# Differential equation mixed RLC-circuit, C parallel to RL

I am having trouble finding the differential equation of a mixed RLC-circuit, where C is parallel to RL. It is a steady-state sinusoidal AC circuit. I need it to determine the Power Factor explicitly as a function of the components. I know I am supposed to use the KCL or KVL, but I can't seem to derive the correct one.

I am actually a math student, so I apologize if this question is straight forward.

• Are you required to do it via differential equations or can you use frequency response (or Laplace) representations of the components?
– Chu
May 24, 2016 at 0:18
• Unfortunately I have to do it using differential equations. I can use $$u(t) = V_m cos(\omega t), i(t) = I_m cos(\omega t - \phi)$$, etc., or phasor representation. May 24, 2016 at 0:33

The voltage, $v$, across $\small C$ is equal to the the voltage across the $\small R, L$ series combination.

The current in $\small C$ is $i_1=\small C\large \frac{dv}{dt}$

The current in $\small RL$ is related to $v$ by; $v=\small R\large i_2+\small L\large\frac{di_2}{dt}$, where $i_2$ is the current through $\small R$ and $\small L$.

Solve both ODEs for $i_1$ and $i_2$, and the total current in the $\small RLC$ circuit is then $i=i_1 + i_2$.

Let $\small t\rightarrow \infty$ in the real parts of the exponentials to remove any transients, and you now have $i$ and $v$ in the required steady-state sinusoidal forms.

• But current resistor = current inductor? So the total current is $$i(t) = i_c + i_r$$ So why is the current in the RL-circuit related by that equation? I would think it is related by $$R i_2 = L \frac{di_2}{dt}$$ May 24, 2016 at 2:16
• I assume your question means that L and R are in series, and this series combination is in parallel with C. The voltage across a series circuit is the sum of the voltages across each individual element. Try two resistors in series and apply Ohm's Law.
– Chu
May 24, 2016 at 6:50
• Oops yes that was what I meant. After you edited your original post I found the correct total current and PF. Thank you very much! May 26, 2016 at 21:30