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Review. Review of Phasors. Goal of phasor analysis is to simplify the analysis of constant frequency ac systems: v ( t ) = V max cos ( w t + q v ), i ( t ) = I max cos ( w t + q I ), where: v ( t ) and i ( t ) are the instantaneous voltage and current as a function of time t ,
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Review of Phasors Goal of phasor analysis is to simplify the analysis of constant frequency ac systems: v(t) = Vmaxcos(wt + qv), i(t) = Imax cos(wt + qI), where: • v(t) andi(t) are the instantaneous voltage and current as a function of time t, • w is the angular frequency (2πf, with f the frequency in Hertz), • Vmax andImax are the magnitudes of voltage and current sinusoids, • qv and qI are angular offsets of the peaks of sinusoids from a reference waveform. Root Mean Square (RMS) voltage of sinusoid:
Phasor Analysis (Note: Z is a complex number but not a phasor).
Complex Power (Note: S is a complex number but not a phasor.)
example ZL=jwL=j*1000*1*10^-3 =j
Example Power flowing from source to load at bus Earlier we found I = 20-6.9 amps = 1600W + j1200VAr
Example First solve basic circuit I
Example, cont’d Now add additional reactive power load and re-solve, assuming that load voltage is maintained at 40 kV.
Power System Notation Power system components are usually shown as “one-line diagrams.” Previous circuit redrawn. Arrows are used to show loads Transmission lines are shown as a single line Generators are shown as circles
Reactive Compensation Key idea of reactive compensation is to supply reactive power locally. In the previous example this can be done by adding a 16 MVAr capacitor at the load. Compensated circuit is identical to first example with just real power load. Supply voltage magnitude and line current is lower with compensation.
Reactive Compensation, cont’d • Reactive compensation decreased the line flow from 564 Amps to 400 Amps. This has advantages: • Lines losses, which are equal to I2 R, decrease, • Lower current allows use of smaller wires, or alternatively, supply more load over the same wires, • Voltage drop on the line is less. • Reactive compensation is used extensively throughout transmission and distribution systems. • Capacitors can be used to “correct” a load’s power factor to an arbitrary value.
Balanced 3 Phase () Systems • A balanced 3 phase () system has: • three voltage sources with equal magnitude, but with an angle shift of 120, • equal loads on each phase, • equal impedance on the lines connecting the generators to the loads. • Bulk power systems are almost exclusively 3. • Single phase is used primarily only in low voltage, low power settings, such as residential and some commercial. • Single phase transmission used for electric trains in Europe.
Advantages of 3 Power • Can transmit more power for same amount of wire (twice as much as single phase). • Total torque produced by 3 machines is constant, so less vibration. • Three phase machines start more easily than single phase machines.
Three Phase - Wye Connection • There are two ways to connect 3 systems: • Wye (Y), and • Delta ().
Vcn Vab Vca Van Vbn Vbc Wye Connection Line Voltages -Vbn (α = 0 in this case) Line to line voltages are also balanced.
Wye Connection, cont’d • We call the voltage across each element of a wye connected device the “phase” voltage. • We call the current through each element of a wye connected device the “phase” current. • Call the voltage across lines the “line-to-line” or just the “line” voltage. • Call the current through lines the “line” current.
Ic Ica Ib Iab Ibc Ia Delta Connection
Three Phase Example Assume a -connected load, with each leg Z = 10020W, is supplied from a 3 13.8 kV (L-L) source
