Chap. 11 Balanced Three-Phase Circuits 2018/11/23 Chap. 11 Balanced Three-Phase Circuits Contents 11.1 Balanced Three-Phase Voltages 11.2 Three-Phase Voltage Sources 11.3 Analysis of the Wye-Wye Circuit 11.4 Analysis of the Wye-Delta Circuit 11.5 Power Calculations in Balanced Three-Phase Ckts 11.6 Measuring Average Power in Three-Phase Circuits Objectives 了解如何分析平衡三相Y-Y連接電路。 2. 了解如何分析平衡三相Y-Δ 連接電路。 3. 具有計算任何三相電路功率(平均功率、無效功率及複數功率) 之能力。

Practical Perspective 2018/11/23 Practical Perspective 為了經濟效益，對於大功率電力的產生、輸送、分配皆以 三相電路來完成。 三相系統的基本架構，是由電壓源經由變壓器及傳輸線連 接至負載。 三相電路電源與負載之連接方式有Y-Y, Y-Δ, Δ-Y 及Δ-Δ 等連接方式。 2 2

11.1 Balanced Three-Phase Voltages 三相電壓： a 相電壓(a-phase voltage) b 相電壓(b-phase voltage) c 相電壓(c-phase voltage) abc 相序(abc phase sequence) 或 正相序(positive phase sequence) ： b 相電壓落後a 相電壓120° c 相電壓領先a 相電壓120° 平衡三相電源: 大小相同、頻率相同、相位角各差120o acb 相序(acb phase sequence) 或 負相序(negative phase sequence) ： b 相電壓領先a 相電壓120° c 相電壓落後a 相電壓120° 3

11.2 Three-Phase Voltage Sources 三相電源由三個分開的繞組所產生： Y 形接法及Δ 形接法。 中性端子(neutral terminal) ： Y 形電源的共同端，接點n。 4

Three-Phase Source with Winding Impedances 三相電壓源及負載間有四種不同的接法 由於其他三種接法皆可化簡成Y-Y 等效電路，故而先行分析Y-Y 電路。 5

11.3 Analysis of the Wye-Wye Circuit 2018/11/23 11.3 Analysis of the Wye-Wye Circuit 圖示三相Y-Y 系統，包含第四條導線連接電源與負載之中性端。 Zga, Zgb, Zgc分別代表各相電壓源之內阻。 Zla, Zlb, Zlc 分別代表各相電源至負載之導線阻抗。 ZA, ZB, ZC則代表各相負載阻抗。 平衡三相? 6 6

Conditions for a Balanced Three-phase Circuit 平衡三相電路的條件： 1. 電壓源Va'n, Vb'n, Vc'n構成一組平衡三相電源。 2. 每相電壓源之阻抗相同，亦即Zga = Zgb =Zgc。 3. 每相導線阻抗相同，亦即Z1a = Z1b = Z1c。 4. 每相負載阻抗相同，亦即ZA = ZB = ZC。 若為平衡三相電路 = 0 兩個中性端沒有電位差 中性導線電流=0 平衡三相電路各線路之電流 在平衡三相系統中，具有平衡的三相電流，所以一旦計算出IaA 電流，就可直接寫出IbB, IcC之電流。 7

Single-phase Equivalent Circuit 單相等效電路中之IaA 並非平衡三相電路的中性導線電流，真正中性導線的電流為 當線電流IaA求出後，便可計算出其他的電壓。 負載端的線電壓VAB, VBC,VCA 可用相電壓VAN, VBN,VCN來表示。 8

Line-to-line Voltages & Line-to-Neutral Voltages abc(正)相序 abc(正)相序 Line-to-Neutral Voltages Line-to-line Voltages 線電壓之大小等於相電壓大小的 倍 線電壓構成一組平衡三相電壓 線電壓組領先相電壓組30° acb(負)相序 注意：若為acb(負)相序，則 線電壓組落後相電壓組30° 9

Line Voltage, Phase Voltage, Line Current & Phase Current 在Y型接法 線電流與相電流相同 線電壓 = line-to-line voltage 相電壓 = line-to-neutral voltage 在型接法 線電壓與相電壓相同 相電壓  line-to-neutral voltage 以表示單相的量，如： V, I, Z, P 及Q 分別代表每相之電壓、 電流、阻抗、實功率及無效功率。 10

EX 11.1 Analyzing a Y-Y Circuit A balanced three-phase Y-connected generator with positive sequence has an impedance of 0.2 + j 0.5 / and an internal voltage of 120 V/ .The generator feeds a balanced three-phase Y-connected load having an impedance of 39 + j 28 / . The impedance of the line connecting the generator to the load is 0.8 + j 1.5 / . The a-phase internal voltage of the generator is specified as the reference phasor. a-phase equivalent Construct the a- phase equivalent circuit of the system. b) Find IaA , IbB , and IcC. 正相序 負相序 11

EX 11.1 Analyzing a Y-Y Circuit (Contd.) c) Find VAN , VBN , and VCN. 正相序 負相序 d) Find VAB , VBC , and VCA. 正相序 負相序 e) Find Van , Vbn , and Vcn. 正相序 負相序 f) Find Vab , Vbc , and Vca. 正相序 負相序 12

11.4 Analysis of the Wye-Delta Circuit 2018/11/23 11.4 Analysis of the Wye-Delta Circuit -to-Y transformation a-phase equivalent 平衡三相 13 13

Line Currents & Phase Currents in  abc(正)相序 Phase Currents abc(正)相序 Line Currents 線電流之大小等於相電流大小的 倍 線電流構成一組平衡三相電流 線電流組落後相電流組30° acb(負)相序 注意：若為acb(負)相序，則 線電流組落後相電流組30° 14

EX 11.2 Analyzing a Y- Circuit The Y-connected source feeds a -connected load through a distribution having an impedance of 0.3 + j 0.9 / . The load impedance is 118.5 + j 85.8 / . Use the a-phase internal voltage as the reference. The a-phase equivalent Y b) The line currents IaA , IbB & IcC c) The phase voltages at the load terminals 15

EX 11.2 Analyzing a Y- Circuit (Contd.) d) The phase currents of the load Check! e) The line voltages at the source terminals 16

11.5 Power Calculations in Balanced Three-Phase Circuits 2018/11/23 11.5 Power Calculations in Balanced Three-Phase Circuits Average Power in a Balanced Wye Load 17 17

Complex Power in a Balanced Wye Load 2018/11/23 Complex Power in a Balanced Wye Load 18 18

Power Calculations in a Balanced Delta Load 2018/11/23 Power Calculations in a Balanced Delta Load 19 19

Instantaneous Power in a Three-Phase Circuit 2018/11/23 Instantaneous Power in a Three-Phase Circuit 若以線至中性點瞬時電壓vAN 為參考，且為正相序， 為相位角差，各單相的瞬時功率可表示為pA, pB, pC，總瞬時功率pT 為三者之和。 20 20

EX 11.3 Calculating Power in a Three-Phase Y-Y Circuit Consider the circuit described in EX 11.1 . a) The average power per phase delivered to the load d) The total internal power lost in the generator e) The total number of magnetizing vars absorbed by the load 或 b) The total average power delivered to the load f) The total complex power delivered by the source 或 c) The total power lost in the line Check! 21

EX 11.4 Calculating Power in a Three-Phase Y- Circuit Consider the circuit described in EX 11.2 . a) The total complex power delivered to the -connected load From EX 11.2 b) The percentage of the average power reaching the load 22

EX 11.5 Three-Phase Power with an Unspecified Load A balanced three-phase load requires 480 kW at a lagging power factor of 0.8. The load is fed from a line having an impedance of 0.005 + j 0.025 /. The line voltage at the terminals of the load is 600 V. The a-phase equivalent b) The magnitude of the line currents 或 23

EX 11.5 Three-Phase Power with an Unspecified Load (Contd.) c) The magnitude of the line voltage at the sending end of the line d) The power factor at the sending end of the line 或 24

11.6 Measuring Average Power in Three-Phase Circuits 2018/11/23 11.6 Measuring Average Power in Three-Phase Circuits 用來測量三相電路功率的基本儀表，稱為電動力瓦特表(electrodynamometer wattmeter)，內含兩個線圈，一為電流線圈(current coil)，通過此線圈之電流與負載電流成正比；另一個稱為電壓線圈(potential coil)，通過此線圈之電流與負載電壓成正比。 瓦特計指針的偏轉量與通過電流線圈電流有效值、電壓線圈兩端電壓有效值、電流和電壓間相角的餘弦值三者之乘積成正比。 右圖由n 條導線供應功率的一般網 路，以n 為參考電壓點，其總功率 p = v1i1 +v2i2 + … + vn-1in-1。 對三相平衡電路而言，因為中性線沒有電流通過，故只需兩個瓦特計，即可量測總功率。 25 25

The Two-Wattmeter Method 兩個瓦特計量測功率之特性： 若功率因數大於0.5 時，兩個瓦特 計皆是正值。 若功率因數等於0.5 時，其中一個 瓦特計的值為零。 若功率因數小於0.5 時，其中一個 瓦特計為負值。 當相序相反時，則兩個瓦特計之量 測值剛好對換。 VCB 正向序 26

EX 11.6 Computing Wattmeter Readings in Three-Phase Circuits Calculate the reading of each wattmeter in the previous circuit if the phase voltage at the load is 120 V and (a) Z = 8 + j 6 ; (b) Z = 8  j 6 ; (c) Z = 5+j5√3 ; and (d) Z = 10∠−75◦ . a) c) Check! b) d) 27

Transmission and Distribution of Electric Power 2018/11/23 Transmission and Distribution of Electric Power 變電所三相的實功率為3.6 MW，無效功率為3.6 MVAR，線電壓為13.8 KV，配電線阻抗為0.6 + j 4.8  。 若電力系統必須將電壓準位維持在公稱值(nominal value) 之5.8 % 範圍內，則代表發電廠之線電壓範圍應介於13 kV 至14.6 kV 之間。 利用連接於變電所之電容器提供無效功率 28 28