mirror of
https://github.com/peter-tanner/Power-and-Machines-notes.git
synced 2024-11-30 17:10:24 +08:00
145 lines
6.2 KiB
Markdown
145 lines
6.2 KiB
Markdown
> Why are the drawings bad?
|
|
|
|
I draw them with a mouse
|
|
|
|
### Types of power factors (From `ENSC2003`)
|
|
|
|
Where $\bar{S}=|\bar{S}|\angle\varphi$:
|
|
|
|
$$ \varphi = \arctan\left(\frac{Q}{P}\right) = \theta_v-\theta_i$$
|
|
|
|
| | Lagging | Leading | Unity |
|
|
| ----------- | -------------- | ------------- | ------------ |
|
|
| Voltage | Current behind | Current ahead | In phase |
|
|
| Load type | Inductive | Capacitive | Resistive |
|
|
| $Q$ | $Q>0$ | $Q<0$ | $Q=0$ |
|
|
| $\varphi$ | $\varphi>0°$ | $\varphi<0°$ | $\varphi=0°$ |
|
|
| PF [Load] | $[0,1)$ | $[0,1)$ | $1$ |
|
|
| PF [Source] | $[0,-1)$ | $[0,-1)$ | $-1$ |
|
|
|
|
## Power types in motor
|
|
|
|
| Type | Description | Equivalent terms |
|
|
| ------------------ | ---------------------------------------------------------------------------------------------------------------- | -------------------------------------------------------------------------------------------------------------- |
|
|
| Input power | Power into machine. $V_T=V_{3\phi}$, $I_L=I_{3\phi}$ | $P_\text{in}$, $\sqrt{3}V_TI_L\cos(\theta)$ |
|
|
| Output power | Mechanical output power of the machine, excludes losses | $P_\text{out}$, $P_\text{load}$ |
|
|
| Converted power | Total electrical power converted to mechanical power, includes useful power and mechanical losses inside machine | $P_\text{conv}$, $P_\text{converted}$, $P_\text{mech}$, $P_\text{developed}$, $\tau_\text{mech}\times\omega_m$ |
|
|
| Airgap power | Power transmitted over airgap. | $P_\text{AG}$, $\tau_\text{mech}\times\omega_s$ |
|
|
| Mechanical loss | Power lost to friction and windage | $P_\text{mechanical loss}$, $P_\text{F\\\&W}$, $P_\text{friction and windage}$ |
|
|
| Core loss | Power lost in machine magnetic material due to hysteresis loss and eddy currents | $P_\text{core}$ |
|
|
| Rotor copper loss | Due to resistance of rotor windings | $P_r$, $P_\text{RCL}$ |
|
|
| Stator copper loss | Due to resistance of stator windings | $P_s$, $P_\text{SCL}$ |
|
|
| Miscellaneous loss | Add 1% to losses to account for other unmeasured losses | $P_\text{misc}$, $P_\text{stray}$ |
|
|
|
|
|
|
|
|
$$
|
|
\begin{align}
|
|
P_\text{in}&=P_\text{SCL}+P_\text{RCL}+P_\text{core}+P_\text{F\\\&W}+P_\text{misc}+P_\text{out}\\
|
|
P_\text{AG}&=P_\text{RCL}+P_\text{F\\\&W}+P_\text{misc}+P_\text{out}\\
|
|
P_\text{mech}&=P_\text{F\\\&W}+P_\text{misc}+P_\text{out}
|
|
\end{align}
|
|
$$
|
|
|
|
Note - assume loss is 0 if not mentioned!
|
|
|
|
| Type | Description | Symbols |
|
|
| ------------------------- | ------------------------------------------- | --------------------------------------- |
|
|
| Load torque, Shaft torque | Torque experienced by load after all losses | $\tau_\text{load}$, $\tau_\text{shaft}$ |
|
|
|
|
## $3\phi$ induction motor
|
|
|
|
### Etc.
|
|
|
|
- Slip speed $N_\text{slip}=N_{s\text{ (sync)}}-N_r=sN_{s\text{ (sync)}}$
|
|
|
|
- "1/4 of rated load" != "1/4 times full load"
|
|
- Means 1/4 of full load slip as it is in the linear region. Accounts for the minimum load.
|
|
- Rated power stated in machine specification refers to the output power $P_\text{out}$, and excludes all losses.
|
|
- Speed regulation using machine speed: $$\text{SR}=\frac{N_{r,\text{NL}}-N_{r,\text{FL}}}{N_{r,\text{FL}}}$$
|
|
|
|
### Diagram
|
|
|
|
|
|
|
|
### Equivalent model
|
|
|
|
#### Assumptions
|
|
|
|
- $x_m\approx X_m$
|
|
- $R_c\ggg X_m\Rightarrow r_c\lll x_m$
|
|
- $x_m=\frac{{R_c}^2}{{R_c}^2+{X_m}^2}X_m\approx\frac{\cancel{{R_c}^2}}{\cancel{{R_c}^2}}X_m=X_m$
|
|
- $r_c\approx {X_m}^2/R_c$
|
|
- $R_c\ggg X_m\Rightarrow r_c\lll x_m$
|
|
- $r_c=\frac{{X_m}^2}{{R_c}^2+{X_m}^2}R_c\approx\frac{{X_m}^2}{{R_c}^2}R_c=\frac{{X_m}^2}{R_c}$
|
|
|
|
### Diagram
|
|
|
|
|
|
|
|
### DC test
|
|
|
|
#### $\Delta$ machine
|
|
|
|
$$R_s=\frac{3}{2}\cdot\frac{V_{\text{DC},3\phi}}{I_{\text{DC},3\phi}}$$
|
|
|
|
|
|
#### Y machine
|
|
|
|
$$R_s=\frac{1}{2}\cdot\frac{V_{\text{DC},3\phi}}{I_{\text{DC},3\phi}}$$
|
|
|
|
|
|
### No-load test
|
|
|
|
#### Assumptions
|
|
|
|
- $P_\text{out}=0$
|
|
- No output power as no load.
|
|
- $R_r/s=\infty$ and $I_r=0$
|
|
- Infinite rotor resistance, ignore rotor path.
|
|
|
|
#### Diagram
|
|
|
|
Using assumptions, remove rotor part of circuit and only consider stator and magnetizing path.
|
|
|
|
|
|
|
|
### Blocked rotor test
|
|
|
|
#### Assumptions
|
|
|
|
- Ignore magnetizing path, $I_m=0$
|
|
- $I_r\ggg I_m$ as $R_r/s\ggg Z_m$
|
|
- $R_r/s=R_r$, $s=1$
|
|
- Slip is $1$ as rotor is blocked.
|
|
- $x_s=x_r'$
|
|
- Same number of turns in stator and rotor
|
|
- and $x_r=f_0/f_\text{BL} \times x_r'$
|
|
- Note: $x_r'$ is the inductance at $f_\text{BL}$, the blocked rotor test frequency which is less than the nominal frequency $f_0$
|
|
|
|
#### Diagram
|
|
|
|
Ignore magnetizing path
|
|
|
|
|
|
|
|
---
|
|
|
|
## Single-phase induction motor
|
|
|
|
### Diagram
|
|
|
|
|
|
|
|
### Blocked-rotor
|
|
|
|
#### Diagram
|
|
|
|
|
|
|
|
### No-load
|
|
|
|
#### Diagram
|
|
|
|
|