Matlab

Simultaneous Equations

Theory

I1 = I2 + I3

V1 = I1R1 + I3R3
V2 = I2R2 - I3R3

V1 = (I2 + I3)R1 + I3R3
V2 = I2R2 - I3R3

|R1 R1+R3||I2| |V1|
|R2  -R3 ||I3|=|V2|

Computation

I = inv([20 30; 5 -10])*[15; 7]

I =
    1.0286
   -0.1857

I3 = -0.1857A

Plotting Exponentials

1.

tau1=100; tau2=200; t=linspace(0, 5*tau2); circA=2*exp(-t/tau1); circB=2*exp(-t/tau2);
plot(t,circA,t,circB)



Circuit A has faster response.

2.

tau1=100; tau2=200; t=linspace(0, 5*tau2); circA=2*(1-exp(-t/tau1)); circB=2*(1-exp(-t/tau2));
plot(t,circA,t,circB)

Adding Sinusoids

1. Theory



Where b = pi/36, a = pi/12, B = 3, A = 5

Computation

omega=2; amp1=3; amp2=5; phase1=10; phase2=30; t=linspace(0,(4*pi)/omega,500);
conver=(2*pi)/(360*omega);
f1=amp1*sin(omega*t+phase1*conver);
f2=amp2*cos(omega*t+phase2*conver);
fnet = f1+f2;
plot(t,f1,t,f2,t,fnet)



2.

freq=10; omega=2*pi*freq; amp1=3; amp2=5; phase1=10; phase2=30;
t=linspace(0,(4*pi)/omega,500); conver=(2*pi)/(360*omega);
f1=amp1*sin(omega*t+phase1*conver);
f2=amp2*cos(omega*t+phase2*conver);
fnet = f1+f2;
plot(t,f1,t,f2,t,fnet)

Complex Numbers

1. Theory

C = (A1 x B) / A2 = [(3*2-2*(-2)) + j((3+2)(2-2)-(3*2+2*(-2)))] / A2 = (10-j2) / A2
= (10*(-1)+(-2)*4)+j((-2)*(-1)-10*4)) / ((-1)^2+4^2) = -18/17 j(-38/17) = -1.059 - j2.235

Computation

A1=3+2j; A2=-1+4j; B=2-2j;
C=(A1*B)/A2

C =
  -1.0588 -  2.2353i

2.

Polar to Rectangular
rect=polMag*exp(j*polPhase*pi/180)

Rectangular to Polar
polPhase=angle(rect)*180/pi
polMag=abs(rect)

rect =
   3.0000 +  2.0000i
polPhase =
   33.6901
polMag =
    3.6056

rect =
  -1.0000 +  4.0000i
polPhase =
  104.0362
polMag =
    4.1231

rect =
   2.0000 -  2.0000i
polPhase =
 -45
polMag =
    2.8284

rect =
  -1.0588 -  2.2353i
polPhase =
 -115.3462
polMag =
    2.4734

3.1 Theory

D = (A1 + B)*A2 = 5(-1+j4) = -5+j20

Computation

D=(A1+B)*A2

D =
  -5.0000 + 20.0000i

3.2

rect =
  -5.0000 + 20.0000i
polPhase =
  104.0362
polMag =
   20.6155

4.

I=inv([8+8j, 2j; 2j, 4-4j])*[50j; -30j];

rect =
   2.0588 +  2.9412i
   5.0000 -  3.5294i
polPhase =
   55.0080
  -35.2176
polMag =
    3.5902
    6.1202

Roots

1.

p=[1 1 4]; r=roots(p)

r =
  -0.5000 +  1.9365i
  -0.5000 -  1.9365i

p=[1 3 0 3]; r=roots(p)

r =
  -3.2790 +  0.0000i
   0.1395 +  0.9463i
   0.1395 -  0.9463i

p=[1 3 4 2 7]; r=roots(p)

r =
  -1.8222 +  1.2680i
  -1.8222 -  1.2680i
   0.3222 +  1.1474i
   0.3222 -  1.1474i

2.

p=[1 5 7 3]; r=roots(p)

r =
  -3.0000 +  0.0000i
  -1.0000 +  0.0000i
  -1.0000 -  0.0000i

F(s) = (s+7)/[(s+3)(s+1)^2] = C1/(s+3) + C2/(s+1) + C3/(s+1)^2

s+7 = C1(s+1)^2 + C2(s+1)(s+3) + C3(s+3)

0 = C1 + C2
1 = 2C1 + 4C2 + C3
7 = C1 + 3C2 + 3C3

C = inv([1 1 0; 2 4 1; 1 3 3])*[0; 1; 7]

C =
  1
 -1
  3

f(t) = e^(-3t) - e^(-t) + 3e^(-t)