FE Electrical & Computer Practice Question Solution included after investigation
CASE E-022 Β· FE

The Transfer Function That Settled at 5, Not 20

The numerator says 20. The unit-step output settles at 5. The model is working exactly as written.

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Case E-022 engineering failure visual
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3

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4

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5

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Case Brief

SAME PRINCIPLES. HIGHER STANDARDS.
PROBLEM STATEMENT

A stable first-order system has transfer function H(s) = 20/(s + 4). A technician applies a unit-step input and expects the output to settle at 20 because the numerator is 20. The measured output instead settles at 5. Determine the DC gain, relate the pole to the time constant, and modify the numerator so the same pole gives a steady-state output of 2 for a unit-step input.

Linear Systems Electrical & Computer Design Verification Root Cause Analysis Safe Correction
E-022
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CASE CLOSED Β· ENGINEERING VERDICT

What actually failed β€” and what should change.

What caused the mismatch

The numerator 20 was mistaken for the DC gain; the actual gain is 20/4 = 5.

What was missed

The denominator must also be evaluated at s = 0 when finding DC gain.

Corrected model

Use H(0) for DC gain and pole location for response speed; for this system Ο„ = 0.25 s.

QUICK ANSWER

For a stable transfer function, the DC gain is H(0), not simply the numerator. For H(s) = 20/(s + 4), H(0) = 20/4 = 5, so a unit-step output settles at 5. The pole at -4 gives a 0.25 s time constant.

ENGINEERING TAKEAWAY

For a stable system, evaluate the transfer function at s = 0 to obtain DC gain, or use the final value theorem with the actual input. The pole at -4 sets a 0.25 s time constant independently of the DC gain scaling.

CASE QUESTIONS

Questions this case should settle.

How do you find the DC gain of a transfer function?

Evaluate the transfer function at s = 0, provided the relevant final value exists for the stable system.

What is the DC gain of 20/(s+4)?

H(0) = 20/4 = 5.

What time constant corresponds to a pole at -4?

For a first-order pole at -a, Ο„ = 1/a, so Ο„ = 0.25 seconds.

NEXT FE PRACTICE QUESTION

The RLC Circuit That Peaked Instead of Filtering

Linear Systems Β· 5–10 min

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