Basic Electronics on the Go - Star Delta Transformation

From http://www.electronics-tutorials.ws/dccircuits/dcp_10.html

Star Delta Transformation

 We can now solve simple series, parallel or bridge type resistive networks using Kirchhoff´s Circuit Laws, mesh current analysis or nodal voltage analysis techniques but in a balanced 3-phase circuit we can use different mathematical techniques to simplify the analysis of the circuit and thereby reduce the amount of math’s involved which in itself is a good thing.

 Standard 3-phase circuits or networks take on two major forms with names that represent the way in which the resistances are connected, a Star connected network which has the symbol of the letter, Υ (wye) and a Delta connected network which has the symbol of a triangle, Δ (delta).

 If a 3-phase, 3-wire supply or even a 3-phase load is connected in one type of configuration, it can be easily transformed or changed it into an equivalent configuration of the other type by using either the Star Delta Transformation or Delta Star Transformation process.
A resistive network consisting of three impedances can be connected together to form a T or “Tee” configuration but the network can also be redrawn to form a Star or Υ type network as shown below.

T-connected and Equivalent Star Network

 

As we have already seen, we can redraw the T resistor network above to produce an electrically equivalent Star or Υ type network. But we can also convert a Pi or π type resistor network into an electrically equivalent Delta or Δ type network as shown below.

Pi-connected and Equivalent Delta Network.

 

  Having now defined exactly what is a Star and Delta connected network it is possible to transform the Υ into an equivalent Δ circuit and also to convert a Δ into an equivalent Υ circuit using a the transformation process. This process allows us to produce a mathematical relationship between the various resistors giving us a Star Delta Transformation as well as a Delta Star Transformation.

 

Delta Star Transformation

To convert a delta network to an equivalent star network we need to derive a transformation formula for equating the various resistors to each other between the various terminals. Consider the circuit below.

Delta to Star Network.

 

 Compare the resistances between terminals 1 and 2.

 Resistance between the terminals 2 and 3.

 

 

 Resistance between the terminals 1 and 3.

 This now gives us three equations and taking equation 3 from equation 2 gives:

 

Then, re-writing Equation 1 will give us:

Adding together equation 1 and the result above of equation 3 minus equation 2 gives:

 From which gives us the final equation for resistor P as:

Similarly, resistor Q and R  can be found  :-

When converting a delta network into a star network the denominators of all of the transformation formulas are the same: A + B + C, and which is the sum of ALL the delta resistances. 

Star Delta Transformation

Star Delta transformation is simply the reverse of above. We have seen that when converting from a delta network to an equivalent star network that the resistor connected to one terminal is the product of the two delta resistances connected to the same terminal, for example resistor P is the product of resistors A and B connected to terminal 1.

The value of the resistor on any one side of the delta, Δ network is the sum of all the two-product combinations of resistors in the star network divide by the star resistor located “directly opposite” the delta resistor being found. For example, resistor A is given as:

 Resistor B is given as:

Resistor C given as:

Basic Electronics on the Go - Maximum Power Transfer

From http://www.electronics-tutorials.ws/dccircuits/dcp_9.html

We have seen in the previous tutorials that any complex circuit or network can be replaced by a single energy source in series with a single internal source resistance, R_S.

 When we connect a load resistance, R_L across the output terminals of the power source, the impedance of the load will vary resulting in the power being absorbed by the load becoming dependent on the impedance of the actual power source. Then for the load resistance to absorb the maximum power possible it has to be “Matched” to the impedance of the power source and this forms the basis of Maximum Power Transfer.

The Maximum Power Transfer Theorem is another useful circuit analysis method to ensure that the maximum amount of power will be dissipated in the load resistance when the value of the load resistance is exactly equal to the resistance of the power source. The relationship between the load impedance and the internal impedance of the energy source will give the power in the load. Consider the circuit below.
 

Thevenins Equivalent Circuit.

In our Thevenin equivalent circuit above, the maximum power transfer theorem states that “the maximum amount of power will be dissipated in the load resistance if it is equal in value to the Thevenin or Norton source resistance of the network supplying the power“.

In other words, the load resistance resulting in greatest power dissipation must be equal in value to the equivalent Thevenin source resistance, then R_L = R_S but if the load resistance is lower or higher in value than the Thevenin source resistance of the network, its dissipated power will be less than maximum.

For example, find the value of the load resistance, R_L that will give the maximum power transfer in the following circuit.

Maximum Power Transfer Example No1.

 

  Where:
  R_S = 25Ω
  R_L is variable between 0 – 100Ω
  V_S = 100v

Then by using the following Ohm’s Law equations:

We can now complete the following table to determine the current and power in the circuit for different values of load resistance.

Table of Current against Power

RL (Ω) I (amps) P (watts) 0 4.0 0 5 3.3 55 10 2.8 78 15 2.5 93 20 2.2 97

RL (Ω) I (amps) P (watts) 25 2.0 100 30 1.8 97 40 1.5 94 60 1.2 83 100 0.8 64

Using the data from the table above, we can plot a graph of load resistance, R_L against power, P for different values of load resistance. Also notice that power is zero for an open-circuit (zero current condition) and also for a short-circuit (zero voltage condition).

Graph of Power against Load Resistance

 From the above table and graph we can see that the Maximum Power Transfer occurs in the load when the load resistance, R_L is equal in value to the source resistance, R_S that is: R_S = R_L = 25Ω. This is called a “matched condition”.

One good example of impedance matching is between an audio amplifier and a loudspeaker. The output impedance, Z_OUT of the amplifier may be given as between 4Ω and 8Ω, while the nominal input impedance, Z_IN of the loudspeaker may be given as 8Ω only.

Then if the 8Ω speaker is attached to the amplifiers output, the amplifier will see the speaker as an 8Ω load. Connecting two 8Ω speakers in parallel is equivalent to the amplifier driving one 4Ω speaker and both configurations are within the output specifications of the amplifier.

Improper impedance matching can lead to excessive power loss and heat dissipation. But how could you impedance match an amplifier and loudspeaker which have very different impedances. Well, there are loudspeaker impedance matching transformers available that can change impedances from 4Ω to 8Ω, or to 16Ω’s to allow impedance matching of many loudspeakers connected together in various combinations such as in PA (public address) systems.

Transformer Impedance Matching

 

 The maximum power transfer can be obtained even if the output impedance is not the same as the load impedance. This can be done using a suitable “turns ratio” on the transformer with the corresponding ratio of load impedance, Z_LOAD to output impedance, Z_OUT matches that of the ratio of the transformers primary turns to secondary turns as a resistance on one side of the transformer becomes a different value on the other.

 If the load impedance, Z_LOAD is purely resistive and the source impedance is purely resistive, Z_OUT then the equation for finding the maximum power transfer is given as:

 Where: N_P is the number of primary turns and N_S the number of secondary turns on the transformer. Then by varying the value of the transformers turns ratio the output impedance can be “matched” to the source impedance to achieve maximum power transfer.