Study Notes on Kirchhoff's Voltage Law and Loop Analysis

Kirchhoff's Voltage Law (KVL)

• Definition: Kirchhoff's Voltage Law states that if you traverse any closed loop in a circuit, the sum of all the voltages around that loop equals zero.

  • This holds true for both the time domain and phasor domain expressions.

• Polarity Scheme: When applying KVL, a consistent polarity scheme must be established for counting voltages as positive or negative.

  • One example of such a scheme is as follows:

  • Positive Voltage Counting: If you encounter a positive terminal first as you traverse through a circuit element (moving from positive to negative), count it as a positive voltage.

  • Negative Voltage Counting: Conversely, if you enter through the negative terminal and leave through the positive terminal, count that as a negative voltage.

  • It is crucial to remain consistent throughout the analysis; switching conventions mid-calculation is not permitted.

Example of KVL Application

• Circuit Configuration: Consider a basic circuit consisting of a voltage source, an inductor, and a resistor.

  • Direction of Traversal: The clockwise direction is selected to match the direction of current flow.

• Traversal Steps:

  1. Starting from the bottom left at the voltage source:

  • Entering on the negative side and leaving on the positive side results in: Voltage Contribution: $-V_s$.

  1. Next, moving to the inductor:

  • Entering on the positive side and leaving on the negative side yields: Voltage Contribution: $+V_x$.

  1. Finally, passing through the resistor:

  • Entering on the positive side and leaving on the negative yields: Voltage Contribution: $+V_R$.

    • Upon returning to the starting point:

  • The overall KVL equation becomes: $-V<em>s + V</em>x + V_R = 0$.

• Alternate Convention:

  • Should the opposite polarity scheme be adopted, where entering a terminal is considered a plus:

  • The resulting expression would be: $V<em>s - V</em>x - V_R = 0$.

  • Both representations are valid as long as the chosen convention is consistently applied.

Combining KVL and Ohm's Law

• Expression of Voltages: By integrating Ohm's Law with KVL:

  • The source voltage ($V_s$) equals the sum of the voltage drops across passive components:

  • $V<em>s = V</em>R + V_x$.

• Voltage Divider Principle:

  • In the context of a series configuration, the voltage divider equation is made possible:

  • Referring to the voltage drop across the k-th element in a series circuit with a total voltage $V$:

  • $V<em>k = V rac{Z</em>k}{ ext{Z_total}}$,

  • where $Zk$ is the impedance of the k-th element and $ ext{Ztotal}$ is the sum of impedances in the series.

Systematic Procedure: Loop Analysis

• Just like Nodal Analysis, Loop Analysis can be employed as a systematic approach to solve circuits by applying KVL and Ohm's Law.

• Steps of Loop Analysis:

  1. Identification and Labeling: Identify loops in the circuit and label them with loop (mesh) currents.

  2. Writing KVL Expressions: Write KVL equations for each loop based on the loop currents designated in step one.

  3. Solving the Equations: Solve the system of KVL equations to find the unknown loop currents.

• Distinct Example: Using the same circuit example, apply Loop Analysis as follows:

  • Step 1: Identify two meshes and label their currents (e.g., $I{L1}, I{L2}$).

  • Step 2: Apply KVL to the left loop:

  • KVL expression becomes: $V<em>1 - I</em>{L1}(Z<em>{R1} + Z</em>{X1}) - I<em>3 imes Z</em>C = 0$,

  • where $I3 = I{L1} + I_{L2}$.

  • For the right loop, the KVL expression similarly reads: $V<em>2 - I</em>{L2} (Z<em>{R2} + Z</em>{X2}) - I<em>3 imes Z</em>C = 0$.

  • Step 3: Formulate the system of equations and solve for the unknown currents, potentially resulting in a complex algebraic system due to the involvement of complex numbers.