| Rated angular speed | \(\displaystyle \omega_{mr} = \frac{2\pi N_r}{60}\) | \(N_r\) in rpm; result in rad/s |
| Rated output torque | \(\displaystyle T_{er} = \frac{P_{out}}{\omega_{mr}}\) | \(P_{out}\) = rated output power in W |
| Rated armature current | \(\displaystyle I_{ar} = \frac{T_{er}}{K_b}\) | Valid when flux = rated (sep. excited, const. \(\phi\)) |
| Rated field current | \(\displaystyle I_{fr} = \frac{V_{f,\max}}{R_f}\) | Steady state; full field voltage applied |
| Condition | \(\phi = \phi_r\) (rated, fixed); \(V_a\) varies from 0 to \(V_{a,\max}\) | Armature voltage control |
| Maximum back-EMF at base speed | \(\displaystyle e_1 = V_{a,\max} - I_{ar}\,R_a\) | Limited by maximum armature voltage and rated current |
| Base (rated) speed | \(\displaystyle \omega_{mr} = \frac{e_1}{K_b}\) | Speed where \(V_a = V_{a,\max}\) and \(I_a = I_{ar}\) |
| Max torque in Region 1 | \(T_{e,\max} = K_b\,I_{ar} = T_{er} = 1\,\text{p.u.}\) | Constant throughout Region 1 |
| Normalised torque | \(T_n = 1\,\text{p.u.}\) | Normalised power \(P_n = \omega_n\) (increases linearly) |
| Condition | \(V_a = V_{a,\max}\); \(I_a = I_{ar}\) (1 p.u.); \(\phi\) reduced | Field voltage control above base speed |
| Flux–speed relation (const. \(E_b\)) | \(\displaystyle \phi_{fn} = \frac{e_n}{\omega_{mn}} = \frac{1}{\omega_{mn}}\) | Normalised; \(e_n = 1\) for continuous operation |
| Normalised torque | \(\displaystyle T_{en} = \phi_{fn}\cdot I_{an} = \frac{1}{\omega_{mn}}\) | Torque falls hyperbolically with speed |
| Normalised power | \(P_n = T_{en}\cdot\omega_{mn} = 1\,\text{p.u.}\) | Constant throughout Region 2 |
| Minimum flux (safe limit) | \(\displaystyle I_{f,\min} = \frac{V_{f,\min}}{R_f},\quad \phi_{\min} = \frac{I_{f,\min}}{I_{fr}}\) | Set by minimum safe field voltage \(V_{f,\min}\) |
| Maximum speed | \(\displaystyle \omega_{mn,\max} = \frac{e_n}{\phi_{\min}}\) | Speed where flux reaches its minimum safe value |
| Peak intermittent current | \(I_{\max} = k\,I_{ar}\) (e.g., \(k = 3\) for 300%) | Time-limited; thermally not sustainable continuously |
| Peak torque | \(T_{e,\max} = K_b\,I_{\max} = k\,T_{er} = k\,\text{p.u.}\) | Scales linearly with current (sep. excited, const. \(\phi\)) |
| Reduced back-EMF at peak current | \(\displaystyle e_m = V_{a,\max} - I_{\max}\,R_a\) | Lower than \(e_1\) because \(I_{\max} R_a > I_{ar} R_a\) |
| Intermittent base speed | \(\displaystyle \omega_{m1} = \frac{e_m}{K_b},\quad \omega_{m1n} = \frac{e_m}{e_r}\) | Less than 1 p.u. because \(e_m < e_r\) |
| Intermittent normalised back-EMF | \(\displaystyle e_n = \frac{e_m}{e_r} = \frac{V_{a,\max} - I_{\max}R_a}{V_{a,\max} - I_{ar}R_a}\) | Determines entire field-weakening range |
| Field-weakening region (intermittent) | \(\displaystyle \phi_{fn} = \frac{e_n}{\omega_{mn}},\quad T_{en} = \frac{k\cdot e_n}{\omega_{mn}}\) | Higher torque but narrower speed range than continuous |
| Intermittent maximum speed | \(\displaystyle \omega_{mn,\max} = \frac{e_n}{\phi_{\min}}\) | Less than continuous max speed when \(e_n < 1\) |
| Rated output power | 2.625 hp |
| Supply voltage | 120 V |
| Rated speed | 1313 rpm |
| Armature resistance \(R_a\) | 0.8 Ω |
| Field resistance \(R_f\) | 100 Ω |
| Back-EMF constant \(K_b\) | 0.764 V·s/rad |
| Armature inductance \(L_a\) | 3 mH |
| Field inductance \(L_f\) | 2.2 H |
For each case, identify Region 1 (constant-torque) and Region 2 (field-weakening/constant-power) and compute the key parameters: base speed, maximum speed, peak torque, and the torque–speed equations.
Rated angular speed
\[ \omega_{mr} = \frac{2\pi N_r}{60} = \frac{2\pi \times 1313}{60} = 137.56\,\text{rad/s} \]Rated output torque
\[ T_{er} = \frac{P_{out}}{\omega_{mr}} = \frac{2.625 \times 745.6}{137.56} = \frac{1957.2}{137.56} = 14.23\,\text{N·m} \]Rated armature current
\[ I_{ar} = \frac{T_{er}}{K_b} = \frac{14.23}{0.764} = 18.63\,\text{A} \]Rated field current
\[ I_{fr} = \frac{V_{f,\max}}{R_f} = \frac{120}{100} = 1.2\,\text{A} \]Rated back-EMF (base EMF \(e_r\))
\[ \begin{align*} e_r & = V_{a,\max} - I_{ar}\,R_a \\ &= 120 - (18.63 \times 0.8) = 120 - 14.90 = 105.1\,\text{V} \end{align*} \]Verify rated speed:
\[ \omega_{mr} = \frac{e_r}{K_b} = \frac{105.1}{0.764} = 137.6\,\text{rad/s}\;\checkmark \]| Quantity | Symbol | Value |
|---|---|---|
| Rated speed | \(\omega_{mr}\) | 137.56 rad/s |
| Rated torque | \(T_{er}\) | 14.23 N·m |
| Rated armature current | \(I_{ar}\) | 18.63 A |
| Rated field current | \(I_{fr}\) | 1.2 A |
| Rated back-EMF | \(e_r\) | 105.1 V |
Field voltage held at 120 V \(\Rightarrow \phi = \phi_r\) (rated flux, 1 p.u.). Armature voltage varies from 0 to \(V_{a,\max} = 120\,\text{V}\).
Maximum back-EMF at rated armature current
\[ e_1 = V_{a,\max} - I_{ar}\,R_a = 120 - (18.63 \times 0.8) = \mathbf{105.1\,\text{V}} \]Corresponding base (rated) speed
\[ \omega_{m1} = \frac{e_1}{K_b} = \frac{105.1}{0.764} = 137.56\,\text{rad/s} = \mathbf{1.0\,\text{p.u.}} \]Maximum (rated) torque
\[ T_{er} = K_b\,I_{ar} = 0.764 \times 18.63 = \mathbf{14.23\,\text{N·m} = 1.0\,\text{p.u.}} \]Above base speed: \(V_a = V_{a,\max} = 120\,\text{V}\) (fixed); \(I_a = I_{ar}\) (1 p.u.); field voltage reduced below 120 V.
Normalised back-EMF (reference value)
\[ e_n = \frac{e_1}{e_r} = \frac{105.1}{105.1} = \mathbf{1.0\,\text{p.u.}} \]Flux required to maintain \(e_n = 1\) at speed \(\omega_{mn}\)
\[ \phi_{fn} = \frac{e_n}{\omega_{mn}} = \frac{1}{\omega_{mn}} \]Minimum field current and minimum flux
\[ I_{f,\min} = \frac{V_{f,\min}}{R_f} = \frac{12}{100} = 0.12\,\text{A} \] \[ \phi_{\min} = \frac{I_{f,\min}}{I_{fr}} = \frac{0.12}{1.2} = \mathbf{0.1\,\text{p.u.}} \]Maximum normalised speed
\[ \omega_{mn,\max} = \frac{e_n}{\phi_{\min}} = \frac{1.0}{0.1} = \mathbf{10\,\text{p.u.}} \] \[ \omega_{m,\max} = 10 \times 137.56 = 1375.6\,\text{rad/s} \]| Region | Speed range | \(T_n\) | \(P_n\) |
|---|---|---|---|
| 1 | 0 → 1 p.u. | 1 (const.) | \(\omega_n\) |
| 2 | 1 → 10 p.u. | \(1/\omega_n\) | 1 (const.) |
Armature current is now permitted to reach 300% of rated value: \(I_{\max} = 3\,I_{ar}\). Field voltage still held at 120 V \(\Rightarrow \phi = \phi_r\).
Peak armature current and torque
\[ \begin{aligned} I_{\max} &= 3\,I_{ar} \\ &= 3 \times 18.63 \\ &= 55.89\,\text{A} \\ T_{e,\max} &= K_b\,I_{\max} \\ &= 0.764 \times 55.89 \\ &= 42.7\,\text{N·m} \\ &= \mathbf{3.0\,\text{p.u.}} \end{aligned} \]Maximum back-EMF at peak current
The large \(I_{\max}\) creates a bigger \(R_a\) drop, reducing available back-EMF:
\[ \begin{aligned} e_m &= V_{a,\max} - I_{\max}\,R_a \\ &= 120 - (55.89 \times 0.8) \\ &= 120 - 44.71 = \mathbf{75.29\,\text{V}} \end{aligned} \]Intermittent base speed
\[ \omega_{m1} = \frac{e_m}{K_b} = \frac{75.29}{0.764} = 98.54\,\text{rad/s} \] \[ \omega_{m1n} = \frac{\omega_{m1}}{\omega_{mr}} = \frac{98.54}{137.56} = \mathbf{0.716\,\text{p.u.}} \]| Parameter | Continuous | Intermittent |
|---|---|---|
| \(I_a\) (A) | 18.63 | 55.89 |
| \(T_e\) (N·m) | 14.23 | 42.70 |
| \(T_n\) (p.u.) | 1.0 | 3.0 |
| Back-EMF \(e\) (V) | 105.1 | 75.29 |
| Base speed (p.u.) | 1.0 | 0.716 |
Above the intermittent base speed: \(V_a = 120\,\text{V}\) (fixed); \(I_a = 3\,I_{ar} = 55.89\,\text{A}\); flux reduced.
Normalised intermittent back-EMF
\[ e_n = \frac{e_m}{e_r} = \frac{75.29}{105.1} = \mathbf{0.716\,\text{p.u.}} \]This is also the normalised base speed: \(\omega_{m1n} = e_n = 0.716\,\text{p.u.}\)
Field-weakening flux requirement
To maintain \(E_b = e_m\) at speed \(\omega_{mn}\):
\[ \phi_{fn} = \frac{e_n}{\omega_{mn}} = \frac{0.716}{\omega_{mn}} \]Normalised torque in Region 2
\[ T_{en} = I_{an}\cdot\phi_{fn} = 3 \times \frac{0.716}{\omega_{mn}} = \frac{2.148}{\omega_{mn}} \]Maximum intermittent speed
Flux limit is the same: \(\phi_{\min} = 0.1\,\text{p.u.}\)
\[ \begin{aligned} \omega_{mn,\max} &= \frac{e_n}{\phi_{\min}} \\ &= \frac{0.716}{0.1} \\ &= \mathbf{7.16\,\text{p.u.}} \\ \omega_{m,\max} &= 7.16 \times 137.56 \\ &= 984.9\,\text{rad/s} \end{aligned} \]
| Parameter | Value |
|---|---|
| Rated speed \(\omega_{mr}\) | 137.56 rad/s |
| Rated torque \(T_{er}\) | 14.23 N·m |
| Rated armature current \(I_{ar}\) | 18.63 A |
| Rated field current \(I_{fr}\) | 1.2 A |
| Rated back-EMF \(e_r\) | 105.1 V |
| Base speed | 1.0 p.u. |
| Maximum speed | 10.0 p.u. |
| Minimum flux \(\phi_{\min}\) | 0.1 p.u. |
| Parameter | Value |
|---|---|
| Peak current \(I_{\max}\) | 55.89 A (3 p.u.) |
| Peak torque \(T_{e,\max}\) | 42.70 N·m (3 p.u.) |
| Peak back-EMF \(e_m\) | 75.29 V (0.716 p.u.) |
| Intermittent base speed | 0.716 p.u. (98.54 rad/s) |
| Maximum speed | 7.16 p.u. |
| Minimum flux \(\phi_{\min}\) | 0.1 p.u. (same) |
| Const. power (Region 2) | 2.148 p.u. |
| Region | Speed Range (p.u.) | Torque \(T_n\) | Power \(P_n\) | Flux \(\phi_{fn}\) |
|---|---|---|---|---|
| 1 — Continuous | 0 → 1.0 | 1.0 (constant) | \(\omega_n\) (increasing) | 1.0 (constant) |
| 2 — Continuous | 1.0 → 10.0 | \(1/\omega_n\) (hyperbolic) | 1.0 (constant) | \(1/\omega_n\) |
| 1 — Intermittent | 0 → 0.716 | 3.0 (constant) | \(3\omega_n\) (increasing) | 1.0 (constant) |
| 2 — Intermittent | 0.716 → 7.16 | \(2.148/\omega_n\) (hyperbolic) | 2.148 (constant) | \(0.716/\omega_n\) |
Continuous Region 1 (\(0 \leq \omega_n \leq 1\)):
\[T_n = 1\,\text{p.u.}\]Continuous Region 2 (\(1 \leq \omega_n \leq 10\)):
\[T_n = \frac{1}{\omega_n}\]Intermittent Region 1 (\(0 \leq \omega_n \leq 0.716\)):
\[T_n = 3\,\text{p.u.}\]Intermittent Region 2 (\(0.716 \leq \omega_n \leq 7.16\)):
\[T_n = \frac{2.148}{\omega_n}\]