AC circuit analysis requires specialized methods different from DC circuits. Unlike DC circuits with constant voltages and currents, AC circuits involve sinusoidal quantities that vary with time. We use phasor representation to convert time-domain sinusoids into complex numbers, making calculations easier. The two primary systematic approaches are nodal analysis, which focuses on finding node voltages, and mesh analysis, which determines loop currents. Both methods work in the phasor domain using impedance concepts.
The phasor domain transforms time-domain sinusoidal quantities into complex numbers, greatly simplifying AC circuit analysis. In this domain, resistors have impedance R, inductors have impedance j omega L, and capacitors have impedance one over j omega C. Ohm's law becomes V equals I times Z, where all quantities are phasors. Kirchhoff's voltage and current laws apply directly in the frequency domain. Phasors can be expressed in rectangular form as a plus j b, or in polar form as magnitude times e to the j phi. This mathematical framework enables systematic analysis of AC circuits.
Nodal analysis is a systematic method for solving AC circuits by finding node voltages. The procedure involves four key steps. First, identify all nodes in the circuit. Second, select a reference node, typically called ground. Third, write Kirchhoff's current law equations at each non-reference node using impedances. Finally, solve the resulting matrix equations to find the unknown node voltages. In this example circuit, we have two unknown node voltages V1 and V2. At node 1, the sum of currents equals zero, giving us the first KCL equation. At node 2, we write another KCL equation. These equations form a system that can be solved using matrix methods to find the phasor voltages.
Mesh analysis provides an alternative systematic approach to solving AC circuits by focusing on mesh currents instead of node voltages. The procedure involves identifying independent meshes, assigning mesh current directions, and writing Kirchhoff's voltage law equations around each mesh using impedances. For the same circuit analyzed previously, we define two mesh currents I1 and I2. Around mesh 1, we write KVL including the voltage source, inductor, and shared resistor. Around mesh 2, we include the shared resistor and capacitor. Mesh analysis often requires fewer equations than nodal analysis, especially for circuits with many nodes but few meshes, making it more efficient in such cases.
For complex AC circuits with multiple sources and various impedance types, the choice between nodal and mesh analysis depends on circuit structure. This example circuit has four unknown node voltages but only three independent meshes, making mesh analysis more efficient. The systematic approach involves writing KVL equations for each mesh, forming a matrix equation with impedance coefficients. Dependent sources require careful handling by expressing their controlling variables in terms of mesh currents. After solving the matrix equation, results should be verified using power balance or an alternative analysis method. The final step involves converting phasor results back to time domain for practical interpretation.