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Circuit analysis begins with precise definitions of voltage, current, power, and energy as circuit variables. Ideal circuit elements—resistors, capacitors, inductors, and independent or dependent sources—are mathematical models that approximate real component behavior. The passive sign convention establishes a consistent framework for assigning reference polarities and current directions. Power absorbed by an element equals voltage times current under the passive sign convention; energy is power integrated over time.
Practice assigning reference directions and applying the passive sign convention to multi-element circuits before writing any equations. Work through examples involving both independent and dependent sources, tracking polarity carefully. Draw complete circuit diagrams with all labeled variables as a habit.
Circuit analysis is built on three circuit variables — voltage, current, and power — and a set of idealized component models. Before you can write a single equation, you need to understand what these variables mean and how to keep their signs straight.
Voltage is the potential difference between two nodes — the energy per unit charge that a charge carrier gains or loses moving between them. Current is the rate at which charge flows past a cross-section, measured in amperes. These two quantities are independent: knowing the voltage across a resistor tells you the current (via Ohm's Law), but knowing the voltage across a capacitor tells you only the *rate of change* of current, not the current itself.
The passive sign convention is the bookkeeping rule that makes multi-element circuits tractable. For any element, you define a positive current direction and a positive voltage polarity together: current enters the terminal marked +. Then power absorbed equals P = V × I. If P comes out positive, the element is absorbing power (load behavior). If P is negative, the element is delivering power (source behavior). This single convention applies to every element — resistors, capacitors, inductors, and sources alike — eliminating the need for separate sign rules for each type.
Ideal circuit elements are mathematical abstractions. A resistor satisfies v = iR for all time and cannot store energy. A capacitor satisfies i = C dv/dt and stores energy in its electric field. An inductor satisfies v = L di/dt and stores energy in its magnetic field. Independent sources impose a fixed voltage or current regardless of what the rest of the circuit does. Dependent sources (controlled sources) impose a voltage or current proportional to some other circuit variable, and they appear in transistor models and op-amp circuits. A key point students miss: a voltage source fixes voltage but not current; a current source fixes current but not voltage. The unconstrained variable is determined entirely by the surrounding network.
Getting these fundamentals right — particularly the passive sign convention and the distinction between reference direction and actual direction — is what makes Kirchhoff's Voltage Law and Kirchhoff's Current Law work cleanly. Every node-voltage or mesh-current analysis you do later depends on applying these conventions consistently.