Circuit laws — applicability: Does Kirchhoff’s Voltage Law (KVL) fail to apply to a simple series R–L circuit driven by a source in lumped-element analysis?

Introduction / Context:
Kirchhoff’s Voltage Law (KVL) states that the algebraic sum of voltage drops around any closed loop is zero at every instant. This law is widely used in analyzing RL, RC, and RLC circuits for both transient and steady-state conditions. The claim that KVL does not apply to series RL circuits is a misconception and is corrected here.

Given Data / Assumptions:

• Lumped-element model (component dimensions small vs. wavelength; negligible radiation).
• Ideal components: resistor R, inductor L, and a voltage source.
• Quasi-static field conditions without distributed transmission-line effects.

Concept / Approach:
KVL emerges from electrostatic field conservativeness in lumped circuits and is valid instant-by-instant. In a series RL loop driven by v_s(t), the loop equation is v_s(t) − v_R(t) − v_L(t) = 0, where v_R(t) = i(t) * R and v_L(t) = L * di/dt. This is the standard starting point for deriving the RL differential equation and its step response.

Step-by-Step Solution:

Verification / Alternative check:
Measure with an oscilloscope: the instantaneous source voltage equals the instantaneous sum of resistor and inductor voltages. Simulations (SPICE) also confirm v_s = v_R + v_L for arbitrary inputs (steps, sinusoids, pulses).

Why Other Options Are Wrong:
Limiting KVL to resistive circuits, steady state, or low frequency is incorrect within lumped-element validity. Only when circuits become distributed (high-frequency transmission lines, significant radiation) do we need full Maxwell’s equations instead of simple KVL.

Common Pitfalls:
Confusing instantaneous addition with RMS magnitudes; in AC, phasor magnitudes do not add arithmetically, but the time-domain KVL always holds.

Final Answer:
Incorrect