For an RC differentiator (series C, shunt R), what must the instantaneous sum of the capacitor voltage and the resistor voltage equal?

Introduction / Context:
KVL (Kirchhoff's Voltage Law) governs series circuits, including the classic RC differentiator configuration. Recognizing how the individual element voltages relate to the source is fundamental to predicting output behavior during rising and falling edges.

Given Data / Assumptions:

• Series connection: input source → capacitor → resistor → return.
• Output is across R, but analysis concerns instantaneous voltages across C and R.
• Linear time-invariant behavior; no parasitics considered beyond RC.

Concept / Approach:
In any single-loop series circuit, KVL states that the algebraic sum of voltages around the loop is zero. Equivalently, the source voltage equals the sum of the drops across the series elements at every instant in time, regardless of transient or steady-state conditions.

Step-by-Step Solution:

Verification / Alternative check:
SPICE simulations of a differentiator responding to a pulse show that at each time point, v_C + v_R tracks the excitation exactly (subject to numerical precision), confirming KVL in time domain.

Why Other Options Are Wrong:

• Zero: Only true if the source were zero or defined with opposite polarity and summation convention.
• Less than the applied voltage: Violates KVL for a single series loop.
• Cannot be determined: It is precisely determined by KVL.

Common Pitfalls:

• Dropping sign conventions and mixing polarities when summing voltages.
• Assuming energy storage elements 'hide' voltage; KVL still applies instant by instant.

Final Answer:
The applied input voltage