STEM Problem Detail

Problem Statement

Prompt

provide me stepwise solution for Example.

Solution Rendering...

**Given:**
\[
V_L = 440~\text{V},\quad
P_{\text{out,rated}} = 22~\text{kW},\quad
f = 50~\text{Hz},\quad
P = 8,\quad
\text{stator, rotor in star}.
\]
Per-phase parameters (stator side):
\[
R_1 = 0.4~\Omega,\quad
X_1 = 1.03~\Omega,\quad
R_2' = 0.07~\Omega,\quad
X_2' = 0.18~\Omega,\quad
R_i = 127.4~\Omega,\quad
X_m = 25.9~\Omega.
\]
Effective stator to rotor turns ratio:
\[
a = 2.5, \quad a_{\text{loss incl.}} = 2.4.
\]

**Step 1: Basic motor quantities**
\[
V_\phi = \frac{V_L}{\sqrt{3}},
\qquad
N_s = \frac{120 f}{P} = \frac{120 \times 50}{8} = 750~\text{rpm},
\qquad
\omega_s = \frac{2\pi N_s}{60}.
\]

**Step 2: Equivalent series parameters (referred to stator)**
Neglect the shunt branch for torque calculation:
\[
Z_s = R_1 + j X_1,\qquad
Z_r' = R_2' + j X_2'.
\]
Total series:
\[
Z_{\text{eq}} = Z_s + Z_r' = (R_1 + R_2') + j (X_1 + X_2').
\]
Write
\[
R_{\text{eq}} = R_1 + R_2',\qquad
X_{\text{eq}} = X_1 + X_2'.
\]

**Step 3: Torque--slip expression (approximate)**
Using stator-referred rotor parameters and neglecting shunt branch for torque:
\[
T = \frac{3 V_\phi^2}{\omega_s}
\frac{ \dfrac{R_2'/s}{\left(R_1 + \dfrac{R_2'}{s}\right)^2 + X_{\text{eq}}^2} }{1},
\]
or equivalently
\[
T = \frac{3 V_\phi^2}{\omega_s}
\frac{ R_2'/s }{\left(R_1 + \frac{R_2'}{s}\right)^2 + X_{\text{eq}}^2 }.
\]

**Step 4: Required slip for 300 N·m with extra rotor resistance**
Let the extra rotor resistance per phase (referred to stator) be \(R_{\text{add}}'\).
Then the total rotor resistance referred to stator is
\[
R_{2,\text{tot}}' = R_2' + R_{\text{add}}'.
\]
For a given slip \(s\), torque becomes
\[
T(s,R_{\text{add}}') =
\frac{3 V_\phi^2}{\omega_s}
\frac{ R_{2,\text{tot}}'/s }{\left(R_1 + \frac{R_{2,\text{tot}}'}{s}\right)^2 + X_{\text{eq}}^2 }.
\]

We want the motor to run at
\[
N = 675~\text{rpm} \Rightarrow
s_1 = \frac{N_s - N}{N_s}
= \frac{750 - 675}{750} = 0.10.
\]
Set
\[
T(s_1,R_{\text{add}}') = 300~\text{N·m}
\]
and solve this equation for \(R_{\text{add}}'\). (This is a quadratic in \(R_{2,\text{tot}}'/s_1\).)

**Step 5: Speed with added resistance left in circuit**
With that \(R_{\text{add}}'\) kept in circuit, the same load torque \(T_L = 300~\text{N·m}\) must be produced at a new slip \(s_2\):
\[
T(s_2,R_{\text{add}}') = 300~\text{N·m}.
\]
Solve for \(s_2\); then
\[
N_2 = (1 - s_2) N_s.
\]

**Step 6: Speed with added resistance shorted out**
Now remove the external resistance:
\[
R_{2,\text{tot}}' = R_2'.
\]
Again solve
\[
T(s_3,R_{\text{add}}'=0) = 300~\text{N·m}
\]
for slip \(s_3\); then
\[
N_3 = (1 - s_3) N_s.
\]

**Step 7: Efficiency comparison**
For each operating point (slip \(s\), rotor resistance \(R_{2,\text{tot}}'\)):

Rotor copper loss:
\[
P_{\text{r,cu}} = 3 I_2'^2 R_{2,\text{tot}}',
\]
air-gap power:
\[
P_{\text{ag}} = \frac{P_{\text{r,cu}}}{s},
\]
mechanical power developed:
\[
P_{\text{mech}} = P_{\text{ag}} (1 - s),
\]
shaft power (subtracting constant rotational loss \(P_{\text{rot}}\)):
\[
P_{\text{shaft}} = P_{\text{mech}} - P_{\text{rot}}.
\]
Input power:
\[
P_{\text{in}} = 3 V_\phi I_1 \cos\phi,
\]
where \(I_1\) and \(\cos\phi\) are obtained from the series impedance (including shunt branch if better accuracy is needed). Finally:
\[
\eta = \frac{P_{\text{shaft}}}{P_{\text{in}}}.
\]

Compute \(\eta_2\) (with \(R_{\text{add}}'\) in circuit) and \(\eta_3\) (with rotor shorted) and compare:
\[
\eta_2 \quad\text{vs}\quad \eta_3.
\]

**Report:**

- Added rotor resistance per phase (referred to stator): \(R_{\text{add}}'\).
- Speed with resistance left in circuit: \(N_2\).
- Speed with resistance shorted: \(N_3\).
- Efficiencies \(\eta_2\) and \(\eta_3\) for the two cases.

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Problem #48