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2.0 Transformers, P = IV, P = IV, 变形金刚很重（Important）是因为它的构造（construction）太过…
2.0 Transformers
2.3 The ideal transformer (Pg 13)
Assumption
No magnetic flux leaks out between the two windings
The winding resistance is zero
It is infinitely easy to set up a magnetic flux in the core
the reluctance of the core is zero
the magnetizing current required is almost zero
core permeability is infinite
There is no hysteresis or eddy-current power loss in the core
Formula
Power, P in transformer
apparent power, S = VI
power factor, pf = cosθ
reactive power, Q = VI cosθ
Impedance transformation through a transformer
Apparent impedance of the primary circuit if trasformer
Z'L = Vp / Ip = a^2 ZL
Load from secondary part
ZL = Vs / Is
Thevenin equivalent of transformer circuit
Primary part
Equivalent circuit when secondary impedance is transferred to primary side
Secondary part
Equivalent circuit when primary source is transferred to secondary side
Relationship between vp&vs and ip&is
vp(t) / vs(t) = Np / Ns = a
ip(t) / is(t) = Ns / Np = 1/a
2.1 Why transformers are important to modern life (Pg 4)
After transformer
Eliminated these restriction on the range and power level of power systems
it changes one AC voltage level to another level voltage without affecting the actual power supplied
If the transformer steps up the voltage, it must decrease the current to keep the power into the device equal to the power out of it.
Transmission losses
transmission losses in the lines of a power system are proportional to the square of the current in the lines
P = I^2 R
Previous problem
Power system generated and transmitted power at such low voltages that very large currents were necessary to supply significant amounts of power
High currents caused huge voltage drops and power losses in transmission line
2.2 Type and construction of transformers (Pg 7)
Type of construction
Core form
Consists a simple rectangular laminated piece of steel
with transformer windings wrapped around two sides of the rectangle
Shell form
Consists three-legged laminated core
with the winding wrapped around the center leg
Type of special-purpose transformer
Potential transformer
Sampling a high voltage and produce a low secondary voltage proportionally
To handle very small current
Current transformer
Provide much smaller secondary current than but directly proportional to its primary current
2.4 Theory of operation of real single phase transformers (Pg 30)
Characteristic
The magnetizing current is not zero
but it may be as little as 3% of the load currents
There is power loss in the iron core
due to hysteresis and eddy current effects
Flux leakage between the windings
leading to inductive reactance effects
Capacitive effects are present in high voltage transformers
affecting their ability to withstand strikes of lightning
Resistance of the windings
Formula
Voltage supply (Faraday's Law)
Primary
vp(t) = e(P) + e(LP)
Secondary
vs(t) = e(S) + e(LS)
Excitation current, iex
i(ex) = im + i(h+e)
Induced emf (Faraday's Law)
e(ind) = N dΦ/dt
Average flux in the core
arrange and integration of vp(t) = Np dΦ/dt
Φ = (Vm/ωNp) sin ωt
2.6 The equivalent circuit of a transformers (Pg 41)
Primary part
resistance, Req = Rp + a^2*Rs
reactance, Xeq = Xp + a^2*Xs
Secondary part
resistance, Req = Rp/a^2 + Rs
reactance, Xeq = Xp/a^2 + Xs
experimentally determine the values of the reactance and resistances in the transformer model
Open circuit test
rectangular form, Y(E) = 1/R(C) + j 1/X(M)
polar form, Y(E) = [ I(OC)/V(OC) ] < - cos-1 PF
Short circuit test
rectangular form, Z(SE) = Req + jXeq
polar form, Z(SE) = [ I(SC)/V(SC) ] < cos-1 PF
2.5 Transformer losses (Pg 40)
Hysteresis losses
associated with the rearrangement of the magnetic domains in core
Eddy current losses
resistive heating losses in the core
Leakage flux
fluxes ΦLP and ΦLS that escape the core
Copper (I^2R) losses
resistive heating losses in the primary and secondary windings
2.8 Transformer voltage regulation and efficiency (Pg 59)
In secondary circuit
Vp/a = Vs + Req Is + j Xeq Is
Transformer efficiency, η
η = {(Vs Is cosθ) / [Vs Is cosθ + I^2 R(eq,s) + (Vs^2 / Rc)]} x 100%
Voltage Regulation, VR
VR = {[V(s, nl) - V(s, fl)] / V(s, fl)} x 100%
2.7 The per-unit system (Pg 55)
Per unit basis equation
quantity per unit = actual value / base value of quantity
In single-phase
Reactive Power, Q(base)
Apparent Power, S(base)
Impedance, Z(base) = V(base) / I(base) = [V(base)]^2 / S(base)
Admittance, Y(base) = I(base) / V(base)
S(base) = V(base)*I(base)
Power, P(base)
2.9 Three phase transformer (Pg 65)
P = IV
P = IV
变形金刚很重（Important）是因为它的构造（construction）太过完美（ideal）
不过身边有一只真的喜欢运动的单身狗（operation real single...）流了很多汗（losses）导致电路损坏（circuit）
所以就分件拿去卖（per-unit system）有调节电压和效率高的比较好卖（voltage regulation and efficiency）卖了三个部分（three phase）
Φ = Ni / R
R is reluctance of circuit = lc / μA
P = VI cosθ
Q = S cosθ
Vp = a Vs
Ip = Is / a
vp(t) = Np dΦM/dt + Np dΦ(LP)/dt
Φ = ΦM + ΦL
im = magnetization current to produce flux in transformer core
i(h+e) = Core-loss current to make up for hysteresis and eddy current losses
Vp(t) = Vm cos ωt
PF = P(OC)/[V(OC)*I(OC)]
Negative θ
PF = P(SC)/[V(SC)*I(SC)]
Positive θ
