Maximum efficiency in a DC machine occurs when:

Question

BlackTom AI · Accepted Answer

Variable losses are equal to constant losses.

1. Relevant concepts:
- Efficiency η of a machine = output power Pout divided by input power Pin = Pout / (Pout + Total losses).
- In a DC machine losses are commonly split into constant losses (C) — e.g., core (iron) loss, constant field loss (if field current fixed), and mechanical losses (approximately constant) — and variable losses (Pv) — mainly copper (I^2R) losses in the armature which depend on load current.
- For a given machine, armature current (and therefore copper loss) is proportional to output power, so Pv may be expressed as k·Pout^2 (k is a constant relating output power to current and the I^2R factor).

2. Step-by-step reasoning:
- Let P denote output power (Pout). Let constant losses = C. Let variable (copper) losses = Pv = k P^2.
- Input power Pin = P + C + k P^2.
- Efficiency η(P) = P / (P + C + k P^2).

3. Calculus and intermediate steps:
- Differentiate η with respect to P and set derivative to zero for maximum:
  η = P / D, where D = P + C + kP^2.
  dη/dP = (D·1 - P·dD/dP) / D^2.
  dD/dP = 1 + 2kP.
  So numerator = (P + C + kP^2) - P(1 + 2kP) = P + C + kP^2 - P - 2kP^2 = C - kP^2.
- Set dη/dP = 0 ⇒ C - kP^2 = 0 ⇒ kP^2 = C.
- But kP^2 is exactly the variable loss Pv, hence Pv = C.

4. Why this is correct:
- The calculus shows the extremum occurs when the rate at which variable losses grow with load exactly balances the constant losses. At that point any small increase or decrease in load would decrease efficiency; therefore it is the maximum.

5. Why other choices are incorrect:
- Any statement that maximum efficiency occurs when Pv > C or Pv < C contradicts the derivative condition found above. Only Pv = C satisfies dη/dP = 0 and corresponds to a maximum (second derivative test or inspection of η curve confirms maximum).

6. Conclusion:
- Thus maximum efficiency in a DC machine occurs when variable losses equal constant losses, matching the provided answer: "variable losses is constant losses."

Maximum efficiency in a DC machine occurs when:单项选择题

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