.restore();
```
```
 
```
```
// draw a sine wave
```
```
ctx.beginPath();
```
```
ctx.moveTo(0,h/2);
```
```
for (var x=1; x<w; x++) {
```

	y=hs/2*Math.sin(2*Math.PI*frequency*x/w)+h/2;

```
	ctx.lineTo(x,y);
```
```
}
```
```
ctx.strokeStyle="red";
```
```
ctx.stroke();
```

3 מאקסימה

3.1 שליפת תדר בסריקה ליניארית

איור 19: סריקת תדרים

/* CAS Maxima to be used in moodle as a STACK question type */

/* Create a random frequency response of an RC circuit - either HPF or LPF */

```
rnd : rand(1.0);
```
```
/* high pass = 1, low pass = 0 */
```
```
high_pass : round(rnd);
```

is_high_pass: if high_pass = 1 then true else false;

f_cutoff: rand(41)+10; /* cutoff between 10 and 50 Hz */

```
/* time constant */
```
```
tau: 1/(2*pi*f_cutoff);
```
```
/* create omega RC (depends on f) */
```
```
wrc: 2*pi*f*tau;
```
```
/* Get the denominator */
```
```
denom: if high_pass=0 then 1 else wrc;
```

/* maximum voltage between 2 to 12 in steps of 0.5 */

```
volt_size : round((10*rnd+2)*2)/2;
```

/* absolute value of filter's output voltage */

v_out: volt_size*abs(denom/sqrt(1+wrc^2));

```
 
```
```
/* cutoff voltage */
```
```
v_cutoff: volt_size / sqrt(2);
```
```
 
```

f0: rand(10)+1; /* start frequency between 1 and 10 */

fT: rand(31)+70; /* stop frequency between 70 and 100 */

T: rand(6)+5; /* total time of frequency sweep between 5 and 10 seconds*/

tC: (f_cutoff-f0)*T/(fT-f0); /* cursor time position */

```
 
```
```
/* calculate f according to time */
```
```
f: (fT-f0)*t/T+f0;
```
```
 
```
```
/* do the plot */
```

p: plot(v_out, [t, 0, T], [ylabel, "Voltage [V]"], [xlabel, "Timing [S]"], grid2d, [y, 0, volt_size], [xtics, tC, T, T], [ytics, 0, v_cutoff, v_cutoff], [label, ["0", 0, 0], [string(T), T, 0]]);

3.2 תדר תהודה

איור 20: תגובה לסריקת תדרים של מעגל RLC

/* CAS Maxima to be used in moodle as a STACK question type */

/* Create a random frequency response of a parallel RLC circuit */

Q: (rand(20)+6)/2; /* Q between 3 and 12.5 */

C: 0.01e-6; /* constant capacitor of 0.01uF */

w0: (rand(21)+10)*500 /* natural frequency between 5k and 15k */

```
L: 1/(C*w0^2) /* w0=1/sqrt(L*C) */
```

R: Q*sqrt(L/C); /* calculate R from Q */

```
alpha: 1/(2*R*C); /* damping ratio */
```

Vin: (rand(4)+1)/2; /* Input voltage between 0.5 and 2 */

```
 
```
```
 
```

Vout: Vin*R/sqrt((R-w^2*R*L*C)^2+(w*L)^2);

```
 
```
```
 
```

p: plot(Vout, [w, 0, w0+5*alpha], grid2d, [xlabel, "Angular frequency [rad/sec]"],[ylabel, "Voltage [V]"] );

@@ שורה 1,090: / שורה 1,090: @@
 /* calculate f according to time */
 f: (fT-f0)*t/T+f0;
-/* create a custom x axes with time instead of frequency */
-custom_x: [label, [string(f0),0,0], [string(float(round(tC*100)/100)), f_cutoff, 0], [string(T), fT, 0]];
 /* do the plot */
-p: plot(v_out, [t, 0, T], [ylabel, "Voltage [V]"], [xlabel, "Timing [S]"], grid2d, [y, 0, volt_size], [xtics, tC, 2*T, 2*T], [ytics, 0, v_cutoff, v_cutoff], [label, ["0", 0, 0], [string(T), T, 0]]);
+p: plot(v_out, [t, 0, T], [ylabel, "Voltage [V]"], [xlabel, "Timing [S]"], grid2d, [y, 0, volt_size], [xtics, tC, T, T], [ytics, 0, v_cutoff, v_cutoff], [label, ["0", 0, 0], [string(T), T, 0]]);
 </syntaxhighlight>

גרפים: הבדלים בין גרסאות