ENSC3016 notes

Etc

FIRST-PASS CHECKS

• 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.
• Check conjugate in current. $\bar{S}=\bar{V}\bar{I}^*$
• Check transformer parameters are referred to the proper side

Y-$\Delta$ transformation (Balanced case)

$Z_\Delta=3Z_Y$

Types of power factors (From ENSC2003)

Where $\bar{S}=|\bar{S}|\angle\varphi$:

$$\varphi = \arctan\left(\frac{Q}{P}\right) = \theta_v-\theta_i$$

Power types in induction motor

$$\begin{align} P_\text{in}&=P_\text{SCL}+P_\text{RCL}+P_\text{core}+P_\text{F\\\&W}+P_\text{misc}+P_\text{out}\\ P_\text{AG}&=P_\text{RCL}+P_\text{F\\\&W}+P_\text{misc}+P_\text{out}\\ P_\text{mech}&=P_\text{F\\\&W}+P_\text{misc}+P_\text{out} \end{align}$$

Note - assume loss is 0 if not mentioned!

$3\phi$ 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

Synchronous machine

Etc

$$E_A=V_{1\phi}-I_A(R_A+j X_s)$$

$$I_A=\text{CONJUGATE}\left(\frac{|S_{3\phi}|\angle\arccos(x)}{3V_{1\phi}}\right), \begin{cases}x=+\text{PF} && \text{lagging} \\ x=-\text{PF} && \text{leading}\end{cases}$$

Voltage regulation

$$\text{VR}=\frac{|V_\text{NL}|-|V_\text{FL}|}{|V_\text{FL}|}=\frac{|E_A|-|V_{1\phi,\text{rated}}|}{|V_{1\phi,\text{rated}}|}$$

• $V_\text{FL}$ is the full-load voltage which is the full-load/maximum rated voltage at the output terminal.
• Calculate $E_A$ at full load by calculating the current as shown above.
• $V_\text{NL}$ is the no-load voltage, which in the no-load case will be $E_A$.

No-load	Full-load

Open and short circuit test

Note - double-check if the axis refers to per-phase or line voltage/current.

Open-circuit test	Short-circuit test

Power flow

$P_\text{out}$ is the rated power

$$P_\text{out}=S_\text{rated}\times \text{PF}$$

$$\begin{align} P_\text{in}&=P_\text{copper}+P_\text{core}+P_\text{F\\\&W}+P_\text{misc}+P_\text{out}\\ P_\text{mech}&=P_\text{F\\\&W}+P_\text{misc}+P_\text{out} \end{align}$$

Magnetic circuit analogy

Transformers

$$Z_P=Z_S\left(\frac{N_P}{N_S}\right)^2=Z_S n^2$$

Maximum power.

If load is resistive ($jX_\text{load}=0$) then for maximum power transfer:

$$R_\text{load}=|{R_\text{src}}^2+j{X_\text{src}}^2|$$

Parameter identification

Open-circuit test	Short-circuit test

Voltage regulation

$$\text{VR}=\frac{|V_\text{NL,P}|-|V_\text{rated,P}|}{|V_\text{rated,P}|}=\frac{|V_\text{in}|-|V_\text{rated,P}|}{|V_\text{rated,P}|}$$

Ignore shunt resistance. Refer from primary side. Use KVL to determine $V_\text{in}$.

Voltage regulation is typically small.

$$|V_\text{in}|=|V_\text{rated,P}+I_\text{L,P}\cdot\bar Z|$$

DC machine

Separately excited machine	Shunt excited	Series excited
	Similar torque-speed characteristic to separately-excited machine	High torque per ampere. Used in high-torque applications
Requires two independent voltage sources		Do not run unloaded - infinite speed at 0 torque as $\omega\propto 1/\sqrt{\tau}$
Motor control using $R_f$	Motor control using $R_F$	Motor control using $V_T$.

Starting DC motors

$R_A$ might need to be adjusted so it is high initially in large DC motors, as the starting current is high since there is no back-emf created by $E_A$.

Magnetizating curve

When a question specifies the field current or $R_\text{adj}$, refer to magnetization curve. Magnetizating curve is valid at a specific speed $n_{m1}$, and the curve is used to find $E_{A1}$. Using the load condition to find the armature current $I_A=\tau_\text{ind}/(K\varPhi)$, $V_A$ can be used to find a second induced EMF $E_{A2}$. Using $E_{A2}$ find the speed $n_{m2}$ by scaling $n_{m1}$ by $E_{A2}/E_{A1}$.

Idk

$$P_\text{mech}=E_AI_A$$

No-load separately excited machine. Assuming no mechanical losses.

$$E_A=V_A\text{ (No load)}$$

$$I_A=0\text{ (No load)}$$

Armature reaction causes increase in speed and causes instability as the core saturates near the poles. Can be reduced with compensating winding which is in series with the armature coil.

$$K\Phi\omega=E_A$$

$$K\Phi I_A=\tau$$

For shunt motor

$$K\Phi=\frac{V_T-R_AI_A}{\omega}$$

$$\tau=K\Phi I_A=\frac{V_T-R_AI_A}{\omega}I_A$$

Assume no saturation, speed locked(?):

This doesn't seem right. We are meant to use the machine constant and the proportionality of current to magnetic flux.

$$\frac{E_{A2}}{E_{A1}}=\frac{I_{f2}}{I_{f1}}$$