There are many popular circuit configurations. Star-connected networks and delta-connected networks are commonly used configurations. A star-connected network, shown in Fig. 14, is referred to as the T network also. Similarly, the delta-connected network in Fig. 14 has another name, and it is p network. The star-connected and delta-connected configurations, shown in Fig. 14, are used commonly. A complex circuit can contain such configurations as its parts. In order to analyze such a complex circuit, it is at times necessary to transform star-connected configuration within the circuit to delta-connected configuration. In some cases, it may be the other way around. It may be easier to analyze the circuit after such transformation.
It is possible to replace a star-connected network by the equivalent delta-connected network. Alternatively, it is possible to replace a delta-connected network by the equivalent star-connected network. There are some memory aids, which can be used to guide the conversion of circuits of one type to the other. It is seen that both the star-connected network and the delta-connected network have each three terminals and three impedances. It is worth remembering that impedances of delta-connected network are larger than impedances of star-connected network, when these two networks are equivalent.
The equations used for converting one type of network to the other may look long and forbidding. The good news is that it is easy to remember them based on the topology of these two networks. At first, we take up converting a delta-connected network to its equivalent star-connected network.
First let us consider the star-connected circuit in Fig. 15. The impedance, as viewed from terminals, a and c, with terminal b left open, is stated below the diagram. It is the complex sum of impedances, Z1 and Z3 . We can repeat the process for the delta-connected network. In this case the impedance as viewed from terminals, a and c, is obtained as follows. When terminal b is open, it is seen that the impedance ZB is in parallel with the network consisting of impedances, ZA and ZC in series. The equivalent impedance as viewed from terminals¸ a and c, is expressed by the equation below the delta-connected network in Fig. 15.
We can repeat the process two more times. Let us find the impedance, as viewed from terminals, a and b, with terminal c left open. First let us consider the star-connected circuit in Fig. 16. The impedance, as viewed from terminals, a and b, with terminal c left open, is stated below the diagram. It is the complex sum of impedances, Z1 and Z2. We can repeat the process for the delta-connected network. In this case the impedance as viewed from terminals, a and b, is obtained as follows. When terminal c is open, it is seen that the impedance ZC is in parallel with the network consisting of impedances, ZB and ZA in series. The equivalent impedance as viewed from terminals¸ a and b, is expressed by the equation below the delta connected network in Fig. 16.
Let us find the impedance, as viewed from terminals, b and c, with terminal a, left open. First let us consider the star-connected circuit in Fig. 17. The impedance, as viewed from terminals, b and c, with terminal a left open, is stated below the diagram. It is the complex sum of impedances, Z2 and Z3. We can repeat the process for the delta-connected network. In this case the impedance as viewed from terminals, b and c, is obtained as follows. When terminal a is open, it is seen that the impedance ZA is in parallel with the network consisting of impedances, ZB and ZC in series. The equivalent impedance as viewed from terminals¸ b and c, is expressed by the equation below the delta-connected network in Fig. 17.
The equations shown in Figs. 15, 16 and 17 are grouped and presented below.
Two sets of equations are displayed. A set of three equations has been obtained for each type of network, and all are combined and presented here. Because the two networks are equivalent, we get the following equivalences. We can sum equations (63), (64) and (65). In the same way, we can sum equations (66), (67) and (68). The sums are equal to each other.
The equivalences are displayed now. It is seen that equation (69) can be subtracted from equation (72), and the result yields an expression for impedance Z2. Similarly, an expression for impedance Z3, is obtained by subtracting equation (70) from equation (72). We get an expression for impedance Z1 , by subtracting equation (71) from equation (72).
The results obtained are shown below.
Subtract equation (69) from equation (72). The result yields impedance Z2, as stated by equation (73). We get equations (74) and (75) in a similar manner, and we can obtain the values of Z3 and Z1, as shown by equations (74) and (75). It is difficult to remember these equations. Fortunately, remembering the equation using the topology of the two configurations is quite easy. That technique is illustrated next.
It is the aim to get the value of star impedance Z1 in terms of the impedances of the delta-connected network. This impedance Z1 is in red colour in Fig. 18. It is seen that this impedance is connected to node a. Two impedances of the delta-connected network are joined at node a. These impedances are in blue colour. The expression for Z1 has a numerator and a denominator. The numerator is the product of the two impedances of delta-connected network, which are connected to node a. The sum of the three impedances of the delta-connected network is the denominator.
The value of star impedance Z2 is obtained as follows. This impedance Z2 is shown in red colour in Fig. 19. It is seen that this impedance is connected to node b. Two impedances of the delta-connected network are joined at node b. These impedances are shown in blue colour. The expression for Z2 has a numerator and a denominator. The numerator is the product of the two impedances of delta-connected network, which are connected to node b. The sum of the three impedances of the delta-connected network is the denominator.
The value of star impedance Z3 is obtained now in terms of the impedances of the delta-connected network. This impedance Z3 is shown in red colour in Fig. 20. It is seen that this impedance is connected to node c. Two impedances of the delta connected network are joined at node c. These impedances are in blue colour. The expression for Z3 has a numerator and a denominator. The numerator is the product of the two impedances of delta-connected network, which are connected to node c. The sum of the three impedances of the delta-connected network is the denominator. It is relatively easy to remember the equations for conversion, based on the topology of the circuits. Next we take up star to delta conversion.
We can replace a star-connected network by its equivalent delta-connected network. First let us consider the star-connected circuit in Fig. 21. Terminals b and c, are shorted with each other. The impedance, as viewed from terminal a and shorted terminals, b and c, is stated below the diagram. The impedances, Z2 and Z3, are in parallel, and this parallel network is in series with the impedance Z1. Hence the impedance as viewed from terminal a and shorted terminals, b and c, is the complex sum of impedance, Z1 and the parallel value of impedances, Z2 and Z3. We can repeat the process for the delta-connected network. In this case the impedance as viewed from terminal a and shorted terminals, b and c, is obtained as the parallel value of impedances, ZB and ZC. It can be seen that impedance ZA plays no part as terminals, b and c, are short-circuited. The expression for the impedance, as viewed from terminal, a and shorted terminals, b and c, is expressed below the delta-connected network.
We can repeat the process two more times. First let us consider the star-connected circuit in Fig. 22. Terminals a and c, are shorted with each other. The impedance, as viewed from terminal b and shorted terminals, a and c, is stated below the diagram. The impedances, Z1 and Z3, are in parallel, and this parallel network is in series with the impedance Z2 . Hence the impedance as viewed from terminal b and shorted terminals, a and c, is the complex sum of impedance Z2 and the parallel value of impedances, Z1 and Z3. We can repeat the process for the delta-connected network. In this case the impedance as viewed from terminal b, and shorted terminals a and c, is obtained as the parallel value of impedances, ZC and ZA. It can be seen that impedance ZB plays no part as terminals, a and c, are short-circuited. The expression for the impedance, as viewed from terminal b and shorted terminals, a and c, is expressed below the delta-connected network.
First let us consider the star connected circuit in Fig. 23. Terminals a and b, are shorted with each other. The impedance, as viewed from terminal c and shorted terminals, a and b, is stated below the diagram. The impedances Z1 and Z2, are in parallel, and this parallel network is in series with the impedance Z3. Hence the impedance as viewed from terminal c and shorted terminals, a and b, is the complex sum of impedance Z3 and the parallel value of impedances, Z1 and Z2. We can repeat the process for the delta-connected network. In this case the impedance as viewed from terminal c and shorted terminals, a and b, is obtained as the parallel value of impedances, ZB and ZA. It can be seen that impedance ZC plays no part as terminals, a and b, are short-circuited. The expression for the impedance, as viewed from terminal c and shorted terminals, a and b, is expressed below the delta-connected network.
The equations shown in Figs. 21, 22 and 23 are grouped and presented below.
Two sets of equations are displayed. A set of three equations has been obtained for each type of network, and all are combined and presented here.
Equation (76) displays expressions for impedance, as viewed terminal a and shorted terminals, b and c, determined for both the delta-connected and the star-connected networks. There are two more equations, for the other two impedances, as shown by equations (77) and (78). These equations are not handy for developing conversion equations. It is preferable to deal with the admittances, as shown next.
Two sets of equations for admittances are displayed. A set of three equations has been obtained for each type of network, and all are combined and presented here. Because the two networks are equivalent, we get the following equivalences. We can see that equation (79) can be equated with (82). Similarly, equation (80) can be equated with equation (83), and then equation (81) can be equated with equation (84). We can sum equations (79), (80) and (81). In the same way, we can sum equations (82), (83) and (84). The sums are equal to each other. and the results are shown next.
Half of the sum of equations (70), (80) and (81) is shown by equation (120). This is equal to half of the sum of equations (82), (83) and (84).
Subtract equation (79) from equation (85). Equate this result to what you obtain by subtracting equation (82) from equation (85). The result yields the reciprocal of impedance ZA, as stated by equation (86). We get equation (87), by subtracting equation (80) from equation (85) and then equating the result with what is obtained by subtracting equation (83) from equation (85). The result is the reciprocal of impedance ZB, as stated by equation (87). Equation (88) is obtained in a similar manner, by subtracting equations (81) and (84) from equation (85) separately, and then equating the results. The result is the reciprocal value of impedance ZC, as shown by equations (88). Once we have expressions for the reciprocals of impedance, we can obtain the impedances, as follows.
The expression for impedance ZA in equation (89) is obtained as the reciprocal of expression in equation (86). Similarly, equations (90) and (91) are obtained from equations (87) and (88). We get the values of impedances ZB, and ZC from equations (90) and (91). It is difficult to remember these equations. Fortunately, remembering the equation using the topology of the two configurations is quite easy. That technique is illustrated next.
It is the aim to get the value of delta impedance ZA in terms of the impedances of the star-connected network. This impedance ZA is shown in red colour in Fig. 24. It is seen that this impedance is connected to nodes b and c . The impedance of the star-connected network which is not connected to these nodes is impedance Z1.
This impedance is connected to node a and it is in blue colour. The expression for ZA has a numerator, and a denominator. The numerator contains the sum of three product terms. The star-connected network has three impedances, It is possible to select three combinations of two impedances out of three impedances. The first combination contains Z1 and Z2, and the product of these two impedances is one of the terms in the numerator. The second combination contains Z2 and Z3, and the product of these two impedances is the second product of the three terms in the numerator. The third combination contains Z3 and Z1, and the product of these two impedances is the third product of the three terms in the numerator. The denominator is Z1, the impedance which is not connected to the nodes to which delta impedance, ZA is connected.
It is the aim to get the value of delta impedance ZB in terms of the impedances of the star-connected network. This impedance ZB is shown in red colour in Fig. 25. It is seen that this impedance is connected to nodes a and c . The impedance of the star-connected network, which is not connected to these nodes is impedance Z2. This impedance Z2 is connected to node b, and it is shown in blue colour. The expression for ZB has a numerator and a denominator. The numerator contains the sum of three product terms, the same as that described earlier for obtaining expression for impedance ZA. The denominator is different and it is the impedance Z2, which is not connected to the nodes to which delta impedance ZB is connected.
It is the aim to get the value of delta impedance ZC in terms of the impedances of the star-connected network. This impedance ZC is in red colour in Fig. 26. It is seen that this impedance is connected to nodes, a, and b . The impedance of the star-connected network, which is not connected to these nodes is impedance Z3. This impedance Z3 is connected to node c, and it is shown in blue colour. The expression for ZC has a numerator and a denominator. The numerator contains the sum of three product terms, the same as that described earlier for obtaining expression for impedance ZA. The denominator is different and it is the impedance Z3, which is not connected to the nodes to which delta impedance ZC is connected. You can now understand how you can carry out star to delta conversion. Remember the topology and you should be able to remember how to get the equivalent delta network.
There is another way to remember star to delta conversion. We can express equations (86) to (88) in terms of admittances. We know that admittance is the reciprocal of impedance. Hence we get the following equations from equations (86) to (88).
Equations (92), (93) and (94) state how the equivalent admittances of the delta-connected network can be obtained from the admittances of the star-connected network. The sketch shown below presents this information in a manner that is easy to remember.
It is the aim to get the value of admittance YA of the delta-connected network in terms of the admittances of the star-connected network. It is seen that this admittance is connected to nodes b and c . The admittances of the star configuration connected to these nodes are Y2 and Y3. The expression for YA has a numerator and a denominator. The numerator contains the product of two star admittances connected to nodes b and c. The denominator is the sum of three star admittances. The denominator is the same in all three cases. Delta admittance YB is connected to nodes a and c , and the numerator for YB is the product of two star admittances connected to nodes a and c. Similarly Delta admittance YC is connected to nodes a and b , and the numerator for YC is the product of two star admittances connected to nodes a and b. Worked examples presented later illustrate the use of these techniques.
This page has shown the equivalence between a star-connected network and a delta-connected network. Replacing one type of network by the other type leads to simplification of analysis in the case of some circuits. It will be possible to use either mesh or nodal analysis too, but that technique may prove to be a bit more cumbersome. The next page is on mesh and nodal analysis.