AC circuits and Kirchoff's laws

[koroljov]

I was trying to solve an exercise.

We are given an AC circuit (see attachment). We had to find the current trough the resistor, the inductor and the capacitor.

This can be solved using the impedance:
The voltage by the source is sinusoidal: V=V0*sin(w*t)=(complex) V0*exp(j*w*t)
The total (complex) impedance is: Z=R+(j*w*C+1/(j*w*L))^-1
where j=sqrt(-1)

The current trough the resistor is I0*exp(j*(w*t+phi))
Since |Z|=V0/I0:
I0=V0/|Z|
phi=arg(Z)=arctan(w*L/(R*(1-w^2*L*C)))

For the current trough the inductor:
Voltage(inductor)=L*dI/dt
thus
I=1/L*intrgral(V0*exp(j*w*t)-I0*exp(j*(w*t+phi))*R)

For the current trough the capacitor:
voltage(capacitor)=Q/C thus Q=C*voltage(capacitor)=C*(V0*exp(j*w*t)-I0*exp(j*(w*t+phi))*R)
and
I=dQ/dt
thus
I=d(C*(V0*exp(j*w*t)-I0*exp(j*(w*t+phi))*R))/dt

Now comes the problem. Kirchoff's laws should apply at any instant of time. Thus, using the junction rule, the algebraic sum of the three currents should be zero. (the current in the resistor + the one in the capacitor + the one in the inductor).
However, using the results obtained above, this is not true.
Can someone shed some light on the subject?

 

[koroljov]

I have managed to solve the problem. More calculations convinced me that the sum is, in fact, zero.

 

[quicknote]

Hello,

Thank you for sharing your exercise and your approach to solving it. Kirchoff's laws are fundamental principles in circuit analysis and are essential for understanding the behavior of AC circuits. Let's take a closer look at your solution and address the issue you have encountered.

Firstly, your calculation of the total impedance is correct. However, when calculating the current through the resistor, inductor, and capacitor, you have assumed that the voltage across each element is the same as the source voltage. This is not always the case in AC circuits, as the voltage across each element can vary depending on the frequency and phase of the source voltage.

In order to accurately calculate the currents through each element, you need to use the voltage divider rule. This rule states that the voltage across an element is equal to the source voltage multiplied by the ratio of the element's impedance to the total impedance of the circuit. This means that the voltage across the resistor is V0*R/(R+Z), the voltage across the inductor is V0*j*w*L/(R+Z), and the voltage across the capacitor is V0/(j*w*C+1/Z). Once you have calculated the voltages across each element, you can then use Ohm's law (V=IR) to calculate the currents.

Now, let's address the issue with Kirchoff's laws. The junction rule states that the algebraic sum of the currents entering and exiting a junction in a circuit should be equal to zero. In your solution, you have correctly calculated the currents through each element, but you have not taken into account the direction of the currents. The currents through the inductor and capacitor are in opposite directions to the current through the resistor, which is why the sum is not equal to zero. However, if you assign a direction to each current and use the correct signs in your calculation, you will find that Kirchoff's laws are satisfied.

In summary, to accurately solve AC circuits using Kirchoff's laws, it is important to consider the voltage divider rule and the direction of the currents in the circuit. I hope this helps to shed some light on the subject. Keep up the good work in your studies of circuit analysis!