Some time we may have a situation , where it is hard to determine that either the circuit is parallel or series. In such cases, solving a circuit to find total resistance or other elements of a network is very hard. Among many solutions to these problem, one solution is DELTA Y conversions

One Port and Two Port Circuits

A one port circuit is a circuit where current entering in any one terminal of the circuit is equal the current leaving from the second terminal of the circuit . For example, resistance, inductance and capacitance acts as a one-port circuit.

How ever we may have situation where we have more than one pair of two terminals

Here in this case we have four terminals. Such configurations are commonly found in form of Delta or Y network

Wye Delta Transforms

Determine if the above circuit have resistors in series or parallel. The answer needs specific conversions.

In many circuit applications, we encounter components connected together in one of two ways to form a three-terminal network the “Delta,” or Δ (also known as the “Pi,” or π) configuration, the “Y” (also known as the “T”) configuration.

So lets start the mathematics

Converting Delta to Y

Although a proper long derivation is required to convert Delta to Y . However the scope of this course is to understand the conversion , so we will be leaving the conversion part.

Sol lets say we have a Delta with resistors RA , RB ad RC and we want to find equivalent Y configurations resistors R1, R2, R3

Finding R1:

Note That if we super impose the two configuration than R1 has two neighboring Delta resistors : Rb and Rc

Find R1 =

Finding R2

Similarly applying the above shortcut technique we have

Find R2 =

 

Finding R3

Find R3 =

Converting Y to Delta

In this approach we will try to convert a Y circuit having resistances R1 R2 and R3 to a delta configuration RA RB and RC

 

Finding RA

Note that here RA is Opposite to R1..

Finding RA

Ra=

Finding RB

For finding Rb notice that R2 lies opposite to Rb

So

Rb=

Finding RC

Using same technique we can find RC

Rc=

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Example 1 :

Find total Voltage

 

Converting First Delta to Y

 

Observe that 5 ohm and 10 Ohm resistors are in series

 

Now using the formulas we can find

 

Now we need to solve other Delta

Observe that 5 ohm and 5 ohm resistors are in series

 

Solving Both give us 10 Ohm

 

 

Redrawing Circuit

 

Solving Circuit in Series

 

Solving Circuit in Parallel

l 9.37 Ohm ||13.65 Ohm =5.5 Ohm

Solving Circuit in series

=1.67 Ohm +5.5 Ohm +3.3 Ohm= 10.47 Ohm

Using Ohms law V= IR

=5 x 10.47

52.35 Volts

Example 2:

Find the Voltage across 35 Ω Resistance

Using the Delta to Y conversion we get following values

Solving Series Circuit

 

Solving Resistors in parallel

Using VDR to find voltage across 19.25 ohm Resistor

=((19.25)/(6.25+19.25))*10

=2.451V

As 19.25 ohm resistor is parallel to 32.5||47.5 ohm resistance. So both resistors will have 2.541 V resistors across them

 

Now our resistor 47.5 Ohm resistor was equal to 35 Ohm and 12.5 Ohm resistor , so the voltage across

Using VDR , we can calculate voltage across 35 ohm resistance

= 0.65 v .