Current-source stiffness (Norton model intuition): A current source delivers an ideal 10 mA and has an internal resistance of 20 MΩ in parallel. For which load resistance values will the source appear “stiff” (i.e., load current remains very close to 10 mA)?

Introduction / Context:
“Stiffness” for a current source means the delivered load current stays close to the ideal source current regardless of reasonable load changes. A Norton current source has an internal resistance in parallel with the ideal current source. Current divides between the internal resistance and the external load; to keep the load current near the nominal value, we want most current to go through the load, not the internal shunt path.

Given Data / Assumptions:

• Ideal current I_N = 10 mA.
• Internal (shunt) resistance R_int = 20 MΩ in parallel.
• Load resistance R_L in parallel with R_int.

Concept / Approach:
For a Norton source, the load current is I_L = I_N * (R_int / (R_int + R_L)). If R_L ≪ R_int, then R_int/(R_int + R_L) ≈ 1, making I_L ≈ I_N. A practical “stiff” guideline is that R_L should be at least one to two orders of magnitude smaller than R_int so that the current through R_int is negligible.

Step-by-Step Solution:
Compute order-of-magnitude: R_int = 20 MΩ; choose R_L ≤ 0.01 * R_int ≈ 200 kΩ for ≲1% error.Check current division: If R_L = 200 kΩ, factor = 20 MΩ / (20 MΩ + 200 kΩ) ≈ 20,000 kΩ / 20,200 kΩ ≈ 0.99 → I_L ≈ 9.9 mA.Thus, loads of 200 kΩ or less keep the source behavior close to ideal 10 mA.

Verification / Alternative check:
Try R_L = 20 kΩ: factor ≈ 20,000 kΩ / 20,020 kΩ ≈ 0.999 → even stiffer. As R_L decreases further, the approximation improves.

Why Other Options Are Wrong:
(a) “20 kΩ or more” includes large R_L values approaching MΩ, which reduce stiffness. (c) “200 kΩ or more” trends the wrong way for stiffness. (d) “20 km” is a units mistake (distance, not resistance).

Common Pitfalls:
Confusing Norton (parallel R_int) with Thevenin (series R_th); forgetting that for current sources, small R_L compared to R_int is desirable for “stiffness.”

Final Answer:
200 kΩ or less