Finding resistances across nodes in a T-network/PI-network

[VinceS]

Homework Statement

Using the known values of R1, R2, R3, R4, R5, R6, and a 5.03kΩ resistor, find the resistance between nodes 1 and 3 in Circuit 1 and Circuit 2.

Circuit 1 is a T-network composed of R1(1.0935kΩ), R2(1.06kΩ), and R3(1.07). Node 2 is connected directly to R2 and Node 3 is connected directly to R3 - the goal is calculate the resistance between nodes 1 and 3 when a 5.11kΩ resistor is connected between them(R5.1)

Circuit 2 is a PI-network composed of R4(2.980kΩ), R5(2.980kΩ), and R6(2.87κΩ). Node 1 is connected directly R4 and R6 while Node 2 is connected to R4 and R5. Node 3 is connected R5 and R6. The goal is the same as above - calculate the resistance between Nodes 1 and 3 when there is nothing connecting Nodes 2 and 3 and then with a 5.1κΩ resistor between them.

Homework Equations

I'll try to model the diagram for a better understanding, don't really have access to a scanner at the moment.

Circuit 1

N1 -----R1(1.0935kΩ) ------R2(1.06kΩ)-----N2
.......|......
........R3(1.07κΩ).....R(5.03) should be here, connected between N2 and N3
.......|......
........N3......

Circuit 2

N1------------R4(2.98κΩ)---------------N2
...|......|...
...|......|...
...R6(2.87κΩ)....R5(2.98κΩ)...R(5.03) would be in parallel with this resistor.
...|--------------------------|...
.......|......
.......N3.....

The Attempt at a Solution

I actually already know the values are supposed to be 2.01kΩ for Circuit 1 and 1.85κΩ and 1.99κΩ for Circuit 2 as this was a lab and I actually measured the values using a breadboard circuit and a multimeter. However, for analysis I'm intended to calculate the expected values, and for the life of me I can't figure out how to approach this.

The first thing I tried for Circuit 1 was to add R2 and R3 as series resistors, then combine them in parallel with R(5.03) which gave an equivalence resistance of~1.5kΩ. Then I combined that in parallel with R1, but that gave me about .67kΩ, which isn't even close to what I measured.

After that, I tried adding just R5.03 in parallel with (R1+R2), and that gave me 1.51kΩ - again considerably less than something around 2.01kΩ.

Is there something I'm missing here? Should I be using γ-Δ transformations, or would that just make a bigger mess of things?

If anyone can help steer me to a more accurate method of calculation, I would really appreciate it.

Also, I apologize if there was something in the formatting I got wrong - First time I've ever used this forum!

 

[gneill]

Homework Statement

Using the known values of R1, R2, R3, R4, R5, R6, and a 5.03kΩ resistor, find the resistance between nodes 1 and 3 in Circuit 1 and Circuit 2.

Circuit 1 is a T-network composed of R1(1.0935kΩ), R2(1.06kΩ), and R3(1.07). Node 2 is connected directly to R2 and Node 3 is connected directly to R3 - the goal is calculate the resistance between nodes 1 and 3 when a 5.11kΩ resistor is connected between them(R5.1)

Circuit 2 is a PI-network composed of R4(2.980kΩ), R5(2.980kΩ), and R6(2.87κΩ). Node 1 is connected directly R4 and R6 while Node 2 is connected to R4 and R5. Node 3 is connected R5 and R6. The goal is the same as above - calculate the resistance between Nodes 1 and 3 when there is nothing connecting Nodes 2 and 3 and then with a 5.1κΩ resistor between them.

Homework Equations

I'll try to model the diagram for a better understanding, don't really have access to a scanner at the moment.

Circuit 1

N1 -----R1(1.0935kΩ) ------R2(1.06kΩ)-----N2
.......|......
........R3(1.07κΩ).....R(5.03) should be here, connected between N2 and N3
.......|......
........N3......

Circuit 2

N1------------R4(2.98κΩ)---------------N2
...|......|...
...|......|...
...R6(2.87κΩ)....R5(2.98κΩ)...R(5.03) would be in parallel with this resistor.
...|--------------------------|...
.......|......
.......N3.....

The Attempt at a Solution

I actually already know the values are supposed to be 2.01kΩ for Circuit 1 and 1.85κΩ and 1.99κΩ for Circuit 2 as this was a lab and I actually measured the values using a breadboard circuit and a multimeter. However, for analysis I'm intended to calculate the expected values, and for the life of me I can't figure out how to approach this.

The first thing I tried for Circuit 1 was to add R2 and R3 as series resistors, then combine them in parallel with R(5.03) which gave an equivalence resistance of~1.5kΩ. Then I combined that in parallel with R1, but that gave me about .67kΩ, which isn't even close to what I measured.

After that, I tried adding just R5.03 in parallel with (R1+R2), and that gave me 1.51kΩ - again considerably less than something around 2.01kΩ.

Is there something I'm missing here? Should I be using γ-Δ transformations, or would that just make a bigger mess of things?

If anyone can help steer me to a more accurate method of calculation, I would really appreciate it.

Also, I apologize if there was something in the formatting I got wrong - First time I've ever used this forum!

If you use the Advanced editing panel ("Go Advanced" button), you'll find an icon in the panel that will wrap "

" tags around a block of text.  The system won't tamper with your space formatting inside code tags.

For clarity, here's your first circuit rendered as a .gif image:

[ATTACH=full]167373[/ATTACH]

Note that R2 and R3 are not in series; R1 shares the same node where they join, so they cannot be in series.  But another pair are in series...

 

[VinceS]

Thanks, both for the advice and the hint! I figured it out a while ago, actually by drawing out the circuit exactly like this. I just combined RL and R2 in series, then combined that in parallel with R3 and finally combined all that in series with R1. The end resistance was about 2.00 κΩ. It was only off by about 10mΩ, which is good enough for me to keep as a final value.