An infinite ladder network is constructed with identical series resistors of unknown value R on both the top and bottom branches, and identical shunt resistors of 20 connecting the branches. If the equivalent resistance of this entire infinite network measured across the input terminals is 5 , what is the value of the individual series resistor R ?

Step-by-step solution

Let the equivalent resistance of the infinite network be R_ eq . Since the network is infinite, removing one repeating unit leaves the equivalent resistance unchanged. The repeating unit consists of two series resistors R (one on the top branch, one on the bottom) and one shunt resistor of 20 . The equivalent resistance can be written as: R_ eq = R + R + 20 R_ eq 20 + R_ eq Given that R_ eq = 5 , we substitute this value into the equation: 5 = 2R + 20 5 20 + 5 5 = 2R + 100 25 5 = 2R + 4 2R = 1 R = 0.5 Answer: 0.5