EEN320 - Power Systems I ( Συστήματα Ισχύος Ι ) Part 2: Single-phase and three-phase AC systems Dr Petros Aristidou Department of Electrical Engineering, Computer Engineering & Informatics Last updated: January 17, 2020
Recap from last lecture - Overview power system , ΕΕΝ 320 — Dr Petros Aristidou — Last updated: January 17, 2020 2/ 84
Learning objectives After this part of the lecture and additional reading, you should be able to . . . . . . calculate complex voltages, currents and powers in single-phase 1 systems using phasors; . . . explain standard configurations of three-phase power systems; 2 . . . calculate complex voltages, currents and powers in balanced 3 three-phase systems using the concepts of phasors and single-phase equivalent circuits. , ΕΕΝ 320 — Dr Petros Aristidou — Last updated: January 17, 2020 3/ 84
Review: Calculations with complex numbers Consider a complex number z = a + jb , where a ∈ R , b ∈ R and j is the imaginary unit satisfying j 2 = − 1 Complex conjugate of z is defined as z ∗ = a − jb Absolute value | z | and argument φ of z are defined as � b � � a 2 + b 2 , | z | = ϕ = arctan a (if a > 0 which is usually the case in power systems; otherwise use atan2-function to determine ϕ ) Polar representation of z via Euler’s formula z = | z | e j ϕ = | z | ϕ Conversion of radians to degrees: 1 rad = 57.3 degrees , ΕΕΝ 320 — Dr Petros Aristidou — Last updated: January 17, 2020 4/ 84
Some general assumptions In this part of the lecture, we will consider single- and three-phase AC networks We will do this under the following assumptions The network only contains passive elements ( R , L , C ) and purely sinusoidal current and voltage sources of identical frequencies The network is in steady-state Example single-phase circuit: R L ˆ ˆ V a sin( θ a + ω t ) C V b sin( θ b + ω t ) , ΕΕΝ 320 — Dr Petros Aristidou — Last updated: January 17, 2020 6/ 84
1 Single-phase AC waveforms Instantanteous values ( στιγμιαίες τιμές ) of voltage and current at a network element given by v ( t ) = ˆ V cos( ω t ) i ( t ) = ˆ I cos( ω t − ϕ ) V and ˆ ˆ I are the amplitudes ( μέγιστη τιμή ) of the respective waveforms ( κυμματομορφές ) ϕ is the phase shift ( διαφορά φάσης ) between voltage and current ω = 2 π f , where f is the stationary ( στάσιμη ) electrical frequency of the network (e.g., 50 Hz in Europe; 60 Hz in US) Instead of peak-to-peak amplitudes ˆ V and ˆ I , often root-mean-square (also called effective) amplitudes V and I used √ √ ˆ ˆ V = 2 V , I = 2 I , ΕΕΝ 320 — Dr Petros Aristidou — Last updated: January 17, 2020 8/ 84
1 Single-phase AC waveforms - Example (time on x-axis) v ( t ) i ( t ) 1 ω = 2 π f 0 f = 50 Hz ϕ = π 4 − 1 0 10 20 30 40 t [ms] , ΕΕΝ 320 — Dr Petros Aristidou — Last updated: January 17, 2020 9/ 84
1 Single-phase AC waveforms - Example (phase on x-axis) v ( t ) i ( t ) 1 ω = 2 π f 0 f = 50 Hz ϕ = π 4 − 1 0 π 2 π 3 π ω t [rad] , ΕΕΝ 320 — Dr Petros Aristidou — Last updated: January 17, 2020 10/ 84
1 Power in AC single-phase systems Instantaneous single-phase AC power p ( t ) = v ( t ) i ( t ) = ˆ V ˆ I cos( ω t ) cos( ω t − ϕ ) (On board) , ΕΕΝ 320 — Dr Petros Aristidou — Last updated: January 17, 2020 11/ 84
1 Individual components of instantaneous power p ( t ) 2 p 1 ( t ) p 2 ( t ) 1 . 5 1 0 . 5 0 − 0 . 5 0 10 20 30 40 t [ms] p ( t ) = 1 + 1 V ˆ ˆ V ˆ ˆ I cos( ϕ )( 1 + cos( 2 ω t )) I sin( ϕ ) sin( 2 ω t ) 2 2 � �� � � �� � p 1 ( t ) p 2 ( t ) , ΕΕΝ 320 — Dr Petros Aristidou — Last updated: January 17, 2020 12/ 84
1 Active power in single-phase systems p 1 ( t ) is oscillating between 0 and ˆ V ˆ I cos( ϕ ) p 1 ( t ) never changes sign → always flows in same direction The average value over time of p 1 ( t ) is T � P = 1 p ( t ) dt = 1 V ˆ ˆ I cos( ϕ ) = VI cos( ϕ ) T 2 0 P is called active power ( ενεργός ισχύς ) P is the only ”useful” component of p ( t ) cos( ϕ ) is called power factor ( συντελεστής ισχύος ) ϕ is called power factor angle The unit of P is the Watt [W] , ΕΕΝ 320 — Dr Petros Aristidou — Last updated: January 17, 2020 13/ 84
1 Reactive power in single-phase systems p 2 ( t ) is oscillating between ± ˆ V ˆ I sin( ϕ ) Average value over time of p 2 ( t ) is zero → no ”useful” work The amplitude of the waveform p 2 ( t ) is Q = 1 V ˆ ˆ I sin( ϕ ) = VI sin( ϕ ) 2 Q is called reactive power ( άεργος ισχύς ) The unit of Q is the Volt-Ampere-reactive [Var] (also used: [var]) , ΕΕΝ 320 — Dr Petros Aristidou — Last updated: January 17, 2020 14/ 84
1 Reactive power in RLC circuits In RLC circuits, Q appears because of the presence of inductors and capacitors In fact, Q is the time derivative of the energy stored in inductors and capacitors These elements continuously accumulate and release energy They never release more energy than they have accumulated (that’s why they are also called passive elements) → The energy is always nonnegative Important: in general networks, it is far more difficult to associate a clear physical meaning to Q , ΕΕΝ 320 — Dr Petros Aristidou — Last updated: January 17, 2020 15/ 84
2 Phasors in electrical power systems Sinusoidal waveforms can also be represented by phasors in the complex plane Phasors are very popular in electric power systems Main reasons: simplify visualisation and calculation of electrical networks This is very useful for analysis, design and operation of power systems , ΕΕΝ 320 — Dr Petros Aristidou — Last updated: January 17, 2020 17/ 84
2 Definition of a phasor ( φασιθέτης ) Consider x ( t ) = ˆ X cos( ω t + θ ) Via Euler’s Formula, we define the phasor corresponding to x ( t ) as 1 ˆ X Xe j θ X = √ (cos( θ ) + j sin( θ )) = X (cos( θ ) + j sin( θ )) = 2 � �� � � �� � exponential form trigonometric form Then √ 2 ℜ{ Xe j ω t } , x ( t ) = i.e., momentary value of x ( t ) corresponds to real part of the phasor X rotating at angular speed ω Alternative common notation for a phasor X = Xe j θ = X θ � �� � angular form 1 Here j denotes the imaginary unit. , ΕΕΝ 320 — Dr Petros Aristidou — Last updated: January 17, 2020 18/ 84
2 Voltages and currents as complex phasors Phasors of voltage and current V = V (cos( ϕ v ) + j sin( ϕ v )) = Ve j ϕ v = V ϕ v I = I (cos( ϕ i ) + j sin( ϕ i )) = Ie j ϕ i = I ϕ i ϕ = ϕ v − ϕ i Note: as we have assumed stationary conditions, it suffices to use X to describe x ( t ) for network calculations Why? Because the term e j ω t cancels out, whenever multiplying two complex quantities � Ve j ω t � � Ie j ω t � ∗ = V I ∗ e j ω t e − j ω t = V I ∗ , where the operator ∗ denotes complex conjugation , ΕΕΝ 320 — Dr Petros Aristidou — Last updated: January 17, 2020 19/ 84
2 Visualization of a phasor V = Ve j θ = V θ � �� � angular form � J. Corda c starting from the origin 0 + j 0 1 2 v ( t ) projection on the real axis is √ the phasor is the position at t = 0 of the rotating vector , ΕΕΝ 320 — Dr Petros Aristidou — Last updated: January 17, 2020 20/ 84
2 Phasor diagrams Phasor diagrams ( φασικά διαγράμματα ): A graphical representation of the phasors ω V ϕ v ϕ i I O , ΕΕΝ 320 — Dr Petros Aristidou — Last updated: January 17, 2020 21/ 84
3 Complex apparent power in single-phase systems Now, we can introduce a third important quantity in power systems - the complex apparent power S = V I ∗ = VIe j ( ϕ v − ϕ i ) = VIe j ϕ = VI (cos( ϕ ) + j sin( ϕ )) Remember that ϕ = ϕ v − ϕ i is called the power factor angle and it’s connected to power factor as PF = cos ϕ The absolute value of the complex apparent power is called apparent power S S = | S | = VI The unit of S and S is Volt-Ampere [VA] Apparent power used to dimension equipment S = VI ⇒ S = P if ϕ = 0 , ΕΕΝ 320 — Dr Petros Aristidou — Last updated: January 17, 2020 23/ 84
3 Relation between S , P and Q Active power P corresponds to real part of S P = ℜ{ S } = VI cos( ϕ ) Reactive power Q corresponds to imaginary part of S Q = ℑ{ S } = VI sin( ϕ ) Hence S = P + jQ and � P 2 + Q 2 S = | S | = p ( t ) , P Watt W kW, MW Q Var (VAr, Var, var) kvar, Mvar S Volt-Ampere VA kVA, MVA , ΕΕΝ 320 — Dr Petros Aristidou — Last updated: January 17, 2020 24/ 84
3 Power triangle in the complex plane ℑ S Q ϕ ℜ P Power factor cos( ϕ ) = P P | S | = √ P 2 + Q 2 , ΕΕΝ 320 — Dr Petros Aristidou — Last updated: January 17, 2020 25/ 84
3 Power conventions For network calculations, it is important to determine whether an element absorbs or delivers power There are 2 conventions: Load convention (standard) Generator convention i ( t ) i ( t ) v ( t ) v ( t ) Current counted positively if Current counted positively if enters leaves circuit element ”by head” of circuit element ”by head” of voltage arrow voltage arrow , ΕΕΝ 320 — Dr Petros Aristidou — Last updated: January 17, 2020 26/ 84
3 Load convention , ΕΕΝ 320 — Dr Petros Aristidou — Last updated: January 17, 2020 27/ 84
3 Generator convention , ΕΕΝ 320 — Dr Petros Aristidou — Last updated: January 17, 2020 28/ 84
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