1.7 DeltaStar Transformation


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1 S Electronic ircuits D ircuits 8.7 DeltaStar Transformation Fig..(a) shows three resistors R, R and R connected in a closed delta to three terminals, and, their numerical subscripts,, and, being opposite to the terminals,, and respectively. It is possible to replace these deltaconnected resistors by three starconnected resistors R a, R b, and R c in Fig..(b) If the starconnected network is said to be equivalent to the deltaconnected network, the resistance between any two terminals in Fig..(b) must be the same as that between the same two terminals in Fig..(a). R R b R c R R R a Fig..(a) Fig..(b) onsider terminals and in Fig..(a), we have a circuit having a resistance R in parallel with a circuit having resistances R and R in series; hence For Fig..(b), we have Equating (.7.) and (.7.) R ( R + R ) R = (.7.) R a a b R = R + R (.7.) RR + RR + Rb = (.7.)
2 S Electronic ircuits D ircuits 9 Similarly, R b RR + RR + Rc = (.7.4) Rearrange (.7.), (.7.4) and (.7.5) to give R a RR + RR + Rc = (.7.5) RR R a = (.7.6) RR R b = (.7.7) RR R c = (.7.8).8 StarDelta Transformation onversely, the starconnected network can be replaced with an equivalent deltaconnected network. Dividing (.7.6) by (.7.7), R Similarly, dividing (.7.6) by (.7.8), = R R a R b (.7.9) R Substituting for R and R into (.7.6), = R R a R c (.7.0) R + = Rb + Rc RbRc Ra (.7.) Similarly, R = R R a c a c b (.7.) R + = Ra + Rb RaRb Rc (.7.)
3 S Electronic ircuits D ircuits 0.9 Maximum Power Transfer onsidering Fig.., a voltage source E with an internal resistance r is connected in series with an external resistance R L. Energy source I r E R L Fig.. The load current I E = (.8.) r + R L The power delivered to the load is P L E = I RL = R L (.8.) ( r + R ) To achieve maximum power transfer, differentiating (.8.) with respect to R L and setting the result equal to zero. L dpl ( r + RL) RL( r + RL) = E = 0 (.8.) dr 4 ( r + R ) L To find the value of RL to satisfy (.8.), setting the numerator of (.8.) equal to zero to obtain L R L = r (.8.4) For maximum power transfer, the load resistance is chosen to be equal to the internal resistance of the source.
4 S Electronic ircuits D ircuits dditional examples () Find the magnitude and direction of the current flowing through 7Ω resistor in the following network. Ω Ω D 8V 4Ω 7Ω 4V E pproach Kirchhoff s Law pproach F G H Using KL, the currents are assigned as in the following diagram. I Ω Ω I I D I I I I 8V 4Ω 7Ω 4V E F G H
5 S Electronic ircuits D ircuits pproach Mesh urrent nalysis Ω Ω D 8V I 4Ω 7Ω 4V I I E F G H pproach Superposition Theorem
6 S Electronic ircuits D ircuits () Obtain the Thevenin s equivalent circuit for the following active network. Ω a Ω 0V 0V D b y KVL, the current flowing through the resistor is I 6 Ω = = (circulate in clockwise direction) + 6 So, V is equal to 0 V = = 0V The Thevenin equivalent resistance R T is 6 R T = + = 5Ω + 6 The Thevenin s equivalent circuit is obtained as 0V 5Ω a b
7 S Electronic ircuits D ircuits 4 () Using Thevenin s Theorem, find the current in the 0Ω resistor of the following network. V Ω 4Ω 4Ω 0Ω pplying Thevenin s Theorem, simplify the networks across  and . I I I V Ω I V I 4Ω 4Ω I The equivalent circuit is shown as follows. I V 0Ω
8 S Electronic ircuits D ircuits 5 (4) network is arranged as shown in the following figure. alculate the equivalent resistance between (a) and, and (b) and N. (a) y using Stardelta transformation, 5Ω 5Ω 4Ω N Ω R R R 0Ω 0Ω
9 S Electronic ircuits D ircuits 6 (b) y using Deltastar transformation,
10 S Electronic ircuits D ircuits 7 (5) alculate the voltage across  in the network by using deltastar transformation. 5Ω 0V (a) Network is rearranged as follows. R a 5Ω I I R c R eq R c R b 0V 0V I 0V () Transform the 5Ω delta connection into R a R b R c star connection.
11 S Electronic ircuits D ircuits 8 () alculate the equivalent resistance R eq and compute the current I. () Using current division principle, compute the currents I and I. (4) alculate the voltage drop across the and resistors which on the LHS in the middle figure. ompute the voltage difference between and.
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