# Find current in a nontrivial circuit

Let us assume that we have this circuit: We know that:

$$I_1 = I_2 + I_3$$ $$I_1R_1 + I_3R_3 = V$$ $$I_1R_1 + I_2R_2 + \dfrac{q_c}{C_1} = V$$

Therefore we can write:

$$I_1 = k_1 - k_2*q_c$$

with:

$$k_1 = \dfrac{V(1/R_2+1/R_3)}{1+R_1/R_2+R_1/R_3}$$ $$k_2 = (R_2*C_1(1+R_1/R_2+R_1/R_3))^(-1)$$

The question is, Can we write the equation:

$$\dfrac{dq(t)}{dt}={k_1-k_2q(t)}$$

I doubt if $q_c=q(t)$ because $I_3$ does not flow through the branch which contains the capacitor, therefore:

$$I_1 = I_2 + I_3 => q_1 = q_2 + q_3$$

I think $q(t)=q_1$ But if it is so $q_2 = q_c \neq q(t)$. How can I write down the differential equation in order to find $q(t)$?

How can I write down the differential equation in order to find q(t)?

There are generally several ways to solve a circuit but some may be far more straightforward than others. The most straightforward approach to find $q_C(t)$ in this case is to transform this circuit into a simple series RC circuit.

By finding the Thevenin equivalent circuit seen from the capacitor terminals, we get by inspection:

$$V_{TH} = V\dfrac{R_3}{R_1 + R_3}$$

$$R_{TH} = R_2 + R_1||R_3$$

Since we now have a simple series RC circuit, the solution for the capacitor current is well known:

$$i_C(t) = i_C(0)e^{-t/\tau}$$

$$\tau = R_{TH}C_1$$

The solution for the capacitor charge follows by integration:

$$q_C(t) = \tau i_C(0)(1 - e^{-t/\tau}) + q_C(0) = C_1V_{TH}(1 - e^{-t/\tau}) + q_C(0)e^{-t/\tau}$$

• I was not familiar with notation $R_1||R_3$, it means "the equivalent resistance for a composed set of $R_1$ and $R_3$ in parallel $(||)$ way". However, You solve $q_c(t)$ and I'm asking for $q_1(t)$, or, are these functions the same? – Phyllipe Oct 21 '13 at 22:23
• @Phyllipe, they are not the same. You're not familiar with the notation for parallel resistances? It is: $R_1||R_3 = \dfrac{1}{\frac{1}{R_1} + \frac{1}{R_3}}$ – Alfred Centauri Oct 21 '13 at 22:30
• I need $I_1(t)$ so in order to find it I asked for $q(t)$ (If I derive $q(t)$ respect to time I'll be able to find $I_1(t)$). However, you find $q_c(t)$ and I dunno how can I get $q(t)$. I was not familiar... – Phyllipe Oct 21 '13 at 22:37
• @Phyllipe, now that you have $i_C(t)$, isn't is clear how to proceed to find $i_1(t)$? – Alfred Centauri Oct 21 '13 at 23:29
• No, it is not clear to me. – Phyllipe Oct 27 '13 at 19:38