diff --git a/README.html b/README.html new file mode 100644 index 0000000..33a5cc8 --- /dev/null +++ b/README.html @@ -0,0 +1,537 @@ + + + + + + + + + + + + +
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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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Y-Δ\Delta transformation (Balanced case)

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ZΔ=3ZYZ_\Delta=3Z_Y

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Types of power factors (From ENSC2003)

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Where Sˉ=∣Sˉ∣∠φ\bar{S}=|\bar{S}|\angle\varphi:

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φ=arctan⁡(QP)=θv−θi \varphi = \arctan\left(\frac{Q}{P}\right) = \theta_v-\theta_i

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LaggingLeadingUnity
VoltageCurrent behindCurrent aheadIn phase
Load typeInductiveCapacitiveResistive
QQQ>0Q>0Q<0Q<0Q=0Q=0
φ\varphiφ>0°\varphi>0°φ<0°\varphi<0°φ=0°\varphi=0°
PF [Load][0,1)[0,1)[0,1)[0,1)11
PF [Source][0,−1)[0,-1)[0,−1)[0,-1)−1-1
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Power types in induction motor

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TypeDescriptionEquivalent terms
Input powerPower into machine. VT=V3ϕV_T=V_{3\phi}, IL=I3ϕI_L=I_{3\phi}PinP_\text{in}, 3VTILcos⁡(θ)\sqrt{3}V_TI_L\cos(\theta)
Output powerMechanical output power of the machine, excludes lossesPoutP_\text{out}, PloadP_\text{load}
Converted powerTotal electrical power converted to mechanical power, includes useful power and mechanical losses inside machinePconvP_\text{conv}, PconvertedP_\text{converted}, PmechP_\text{mech}, PdevelopedP_\text{developed}, τmech×ωm\tau_\text{mech}\times\omega_m
Airgap powerPower transmitted over airgap.PAGP_\text{AG}, τmech×ωs\tau_\text{mech}\times\omega_s
Mechanical lossPower lost to friction and windagePmechanical lossP_\text{mechanical loss}, PF&WP_\text{F\\\&W}, Pfriction and windageP_\text{friction and windage}
Core lossPower lost in machine magnetic material due to hysteresis loss and eddy currentsPcoreP_\text{core}
Rotor copper lossDue to resistance of rotor windingsPrP_r, PRCLP_\text{RCL}
Stator copper lossDue to resistance of stator windingsPsP_s, PSCLP_\text{SCL}
Miscellaneous lossAdd 1% to losses to account for other unmeasured lossesPmiscP_\text{misc}, PstrayP_\text{stray}
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Pin=PSCL+PRCL+Pcore+PF&W+Pmisc+PoutPAG=PRCL+PF&W+Pmisc+PoutPmech=PF&W+Pmisc+Pout +\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} +

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Note - assume loss is 0 if not mentioned!

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TypeDescriptionSymbols
Load torque, Shaft torqueTorque experienced by load after all lossesτload\tau_\text{load}, τshaft\tau_\text{shaft}
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3ϕ3\phi induction motor

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Etc.

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Diagram

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Equivalent model

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Assumptions

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Diagram

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DC test

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Δ\Delta machine

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Rs=32⋅VDC,3ϕIDC,3ϕR_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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Rs=12⋅VDC,3ϕIDC,3ϕR_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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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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Diagram

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Ignore magnetizing path

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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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EA=V1ϕ−IA(RA+jXs)E_A=V_{1\phi}-I_A(R_A+j X_s)

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IA=CONJUGATE(∣S3ϕ∣∠arccos⁡(x)3V1ϕ),{x=+PFlaggingx=−PFleadingI_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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VR=∣VNL∣−∣VFL∣∣VFL∣=∣EA∣−∣V1ϕ,rated∣∣V1ϕ,rated∣\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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No-loadFull-load
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Power factorVoltage regulation
LaggingPositive
UnityNear 0
LeadingNegative
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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 testShort-circuit test
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Power flow

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PoutP_\text{out} is the rated power

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Pout=Srated×PFP_\text{out}=S_\text{rated}\times \text{PF}

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Pin=Pcopper+Pcore+PF&W+Pmisc+PoutPmech=PF&W+Pmisc+Pout +\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} +

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Magnetic circuit analogy

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Magnetic circuitnameElectrical circuitname
F\mathcal F
Magnetomotive force [A-turn]
E\mathcal E
Electromotive force [V]
R\mathcal R
Reluctance [1/H]
RR
Resistance [Ω\Omega]
Φ\Phi
Magnetic flux [Wb]
II
Current [A]
P=1R\mathcal P=\frac{1}{\mathcal R}
Permeance [H]
G=1RG=\frac{1}{R}
Conductivity [℧\mho]
F=ΦR\mathcal F=\Phi\mathcal R
Hopkinson's law
V=IRV=IR
Ohm's law
R=lμA\mathcal R=\frac{l}{\mu A}
R=lσAR=\frac{l}{\sigma A}
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Transformers

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ZP=ZS(NPNS)2=ZSn2Z_P=Z_S\left(\frac{N_P}{N_S}\right)^2=Z_S n^2

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Maximum power.

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

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Rload=∣Rsrc2+jXsrc2∣R_\text{load}=|{R_\text{src}}^2+j{X_\text{src}}^2|

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Parameter identification

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Open-circuit testShort-circuit test
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Voltage regulation

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VR=∣VNL,P∣−∣Vrated,P∣∣Vrated,P∣=∣Vin∣−∣Vrated,P∣∣Vrated,P∣\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}|}

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

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Voltage regulation is typically small.

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∣Vin∣=∣Vrated,P+IL,P⋅Zˉ∣|V_\text{in}|=|V_\text{rated,P}+I_\text{L,P}\cdot\bar Z|

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DC machine

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Separately excited machineShunt excitedSeries excited
Similar torque-speed characteristic to separately-excited machineHigh torque per ampere. Used in high-torque applications
Requires two independent voltage sourcesDo not run unloaded - infinite speed at 0 torque as ω∝1/τ\omega\propto 1/\sqrt{\tau}
Motor control using RfR_fMotor control using RFR_FMotor control using VTV_T.
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Starting DC motors

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RAR_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 EAE_A.

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Magnetizating curve

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When a question specifies the field current or RadjR_\text{adj}, refer to magnetization curve. Magnetizating curve is valid at a specific speed nm1n_{m1}, and the curve is used to find EA1E_{A1}. Using the load condition to find the armature current IA=τind/(KΦ)I_A=\tau_\text{ind}/(K\varPhi), VAV_A can be used to find a second induced EMF EA2E_{A2}. Using EA2E_{A2} find the speed nm2n_{m2} by scaling nm1n_{m1} by EA2/EA1E_{A2}/E_{A1}.

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Idk

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Pmech=EAIAP_\text{mech}=E_AI_A

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

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EA=VA (No load)E_A=V_A\text{ (No load)}

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IA=0 (No load)I_A=0\text{ (No load)}

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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.

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KΦω=EAK\Phi\omega=E_A

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KΦIA=τK\Phi I_A=\tau

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For shunt motor

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KΦ=VT−RAIAωK\Phi=\frac{V_T-R_AI_A}{\omega}

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τ=KΦIA=VT−RAIAωIA\tau=K\Phi I_A=\frac{V_T-R_AI_A}{\omega}I_A

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Assume no saturation, speed locked(?):

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

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EA2EA1=If2If1\frac{E_{A2}}{E_{A1}}=\frac{I_{f2}}{I_{f1}}

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+``` + > Why are the drawings bad? I draw them with a mouse @@ -43,7 +48,7 @@ $$ \varphi = \arctan\left(\frac{Q}{P}\right) = \theta_v-\theta_i$$ | 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 | $P_\text{misc}$, $P_\text{stray}$ | -![](2022-10-25-11-33-40.png) +
$$ \begin{align} @@ -72,7 +77,7 @@ Note - assume loss is 0 if not mentioned! ### Diagram -![](2022-10-26-22-06-19.png) +
### Equivalent model @@ -87,19 +92,21 @@ Note - assume loss is 0 if not mentioned! ### Diagram -![](2022-10-26-21-53-13.png) +
### DC test #### $\Delta$ machine $$R_s=\frac{3}{2}\cdot\frac{V_{\text{DC},3\phi}}{I_{\text{DC},3\phi}}$$ -![](2022-10-26-22-43-25.png) + +
#### Y machine $$R_s=\frac{1}{2}\cdot\frac{V_{\text{DC},3\phi}}{I_{\text{DC},3\phi}}$$ -![](2022-10-26-22-48-09.png) + +
### No-load test @@ -114,7 +121,7 @@ $$R_s=\frac{1}{2}\cdot\frac{V_{\text{DC},3\phi}}{I_{\text{DC},3\phi}}$$ Using assumptions, remove rotor part of circuit and only consider stator and magnetizing path. -![](2022-10-25-11-45-26.png) +
### Blocked rotor test @@ -133,7 +140,7 @@ Using assumptions, remove rotor part of circuit and only consider stator and mag Ignore magnetizing path -![](2022-10-25-11-46-04.png) +
--- @@ -141,19 +148,19 @@ Ignore magnetizing path ### Diagram -![](2022-10-26-21-47-29.png) +
### Blocked-rotor #### Diagram -![](2022-10-26-21-48-00.png) +
### No-load #### Diagram -![](2022-10-26-21-47-49.png) +
## Synchronous machine @@ -170,9 +177,9 @@ $$\text{VR}=\frac{|V_\text{NL}|-|V_\text{FL}|}{|V_\text{FL}|}=\frac{|E_A|-|V_{1\ - 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 | -| ---------------------------- | ---------------------------- | -| ![](2022-10-27-20-15-13.png) | ![](2022-10-27-20-19-47.png) | +| No-load | Full-load | +| ---------------------------------------------------------------- | ---------------------------------------------------------------- | +|
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| | Power factor | Voltage regulation | | ------------ | ------------------ | @@ -184,9 +191,9 @@ $$\text{VR}=\frac{|V_\text{NL}|-|V_\text{FL}|}{|V_\text{FL}|}=\frac{|E_A|-|V_{1\ #### **Note** - double-check if the axis refers to per-phase or line voltage/current. -| Open-circuit test | Short-circuit test | -| ---------------------------- | ---------------------------- | -| ![](2022-10-27-15-31-49.png) | ![](2022-10-27-15-32-07.png) | +| Open-circuit test | Short-circuit test | +| ----------------------------------------------------------------- | ---------------------------------------------------------------- | +|
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| ### Power flow @@ -227,11 +234,11 @@ $$R_\text{load}=|{R_\text{src}}^2+j{X_\text{src}}^2|$$ #### Parameter identification -![](2022-10-28-15-52-14.png) +
-| Open-circuit test | Short-circuit test | -| ---------------------------- | ---------------------------- | -| ![](2022-10-28-15-53-29.png) | ![](2022-10-28-15-52-58.png) | +| Open-circuit test | Short-circuit test | +| ---------------------------------------------------------------- | ---------------------------------------------------------------- | +|
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| #### Voltage regulation @@ -243,16 +250,16 @@ Voltage regulation is typically small. $$|V_\text{in}|=|V_\text{rated,P}+I_\text{L,P}\cdot\bar Z|$$ -![](2022-10-28-16-30-51.png) +
### DC machine -| Separately excited machine | Shunt excited | Series excited | -| ---------------------------------------- | ----------------------------------------------------------------- | --------------------------------------------------------------------------------- | -| ![](2022-10-28-18-21-53.png) | ![](2022-10-28-18-22-17.png) | ![](2022-10-28-18-28-51.png) | -| | 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 $V_T$. | +| Separately excited machine | Shunt excited | Series excited | +| ---------------------------------------------------- | ----------------------------------------------------------------- | --------------------------------------------------------------------------------- | +|
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| +| | 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