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Homework Statement
Find the currents through ##R_3, i_1(t)## and ##R_2, i_2(t)##.
I attached a diagram of the circuit (diagram.pdf).
In the original diagram the currents I'm trying to find is marked going top to bottom, there's also no directions on the voltages in the original diagram but from the currents I guess they're in the direction drawn
##u_1(t) = 2\sin(200t+30^\circ)##
##u_2(t) = 3 \cos(200t + 20^\circ)##
##R_1 = 100 \Omega##
##R_2=300 \Omega##
##R_3 = 250 \Omega##
##C=10 \mu F##
##L=300mH##
Homework Equations
N/A
The Attempt at a Solution
Lets first simplify at bit by calculating the impedance.
We then have
##Z_1 = R_3+\frac{1}{j\omega C} = 250j500##
##Z_2 = R_2+j\omega L = 300+j60##
We then have the diagram (diagram2.pdf)
We also write the voltages in complex form
##u_1(t) = 2e^{200t+30^\circ}##
##u_2(t) = 3e^{200t+110^\circ}.## (add ##90^\circ## to write the ##\cos## in ##\sin## form.)
Node analysis:
I ground the bottom node then I use KCL on the top Node, let's call it ##A## and it's potential be ##U_A##.
We then get the equation
##\frac{U_Au_1(t)}{R_1} + \frac{U_A}{Z_2}+\frac{U_A+u_2(t)}{Z_1}=0.##
Rearanging
##U_A\left( \frac{1}{R_1} +\frac{1}{Z_2} + \frac{1}{Z_1} \right) = \frac{u_1}{R_1}\frac{u_2(t)}{Z_1}.##
Then ##U_A## can be calculatee and the currents are then
##i_1(t) = \frac{U_A+u_2(t)}{Z_1}## and
##i_2(t) = \frac{U_A}{Z_2}##.
To check that I didn't make any mistakes I calculated this in Matlab
Code:
R=100;
Z1 = 250500i;
Z2 = 300+60i;
u1 = 2*exp(i*pi*30/180);
u2 = 3*exp(i*pi*110/180);
R3 =250;
Zc = 500j;
temp = u1/Ru2/Z2;
UA = temp/(1/R+1/Z2+1/Z1)
I1 = (UA+u2)/Z1
I2 = UA/Z2
A1 = abs(I1) %amplitude of i_1(t)
phi1 = 180/pi*atan(imag(I1)/real(I1)) %phase constant of i_1
A2 = abs(I2)
phi2 = 180/pi*atan(imag(I2)/real(I2))
##UA = 1.3377  0.0697i##
##I1=0.0041 + 0.0027i##
##I2 = 0.0042  0.0011i##
##A1 = 0.0049##
##A2 = 0.0044##
##phi1 = 33##
##phi2 = 14.3##
So the currents should be
##i_1(t) =4.9 \sin (200 t 33^\circ) mA## and
##i_2(t) = 4.4 \sin (200t  14.3^\circ mA##.
However the answer says that it should be
##i_1(t) = 5.13\sin(200t + 15.7^\circ )## mA and
##I_2(t) = 3.73 \sin (200t + 26.2^\circ )##mA.
I've been at this question for two days and I can't see where I'm going wrong.
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