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The output Co-efficient and Synchronous Speed of the machine are ____________ proportional to the Size (product of 𝐷²𝐿)

Directly

Inversely

Equally

Square

Correct Answer

⚙️ TE • Technical Concept & PrincipleElectricalMachine

Option B

Inversely

Quick Summary:

The output coefficient ( $C_0$ ) of an electrical machine represents the relationship between the machine's rated output (kVA or kW) and its dimensions (bore diameter $D$ and core length $L$ ) along with the speed. Because the output power $P$ is directly proportional to $D^2 L n_s$ , the output coefficient $C_0$ defined as $P = C_0 D^2 L n_s$ implies that for a fixed output and speed, the physical size $D^2 L$ is inversely proportional to the output coefficient.

⚙️TETechnical SolutionConcept & Principle

💡 Explanation

🔢 Key Formulas

$P = C_0 D^2 L n_s$ — The output equation for electrical machines

$C_0 = 1.11 \pi^2 B_{av} q \times 10^{-3}$ — Expression for output coefficient

⚙️ Working Principle

In machine design, the output equation is expressed as $Q = C_0 D^2 L n_s$ . Here, $C_0$ is a function of specific magnetic loading ( $B_{av}$ ) and specific electric loading ( $q$ ). Since $P \propto C_0 D^2 L n_s$ , if we hold the power rating and rotational speed constant, an increase in the output coefficient allows for a decrease in the volume of the machine ( $D^2 L$ ). Therefore, the required machine volume is inversely proportional to the chosen output coefficient.

📌 Key Points

▸
The term $D^2 L$ is known as the output coefficient or volume constant of the machine.
▸
Higher values of $C_0$ lead to smaller machines for a given output.
▸
$B_{av}$ (Average magnetic flux density) and $q$ (Specific electric loading) are limited by material properties and cooling capabilities.
▸
The synchronous speed $n_s$ is inversely proportional to $D^2 L$ for a constant power output.

✅ Advantages

▸
Allows optimization of material usage (Copper and Iron).
▸
Provides a standardized metric to compare different machine sizes for specific ratings.

❌ Disadvantages / Limitations

▸
Increasing $C_0$ beyond limits causes thermal degradation.
▸
Higher magnetic loading increases core losses and excitation current.

🛠️ Applications / Uses

▸
Design of Polyphase Induction Motors.
▸
Design of Synchronous Generators and Transformers.

📄 Additional Information

▸
For Induction motors, typical values of $C_0$ depend on the number of poles and the cooling method used.
▸
Option B is correct because the design equation $D^2 L = P / (C_0 n_s)$ mathematically proves the inverse relationship.

📊 Diagram / Illustration

✅

B is correct — The physical size of the machine, represented by the product $D^2 L$ , is inversely proportional to the output coefficient $C_0$ and the synchronous speed $n_s$ for a constant power rating.

Core Concepts Used

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Output Equation of Electrical Machines Specific Magnetic Loading ($B_{av}$) Specific Electric Loading ($q$)

💡 EXAM TIP

Always remember that in electrical machine design, power is the product of flux, current, and speed; maximizing one term usually reduces the physical volume requirement ( $D^2 L$ ).

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