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Chapter 3

# Chapter 3 - 3-1 Chapter 3 Review of Basic Electric Circuits...

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Unformatted text preview: 3-1 Chapter 3 Review of Basic Electric Circuits Exit 2001 by N. Mohan Print TOC " ! 3-2 Conventions ! MKS (SI) Units ! lower case v and i for instantaneous quantities ! upper case V and I for average and rms ! voltage and current subscripts + a + vab i − b + va vb − − ! voltage polarities and current directions Exit 2001 by N. Mohan TOC " ! 3-3 Phasor Domain Representation for Sinusoidal Steady State AC φ Imaginary v ωt =0 i positive angles ˆ V = V ∠0 Real ωt −φ ˆ I = I∠ −φ ˆ v(t ) = V cos(ω t ) ˆ i (t ) = I cos(ω t − φ ) Exit 2001 by N. Mohan ⇔ ⇔ ˆ V = V ∠0 ˆ I = I ∠ −φ TOC " ! 3-4 Time-Domain Analysis i( t ) v( t ) ˆ =V cos( ω t ) + L − R C di (t ) 1 ˆ Ri (t ) + L + ∫ i (t ) ⋅ dt = V cos(ω t ) dt C Exit 2001 by N. Mohan TOC " ! 3-5 Phasor Domain Analysis I + ˆ V =V ∠0 − Im jω L = j X L R 1 − j = − j XC ωC − jX c jX L Z Re R 0 Z = R + j X L − j X C = Z ∠φ Z = 1 R2 + ω L − ωC ˆ V i (t ) = cos(ω t − φ ) Z Exit 2001 by N. Mohan 2 ; ⇔ 1 ω L − ωC −1 φ = tan R ˆ V V I = = ∠ −φ Z Z TOC " ! 3-6 Instantaneous Power i (t ) + Subcircuit 1 ˆ v(t ) =V cos(ω t + φv ) Subcircuit 2 v (t ) − ˆ i (t ) = I cos(ω t + φi ) p (t ) = v (t ) i (t ) φ /ω average power p (t ) 0 t v (t ) p (t ) average power 0 t v(t ) i (t ) x v and i in phase (φv = φi ) x power flows in one direction x maximum average power for given V and I Exit 2001 by N. Mohan i (t ) x v and i out of phase (φv ≠ φi ) x power flow reverses periodically x average power lower than maximum possible TOC " ! 3-7 Real Power, Reactive Power and Power factor I " Complex Power S = V I ∗ (S is a complex number) = V I ∠( φv − φi ) = V I ∠φ S = P + jQ = S ∠φ " Real Power (average power) P = V I cos φ " Reactive Power Q = V I sin φ [W ] P 2 + Q 2 = VI " Power Factor = Exit 2001 by N. Mohan Subcircuit 1 [VA] P P = = cos φ S VI Subcircuit 2 V − S = P + jQ ˆ V = V ∠φv Im φv − φi [VAR ] " Apparent Power S = + Re ˆ I = I ∠φi Im Q S φ = φv − φi P Re TOC " ! 3-8 Inductive Load ❏ The impedance is Z = Z ∠φ , where φ is positive ❏ The current lags the voltage by the impedance angle φ ❏ Corresponds to a lagging power factor of operation ❏ In the power triangle, the same angle φ relates P,Q and S ❏ An inductive load draws positive reactive power (VARs) ❏ Most loads are inductive, particularly motors and transformers Exit 2001 by N. Mohan TOC " ! 3-9 Three Phase Circuits (Transmission and Distribution) ❏ One-line diagram of power systems Step up Transformer Generator Transmission line 13.8 kV Feeder Load Exit 2001 by N. Mohan TOC " ! 3-10 Three Phase Circuits ❏ Wye-connection Ia ˆ Van = Vs ∠0 o ˆ Vbn = Vs ∠ − 120 o a + V an V cn + − n ˆ Vcn = Vs ∠ − 240 o − V bn + N c Ib Ic Van + Vbn + Vcn = 0 van (t ) + vbn( t) + vcn ) = 0 (t ZL − a−b−c van (t ) vbn (t ) vcn (t ) V cn 0 ωt b positive sequence 120° 120° 120° V an V bn 2π Exit 2001 by N. Mohan 3 2π 3 TOC " ! 3-11 Per phase Analysis of Balanced Three phase Circuits ˆ Van Vs Ia = = ∠ −φ ZL ZL ˆ Vbn Vs 2π Ib = = ∠− −φ ZL ZL 3 Ia V an V cn + ˆ Vcn Vs 4π Ic = = ∠− −φ ZL ZL 3 a + − − n − V bn + Ic ZL In N c b Ib I n = ( I a + I a + I c ) = 0 ⇒ in ( t ) =[ ia ( t ) + ib ( t ) + ic ( t ) = 0 a + Ia V cn V an (Hypothetical) 2001 by N. Mohan φ Ib − n Exit Ic a N V an Ia V bn TOC " ! 3-12 Line-to-Line Voltages Ia a + V an V cn + − Vc + + − n − V bn + ZL N c Ib Ic ˆ Vab =Va − Vb = 3Vs ∠30 Va − − o ˆ Vbc =Vb − Vc = 3Vs ∠ − 90 o ˆ Vca = Vc − Va = 3Vs ∠ − 210 o Exit 2001 by N. Mohan Vca −Vb 30 o V ab Vb + Vab Va − b Vb Vbc VLL = 3V ph ∠30 o TOC " ! 3-13 Delta Connection Phase Currents in Delta Load Ia I a = I ab − I ca I b = I bc − I ab I c = I ca − I bc V an a I ca + − V cn − n + − V bn + Z∆ Ib c b I ca 1 I∆ = Il ∠30 o 3 I ab 30 o Ib 2001 by N. Mohan Z ∆ I ab Ic Ic Exit Ibc Z∆ Ibc " I ca " Ia TOC " ! 3-14 ∆ − Y Transformation ❏ Allows per phase analysis Ia equivalent to external circuit a I ca c Z∆ I ab Ibc Delta connected load Ia a ZΥ ⇔ b c ZΥ ZΥ b Wye connected load These circuits are indistinguishable to the external circuit, when Z∆ ZΥ = Exit 2001 by N. Mohan 3 TOC " ! 3-15 Summary ❏ Why is it important to always indicate the directions of currents and the polarities of voltages? ˆ ❏ What are the meanings of i, I, I, and I ? ❏ In a sinusoidal waveform voltage, what is the relationship between the peak and the rms values? ❏ How are currents, voltages, resistors, capacitors, and inductors represented in the phasor domain? Express and draw the following as phasors, assuming both φv and φi to be positive: ˆ ˆ v(t ) =V cos(ω t + φv ) and, i (t ) = I cos(ω t + φi ) Exit 2001 by N. Mohan TOC " ! 3-16 Summary ❏ How is the current flowing through impedance Z ∠φ related to the voltage across it, in magnitude and phase? ❏ What are real and reactive powers? What are the expressions for these in terms of rms values of voltage and current and the phase difference between the two? ❏ What is complex power S? How are real and reactive powers related to it? What are the expressions for S, P, and Q, in terms of the current and voltage phasors? What is the power triangle? What is the polarity of the reactive power drawn by an inductive/capacitive circuit? ❏ What are balanced three-phase systems? How can their analyses be simplified? What is the relation between line-to-line and phase voltages in terms of magnitude and phase? What are wye and delta connections? Exit TOC " 2001 by N. Mohan ! ...
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