Vectors addition and Subtraction

Vectors addition and Subtraction This is a very basic and simple review of adding and subtracting vectors graphically. It may be necessary in studying three phase circuits. Students are recommended to solve more exercises analytically and graphically and compare results. The graphical vector representation is very essential in understanding the problem even if you solve analytically only. Vector additions: Parallelogram law Addition of two vectors is accomplished by laying the vectors head to tail in sequence to create a triangle such as is shown in the figure. Sometimes vector addition is obtained using rule called the parallelogram law or triangle law, as shown below. A+B B A Parallelogram law B Vector subtraction: The geometric interpretation of the difference of two vectors is shown. A Think of subtraction as the addition of A and (-B). First, draw (-B), then use parallelogram law to add A+(-B) as shown in figure. A-B -B Sheet 1 Review of Vector Math [1]For t he circuit show n in Fig 1, given t he 3 volt age sources Van , Vbn and Vcn using KVL: 1) Find volt ages bet w een lines a, b, and c (Vab , Vbc and Vca). 4) Com m ent on result s. Hint: Van, Vbn and Vcn are called phase voltage and Vab, Vbc and Vca are called line voltage. 2) Draw t he vect or diagram of t he set of vect ors (Van , Vbn and Vcn ) and (Vab , Vbc and Vca). 3) Com m ent on result s. Fig.on1 result s. 3) Com m ent [2]For t he circuit show n in Fig. 2, given o t he 3 branch current s IAB=10–-37 A, o o IBC=10 –-157 A and ICA=10 –83 A , using KCL: 1) Find lines current s (Ia, Ib and Ic). 2) Draw t he vect or diagram of t he set of vect ors (Ia, Ib and Ic) and (IAB, IBC and ICA). Hint: Ia, Ib and Ic are called line currents and IAB, IBC and ICA are called phase current. Fig. 2 [3] From t he result s of problem [1], answ er t he follow ing M CQ quest ions: (a)The volt age Vab …….t he volt age Van ( )leads ( )lags ( )is in phase w it h ( )…………….. (b)The m agnit ude of volt ages Vab is …….t he m agnit ude of volt age Van ( )equal t o ( )great er t han ( )√3 t im es ( )0.577 t im es ( )…………….. (c)The phase shift bet w een Vab and Van is …….. (t ake Van as a reference) o o o o o o ( )30 lead ( ) 30 lag ( )60 lead ( ) 60 lag ( )90 lead ( ) 90 lag ( )…………….. (d)If w e call t he t hree volt ages Van , Vbn, and Vcn balanced volt ages, t hen w e can call t he t hree current s Ia, Ib , and Ic …….. . ( )balanced ( ) un-balanced ( )…………….. Hint: A set of three vectors variables is said to be balanced if the three vectors have: (1)equal magnitudes, and (2)phase shift of 120o between each other. They are un-balanced if any of the above two conditions is violated. o (e)If Van , Vbn, and Vcn are balanced volt ages, and if Van =200– 50 , t hen Vbn is given by…….. . o o o o o ( ) 200– 50 ( ) 200–- 50 ( ) 200–-70 ( ) 200–+70 ( ) 200–170 ( )…………….. o (f)If Van , Vbn, and Vcn are balanced volt ages, and if Van =220– 50 , t hen Vbc is given by…….. . o o o o o ( ) 381– -80 ( ) 115– 80 ( ) 381–200 ( ) 381–-40 ( ) 381–-200 ( )…………….. o (g)If Van , Vbn, and Vcn are balanced volt ages, and if Van =220– 50 , t hen Vba is given by…….. . o o o o o ( ) 381– -100 ( ) 115– 80 ( ) 381–100 ( ) 381–-40 ( ) 381–-200 ( )…………….. Hint: Vba= -Vab= Vab–+180o [4]For t he circuit show n in Fig 3, find t he (a) Equivalent ∆-circuit at t erm inals ABC. (a) Equivalent Y-circuit at t erm inals ABC. Hint: Use ∆-Y transformation to make the two parallel loads same (delta or star). Fig 3 [5]For t he circuit show n in Fig 4, find t he equivalent Y-circuit at t erm inals abc. Hint: Use ∆-Y transformation to transform ∆-Capacitors at ABC to an equivalent Y-Capacitors. The obtained Y-Capacitors are now in parallel with ZL (=10+j15), and their parallel combination is in series with ZS (=4+j3). v This problem is taken from Reference: doctord.dyndns.org/courses/.../ProbSolv_Chapter12.pdf