# 8.6: Initial and Steady-State Analysis of RLC Circuits

When analyzing resistor-inductor-capacitor circuits, remember that capacitor voltage cannot change instantaneously, thus, initially, capacitors behave as a short circuit. Once the capacitor has been charged and is in a steady-state condition, it behaves like an open. This is opposite of the inductor. As we have seen, initially an inductor behaves like an open, but once steady-state is reached, it behaves like a short. For example, in the circuit of Figure 9.4.1 , initially $$L$$ is open and $$C$$ is a short, leaving us with $$R_1$$ and $$R_2$$ in series with the source, $$E$$. At steady-state, $$L$$ shorts out both $$C$$ and $$R_2$$, leaving all of $$E$$ to drop across $$R_1$$. For improved accuracy, replace the inductor with an ideal inductance in series with the corresponding $$R_{coil}$$ value. Similarly, practical capacitors can be thought of as an ideal capacitance in parallel with a very large (leakage) resistance. Figure 9.4.1 : Basic RLC circuit.

##### Example 9.4.1

Assuming the initial current through the inductor is zero and the capacitor is uncharged in the circuit of Figure 9.4.2 , determine the current through the 2 k$$\Omega$$ resistor when power is applied and after the circuit has reached steady-state. Draw each of the equivalent circuits. Figure 9.4.2 : Circuit for Example 9.4.1 .

For the initial-state equivalent we open the inductor and short the capacitor. The new equivalent is shown in Figure 9.4.3 . The shorted capacitor removes everything to its right from the circuit. All that's left is the source and the 2 k$$\Omega$$ resistor. Figure 9.4.3 : Initial-state equivalent of the circuit of Figure 9.4.2 .

We can find the current through the 2 k$$\Omega$$ resistor using Ohm's law.

$I_{2k} = \frac{E}{R} \nonumber$

$I_{2k} = \frac{14 V}{2k \Omega} \nonumber$

$I_{2k} = 7 mA \nonumber$

Steady-state is redrawn in Figure 9.4.4 , using a short in place of the inductor, and an open for the capacitor. We are left with a resistance of 2 k$$\Omega$$ in series with the parallel combination of 1 k$$\Omega$$ and 4 k$$\Omega$$, or 2.8 k$$\Omega$$ in total. Figure 9.4.4 : Steady-state equivalent of the circuit of Figure 9.4.2 .

$I_{2k} = \frac{E}{R} \nonumber$

$I_{2k} = \frac{14 V}{2.8k \Omega} \nonumber$

$I_{2k} = 5mA \nonumber$