Welcome to RC circuit analysis. An RC circuit is a fundamental electrical circuit containing a resistor and capacitor. The resistor opposes current flow, while the capacitor stores electrical energy. Understanding how voltage and current change over time in these circuits is essential for electrical engineering.
To solve an RC circuit, we start by applying Kirchhoff's Voltage Law. This fundamental law states that the sum of voltages around any closed loop equals zero. For our RC circuit, the source voltage equals the sum of the resistor voltage and capacitor voltage. The resistor voltage equals current times resistance, while the current through the capacitor equals capacitance times the rate of change of capacitor voltage.
By substituting the component equations into Kirchhoff's voltage law, we obtain a first-order differential equation. After rearranging, we get the rate of change of capacitor voltage equals V minus V C, all divided by RC. The general solution shows that the capacitor voltage follows an exponential charging curve, approaching the source voltage asymptotically. The time constant tau equals RC and determines how quickly the capacitor charges.
Initial conditions are crucial for determining the specific solution. At time zero, the capacitor is uncharged so its voltage is zero, while the current is at maximum value V over R. The current equation shows that current equals capacitance times the rate of change of capacitor voltage. This gives us an exponentially decaying current. As we can see, the current starts high and decays while the voltage rises exponentially toward the source voltage.
To summarize RC circuit analysis: First, apply Kirchhoff's voltage law to establish the governing differential equation. Second, solve for the capacitor voltage which follows an exponential charging curve. Third, the current decays exponentially from its initial maximum value. The time constant RC determines how quickly the circuit reaches steady state, and initial conditions specify the particular solution for any given scenario.