Electrotechnology(Electrical Technology) - APS

APS, EE
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1. Circuit

  • built for power or signal transmission

  • real to ideal

  • closed circuit
  • open circuit
  • short circuit

1.1 Basic Components

  • U : Voltage, V
  • $V_x$ : Electric potential to some point
  • R : linear resistance, Ohm $\Omega$
    • series : sum
    • parallel : $\frac{1}{R}=\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}\cdots$
  • I : Current, Ampere A
  • P : Power, Watt
    • >0 load
    • <0 power supply

1.2 Power

  • power source : Electromotive force

  • current source

1.3 Kirchhoff

  • KCL current law : node $\sum i=0$

  • KVL voltage law : circle $\sum u=0$

  • Branch Current Method : use KCL and KVL

    • KCL : node-1 个
    • KVL : branch 个

1.4 Superposition principle

1.5 Thevenin’s Theorem & Norton Theorem

  • TT : Any combination of batteries and resistances with two terminals to power source and resistence

  • NT : to current source

2. Transient analysis

2.1 RLC

  • $L=\dfrac{\Psi({\rm Flux})}{i}$ : inductance, Henry H

    • $\Psi=n\Phi$Flux Weber, Wb
    • $e=-L\dfrac{di}{dt}=u$
    • $W=\int\limits^t_0pdt=\int\limits^t_0uidt\=\int\limits^i_0Lidi=\frac{1}{2}Li^2$
    • save energy to magnetic

  • $C=\dfrac{q}{u}$ : capacitance, Farad F

    • $i=C\dfrac{du}{dt}$
    • $W=\frac{1}{2}Cu^2$
    • save to the Electric field energy

2.2 RC

  • Zero input response : power=0

    • Differential equation : $RC\dfrac{du_c}{dt}+u_c=0$

    • $u_c=\exp(-\dfrac{t}{RC})$

    • Time constant : $\tau=RC$

    • >5$\tau$ stable

  • Zero state response : energy storage element u/i=0

    • $RC\frac{du_c}{t}+u_c=U$
    • $u_c=K-\exp(-\dfrac{t}{RC})$, K putin
    • $u_c=U-\exp(-\dfrac{t}{RC})$

RL : $\tau=\dfrac{L}{R}$

2.3 First Order Response

only 1 energy storage element

$$x(t)=x(\infty)+[f(0_+)-f(\infty)]e^{-t/\tau}$$

  • Initial condition : $i_L(0_+)=i_L(0_-)\u_c(0_+)=u_c(0_-)$

3. Sinusoidal alternating current

3.1 Basic concepts

$$e=E_m\sin(wt+\psi_e)\u=U_m\sin(wt+\psi_u)\i=I_m\sin(wt+\psi_i)$$

  • w : angular frequency (unit : radians per second)
  • $U_m/I_m$ : peak voltage/current
  • $\psi$ : phase
  • $\phi=\psi_1-\psi_2$ : phase shift
    • >0 : 2 lead 1, 1 lagging
    • <0 : 1 lead 2
    • 180 : mirror-image
  • Valid value
    • $A=\dfrac{A}{\sqrt{2}}$

3.2 Phasor Diagrams

use complex number

$$u=U_m\sin(wt+\psi)\Longleftrightarrow\dot{U_m}=U_me^{j\psi}=U_m\angle\psi$$

  • $R =R\angle 0$
    • real power :$P=U^2/R$
  • L : $X_L$
    • $\dfrac{U_m}{I_m}\angle90=jX_L=jwL$
    • P=0
    • reactive power : $Q=I^2X_L$ (unit : var$
  • C : $X_C$
    • $\dfrac{U_m}{I_m}\angle-90=-jX_C=\dfrac{1}{jwC}$
    • P=0
    • $Q=-I^2X_C$

3.3 Impedance

$$Z=R+jX, {\rm unit :} \Omega$$

  • series : same with resistance
  • parallel : same with resistance

$$P=UI\cos\phi$$

  • $\cos\phi$ : Power factor, $\Uparrow$ effective
  • parallel C on L

$$Q=I^2X=I^2(X_L-X_C)=UI\sin\phi$$

$$S=UI=\sqrt{P^2+Q^2}$$

3.4 Three-phase electric power

phase differ 120 degree

$$u_1=U_m\sin wt\u_2=U_m\sin (wt-120)\u_3=U_m\sin(wt+120)$$

$$\dot{U_1}=U\angle 0\\dot{U_2}=U\angle 120\\dot{U_3}=U\angle -120$$

3.4.1 Y Configuration

  • 4 wires : 3 phase and 1 neutral(0)

  • phase voltage : phase to 0

  • line voltage : phase 12, 23, 31

3.4.2 Load

  • 1 phase load

  • 3 phases load

    • when parity

      $P=3U_pI_p\cos\phi_p=\sqrt 3U_LI_L\cos\phi_p$

      p means load

    • $Q=\sqrt 3U_LI_L\sin\phi_p$

    • $S=3U_pI_p=\sqrt 3U_LI_L$

4. Transformer

4.1 Concepts

  • B : Magnetic Induction, 1 Teslas= 1 Wb/m2

  • H : magnetic field, 1 Amperes per meter A/m

  • $\Phi$ : magnetic flux, 1 Weber =1 Vs

  • $\mu=B/H$ : magnetic permeability

    • $\mu_0=4\pi\times10^{-7} {\rm H/m}$ : vaccum
    • $\mu_r=\frac{\mu}{\mu_0}$ : relative

4.2 Material

  • Saturation

    • non magnetic material : linear
    • magnetic material : non-linears

  • Hysteresis : B lagging change after H when AC

4.3 Ampere loop theorem

$$\oint Hdl=\sum I$$

$$H_x=\dfrac{NI}{2\pi x}$$

4.4 Iron core coil

$e=-N\dfrac{\Phi}{dt}\e_\sigma=-N\dfrac{d\Phi_\sigma}{dt}=-L_\sigma\dfrac{di}{dt}$

  • $\Phi_\sigma$ : Leakage flux

$\Downarrow u=-e-e_\sigma+Ri$

$e=-N\frac{d}{dt}(\Phi\sin wt)=2\pi fN\Phi_m\sin(wt-90)$

$U\approx E=4.44fN\Phi_m$

$P=UI\cos \phi=I^2R+\Delta P_{Fe}$

  • $\Delta P_{Fe}=\Delta P_h+\Delta P_e$
    • $\Delta P_h$ hysteresis $\rightarrow$heat
    • $\Delta P_e$ vortex

4.5 Transformer

  • core
  • primary winding
  • secondary winding

when 0 load

$$\dfrac{U_1}{U_2}\approx\dfrac{E_1}{E_2}=\dfrac{N_1}{N_2}=K$$

4.5.1 3 phases transformer

4.5.2 Y/Y Y/Triangle

4.5.3 Load

$|Z_1|=K^2|Z_2|$

4.5.4 Parameters

  • $U_{1N},U_{2N}$
  • $I_{1N},I_{2N}$

4.5.5 Special Transformer

  • Autotransformer

  • VT Voltage t

  • CT Current t

4.5.6 Series and parallel

5. Three-phase asynchronous motor

5.1 Principle

$$n_0=\dfrac{60f_1}{p}$$

  • slip rate : $s=\dfrac{n_0-n}{n_0}$

5.2 Mechanical properties

  • rated torque $T_N=9550\dfrac{P_{2N}(\rm KW)}{n_N(\rm r/min)}$
  • max torque $T_{max}$
    • $s=s_m$
    • Overload coefficient : $\lambda=\dfrac{T_{max}}{T_N}$
  • starting torque $T_{st}$
    • n=0
    • s=1

5.3 Influence

  • $U_1$

  • $R_2$

5.4 Control

5.4.1 Starting

  • direct

  • low voltage (Squirrel cage motor)

    • Y-$\Delta$

    • auto coupling

      • $I_{st}’=x^2I_{st},T_{st}’=x^2T_{st},$

  • add resistance (wound rotor induction motor)

5.4.2 Speed regulation

$n=(1-s)n_0=(1-s)\dfrac{60f_1}{p}$

  • f
  • p
  • s

5.4.3 Stop

  • power
  • reverse
  • feedback

6. Relay

6.1 Types

  • Combination Switch

  • button

    • Normally closed button
    • normally open button
    • combination

  • Air Conditioner Contactor

  • Intermediate relay : for information PLC

  • thermal relay

6.2 Squirrel cage motor starting control circuit

  • SB1 stop
  • SB2 start

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