Find Resistance Req points Rab

Question

Answer 1

To find the equivalent resistance Rab between terminals a and b, we need to simplify the given resistor network.

This circuit contains a "bridge" configuration that cannot be simplified using only standard series and parallel combinations. We can solve this using a Delta-to-Star (Δ-Y) transformation.

Step 1: Identify a Delta (Δ) or Star (Y) configuration.

Let's focus on the inner part of the circuit. We can see a delta (or Pi, Π) configuration formed by the 30 Ω, 10 Ω, and 20 Ω resistors. Let's label the nodes to make it clearer:
Node C: The junction between the 25 Ω, 10 Ω, and 30 Ω resistors.
Node D: The junction between the 10 Ω, 20 Ω, and 5 Ω resistors.
* Node E: The junction between the 30 Ω, 20 Ω, and 15 Ω resistors.

The delta is formed by the resistors connecting nodes C, D, and E.
R_CD = 10 Ω
R_DE = 20 Ω
* R_CE = 30 Ω

Step 2: Convert the Delta (Δ) to a Star (Y).

We will replace the delta network (10 Ω, 20 Ω, 30 Ω) with an equivalent star network. This new network will have a central node (let's call it N) and three new resistors (R_C, R_D, R_E).

The formulas for the conversion are:
R_C (connects C to N) = (R_CD R_CE) / (R_CD + R_DE + R_CE)
R_D (connects D to N) = (R_CD R_DE) / (R_CD + R_DE + R_CE)
R_E (connects E to N) = (R_DE R_CE) / (R_CD + R_DE + R_CE)

Let's calculate the values:
Sum of resistors in delta = 10 Ω + 20 Ω + 30 Ω = 60 Ω
R_C = (10 30) / 60 = 300 / 60 = 5 Ω
R_D = (10 20) / 60 = 200 / 60 = 10/3 Ω (or approx. 3.33 Ω)
R_E = (20 * 30) / 60 = 600 / 60 = 10 Ω

Step 3: Redraw the circuit with the new Star network.

Now, we replace the delta with our new star resistors. The circuit becomes much simpler:

Step 4: Simplify the resulting series and parallel combinations.

Looking at the new circuit:
1. The resistor R_D (10/3 Ω) is in series with the 5 Ω resistor.

*   R_series1 = 10/3 Ω + 5 Ω = 10/3 Ω + 15/3 Ω = **25/3 Ω**

1. The resistor R_E (10 Ω) is in series with the 15 Ω resistor.
   * R_series2 = 10 Ω + 15 Ω = 25 Ω

2. These two series branches (R_series1 and R_series2) are in parallel with each other.
   R_parallel = (R_series1 R_series2) / (R_series1 + R_series2)
   R_parallel = ( (25/3) 25 ) / ( (25/3) + 25 )
   R_parallel = ( 625/3 ) / ( 25/3 + 75/3 )
   R_parallel = ( 625/3 ) / ( 100/3 )
   * R_parallel = 625 / 100 = 6.25 Ω

Step 5: Calculate the final equivalent resistance (Rab).

The circuit is now reduced to three resistors in series: the initial 25 Ω resistor, the new R_C resistor, and the equivalent parallel resistance we just calculated.

• Rab = 25 Ω + R_C + R_parallel
• Rab = 25 Ω + 5 Ω + 6.25 Ω
• Rab = 36.25 Ω

The equivalent resistance between terminals a and b is 36.25 Ω.