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https://github.com/peter-tanner/Power-and-Machines-notes.git
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203 lines
8.2 KiB
Markdown
203 lines
8.2 KiB
Markdown
> Why are the drawings bad?
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I draw them with a mouse
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### Etc
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#### FIRST-PASS CHECKS
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- Double check all are in the correct phase! Multiplications and divisions by $\sqrt{3}$ or $3$ where necessary must be checked! Try annotating everything that does not have an associated phase.
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- Check conjugate in current. $\bar{S}=\bar{V}\bar{I}^*$
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#### Y-$\Delta$ transformation (Balanced case)
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$Z_\Delta=3Z_Y$
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### Types of power factors (From `ENSC2003`)
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Where $\bar{S}=|\bar{S}|\angle\varphi$:
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$$ \varphi = \arctan\left(\frac{Q}{P}\right) = \theta_v-\theta_i$$
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| | Lagging | Leading | Unity |
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| ----------- | -------------- | ------------- | ------------ |
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| Voltage | Current behind | Current ahead | In phase |
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| Load type | Inductive | Capacitive | Resistive |
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| $Q$ | $Q>0$ | $Q<0$ | $Q=0$ |
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| $\varphi$ | $\varphi>0°$ | $\varphi<0°$ | $\varphi=0°$ |
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| PF [Load] | $[0,1)$ | $[0,1)$ | $1$ |
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| PF [Source] | $[0,-1)$ | $[0,-1)$ | $-1$ |
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## Power types in induction motor
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| Type | Description | Equivalent terms |
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| ------------------ | ---------------------------------------------------------------------------------------------------------------- | -------------------------------------------------------------------------------------------------------------- |
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| Input power | Power into machine. $V_T=V_{3\phi}$, $I_L=I_{3\phi}$ | $P_\text{in}$, $\sqrt{3}V_TI_L\cos(\theta)$ |
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| Output power | Mechanical output power of the machine, excludes losses | $P_\text{out}$, $P_\text{load}$ |
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| Converted power | Total electrical power converted to mechanical power, includes useful power and mechanical losses inside machine | $P_\text{conv}$, $P_\text{converted}$, $P_\text{mech}$, $P_\text{developed}$, $\tau_\text{mech}\times\omega_m$ |
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| Airgap power | Power transmitted over airgap. | $P_\text{AG}$, $\tau_\text{mech}\times\omega_s$ |
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| Mechanical loss | Power lost to friction and windage | $P_\text{mechanical loss}$, $P_\text{F\\\&W}$, $P_\text{friction and windage}$ |
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| Core loss | Power lost in machine magnetic material due to hysteresis loss and eddy currents | $P_\text{core}$ |
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| Rotor copper loss | Due to resistance of rotor windings | $P_r$, $P_\text{RCL}$ |
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| Stator copper loss | Due to resistance of stator windings | $P_s$, $P_\text{SCL}$ |
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| Miscellaneous loss | Add 1% to losses to account for other unmeasured losses | $P_\text{misc}$, $P_\text{stray}$ |
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$$
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\begin{align}
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P_\text{in}&=P_\text{SCL}+P_\text{RCL}+P_\text{core}+P_\text{F\\\&W}+P_\text{misc}+P_\text{out}\\
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P_\text{AG}&=P_\text{RCL}+P_\text{F\\\&W}+P_\text{misc}+P_\text{out}\\
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P_\text{mech}&=P_\text{F\\\&W}+P_\text{misc}+P_\text{out}
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\end{align}
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$$
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Note - assume loss is 0 if not mentioned!
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| Type | Description | Symbols |
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| ------------------------- | ------------------------------------------- | --------------------------------------- |
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| Load torque, Shaft torque | Torque experienced by load after all losses | $\tau_\text{load}$, $\tau_\text{shaft}$ |
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## $3\phi$ induction motor
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### Etc.
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- Slip speed $N_\text{slip}=N_{s\text{ (sync)}}-N_r=sN_{s\text{ (sync)}}$
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- "1/4 of rated load" != "1/4 times full load"
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- Means 1/4 of full load slip as it is in the linear region. Accounts for the minimum load.
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- Rated power stated in machine specification refers to the output power $P_\text{out}$, and excludes all losses.
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- Speed regulation using machine speed: $$\text{SR}=\frac{N_{r,\text{NL}}-N_{r,\text{FL}}}{N_{r,\text{FL}}}$$
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### Diagram
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### Equivalent model
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#### Assumptions
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- $x_m\approx X_m$
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- $R_c\ggg X_m\Rightarrow r_c\lll x_m$
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- $x_m=\frac{{R_c}^2}{{R_c}^2+{X_m}^2}X_m\approx\frac{\cancel{{R_c}^2}}{\cancel{{R_c}^2}}X_m=X_m$
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- $r_c\approx {X_m}^2/R_c$
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- $R_c\ggg X_m\Rightarrow r_c\lll x_m$
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- $r_c=\frac{{X_m}^2}{{R_c}^2+{X_m}^2}R_c\approx\frac{{X_m}^2}{{R_c}^2}R_c=\frac{{X_m}^2}{R_c}$
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### Diagram
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### DC test
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#### $\Delta$ machine
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$$R_s=\frac{3}{2}\cdot\frac{V_{\text{DC},3\phi}}{I_{\text{DC},3\phi}}$$
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#### Y machine
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$$R_s=\frac{1}{2}\cdot\frac{V_{\text{DC},3\phi}}{I_{\text{DC},3\phi}}$$
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### No-load test
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#### Assumptions
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- $P_\text{out}=0$
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- No output power as no load.
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- $R_r/s=\infty$ and $I_r=0$
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- Infinite rotor resistance, ignore rotor path.
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#### Diagram
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Using assumptions, remove rotor part of circuit and only consider stator and magnetizing path.
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### Blocked rotor test
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#### Assumptions
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- Ignore magnetizing path, $I_m=0$
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- $I_r\ggg I_m$ as $R_r/s\ggg Z_m$
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- $R_r/s=R_r$, $s=1$
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- Slip is $1$ as rotor is blocked.
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- $x_s=x_r'$
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- Same number of turns in stator and rotor
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- and $x_r=f_0/f_\text{BL} \times x_r'$
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- Note: $x_r'$ is the inductance at $f_\text{BL}$, the blocked rotor test frequency which is less than the nominal frequency $f_0$
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#### Diagram
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Ignore magnetizing path
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---
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## Single-phase induction motor
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### Diagram
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### Blocked-rotor
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#### Diagram
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### No-load
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#### Diagram
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## Synchronous machine
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### Etc
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$$E_A=V_{1\phi}-I_A(R_A+j X_s)$$
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$$I_A=\text{CONJUGATE}\left(\frac{|S_{3\phi}|\angle\arccos(x)}{3V_{1\phi}}\right), \begin{cases}x=+\text{PF} && \text{lagging} \\ x=-\text{PF} && \text{leading}\end{cases}$$
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### Voltage regulation
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$$\text{VR}=\frac{|V_\text{NL}|-|V_\text{FL}|}{|V_\text{FL}|}=\frac{|E_A|-|V_{1\phi,\text{rated}}|}{|V_{1\phi,\text{rated}}|}$$
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- $V_\text{FL}$ is the full-load voltage which is the full-load/maximum rated voltage at the output terminal.
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- Calculate $E_A$ at full load by calculating the current as shown above.
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- $V_\text{NL}$ is the no-load voltage, which in the no-load case will be $E_A$.
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| No-load | Full-load |
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| ---------------------------- | ---------------------------- |
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| Power factor | Voltage regulation |
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| ------------ | ------------------ |
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| Lagging | Positive |
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| Unity | Near 0 |
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| Leading | Negative |
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### Open and short circuit test
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#### **Note** - double-check if the axis refers to per-phase or line voltage/current.
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| Open-circuit test | Short-circuit test |
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| ---------------------------- | ---------------------------- |
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### Power flow
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$P_\text{out}$ is the rated power
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$$P_\text{out}=S_\text{rated}\times \text{PF}$$
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$$
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\begin{align}
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P_\text{in}&=P_\text{copper}+P_\text{core}+P_\text{F\\\&W}+P_\text{misc}+P_\text{out}\\
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P_\text{mech}&=P_\text{F\\\&W}+P_\text{misc}+P_\text{out}
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\end{align}
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$$
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---
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