Question

The four stator conductors (A, A$'$, B and B$'$) of a rotating machine are carrying DC currents of the same value, the directions of which are shown in the figure (i). The rotor coils $a$–$a'$ and $b$–$b'$ are formed by connecting the back ends of conductors $a$ and $a'$, and $b$ and $b'$, respectively, as shown in figure (ii). The e.m.f. induced in coil $a$–$a'$ and coil $b$–$b'$ are denoted by $E_{a\text{–a'}$ and $E_{b\text{–}b'}$, respectively. If the rotor is rotated at uniform angular speed $\omega \, \text{rad/s}$ in the clockwise direction then which of the following correctly describes the $E_{a\text{–}a'}$ and $E_{b\text{–}b'}$?} \begin{center} \includegraphics[width=0.5\textwidth]{07.jpeg} \end{center}

• A. $E_{a\text{–}a'}$ and $E_{b\text{–}b'}$ have finite magnitudes and are in the same phase
• B. $E_{a\text{–}a'}$ and $E_{b\text{–}b'}$ have finite magnitudes with $E_{b\text{–}b'}$ leading $E_{a\text{–}a'}$
• C. $E_{a\text{–}a'}$ and $E_{b\text{–}b'}$ have finite magnitudes with $E_{a\text{–}a'}$ leading $E_{b\text{–}b'}$
• D. $E_{a\text{–}a'} = E_{b\text{–}b'} = 0$

Hint:

In rotating machines, if the stator field is stationary (from DC currents) and the rotor moves, the induced rotor voltages are AC and their relative phase depends on the angular placement of the rotor coils. Symmetrical placement often results in induced e.m.f.s that are in phase.

Answer 1

The Correct Option is A

Step 1: Magnetic field produced by the stator currents.
The stator conductors (A, A$'$, B, B$'$) are carrying equal DC currents, arranged symmetrically. From figure (i), currents in opposite sides create a magnetic field that is approximately sinusoidal and distributed across the air gap. This field is stationary in space but constant in magnitude because the currents are DC.

Step 2: Induced e.m.f. in the rotor conductors.
The rotor is rotating clockwise at angular speed $\omega$. A conductor cutting the stationary flux at speed $\omega r$ will experience an induced e.m.f. according to: \[ e = B \cdot l \cdot v = B l \omega r \] where $B$ is the flux density produced by the stator field. Thus, the induced e.m.f. in each rotor coil is AC in time because of rotation relative to the stator magnetic field.

Step 3: Relative positions of coils $a$–$a'$ and $b$–$b'$.
From the diagram, $a$–$a'$ and $b$–$b'$ are placed at $\;90^{\circ}$ apart (electrical). The stator MMF distribution is such that both $a$ and $b$ conductors see the same polarity of flux simultaneously due to symmetric current placement. Hence, their induced e.m.f.s will be in phase, not displaced.

Step 4: Evaluation of options.
- (A) \; Correct – Both $E_{a\text{–}a'}$ and $E_{b\text{–}b'}$ are finite and in phase. - (B) \; Incorrect – No phase lead exists between the two. - (C) \; Incorrect – No phase lead of $E_{a\text{–}a'}$ over $E_{b\text{–}b'}$. - (D) \; Incorrect – Induced e.m.f.s are not zero because the rotor is rotating in a magnetic field. % Final Answer \[ \boxed{\text{Option (A): $E_{a\text{–}a'}$ and $E_{b\text{–}b'}$ are finite and in phase.}} \]