Efficient Phasor Analysis for AC Systems Simplification
Explore the fundamentals of phasor analysis to simplify constant frequency AC systems. Learn about phasor representation, complex power, reactive compensation, power factor correction, and balanced 3-phase systems. Understand the advantages and applications of 3-phase power distribution.
Efficient Phasor Analysis for AC Systems Simplification
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Presentation Transcript
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
