Microwave passive components – Equivalent circuit of a physical resistor when its physical length l is much smaller than the wavelength λ (l << λ) In the lumped-element region (electrically short body and leads), what is the most appropriate small-signal equivalent circuit for a practical microwave resistor?

Introduction:
Real resistors at microwave frequencies are physical structures with finite lead lengths and electrode areas. However, when the physical length l is much smaller than the operating wavelength λ, the component behaves as an electrically short element. This question tests whether you recognize that, under the l << λ condition, the resistor is well approximated by a lumped resistance R, with any parasitic inductance (from leads) and capacitance (from pads/body) being comparatively small.

Given Data / Assumptions:

• Operating frequency such that l << λ (electrically short).
• Linear, small-signal operation about the bias point.
• Physical resistor realized as a film or chip with short interconnects.

Concept / Approach:

The lumped-element model applies when phase variation across the device is negligible. Under l << λ, distributed effects (standing waves along the body) do not develop, and the element can be modeled by its lowest-order parameters. The dominant parameter of a resistor is its resistance R. Parasitic series inductance Ls and shunt capacitance Cp exist but, for sufficiently small l/λ, they have second-order influence and can often be ignored in first-pass analysis, especially for narrowband designs or moderate Q networks.

Step-by-Step Solution:

Verification / Alternative check:

EM or circuit simulations show that as frequency lowers (or size shrinks), the impedance approaches a constant real value R; reactance magnitudes diminish toward zero, validating the ideal-R approximation.

Why Other Options Are Wrong:

Option B/C: A real resistor cannot be purely inductive or capacitive. Option D: A λ/4 line is a distributed element, contradicting l << λ. Option E: Negative resistance describes active or special tunneling devices, not passive resistors.

Common Pitfalls:

Over-fitting parasitics in the lumped region; confusing the high-frequency distributed regime (where parasitics matter) with the explicitly stated l << λ case.

Final Answer:

An ideal lumped resistance R (dominant), with negligible parasitic inductance and capacitance.