Find the equivalent impedance of an infinite series of resistors and capacitors

[kryzlof]

Homework Statement

Find the equivalent impedance of the infinite series of resistors and capacitors as shown below


-R----R----R----R----...R----...
____C____C____C____B_______C
-r----r----r----r----...r----...

Homework Equations

2.1. Equivalent resistance of resistors in series : R = R1 + R2 + R3 + ..
2.2. Equivalent resistance of resistors in parallel: 1 / R = 1 / R1 + 1 / R2 + 1 / R3 +...
2.3. Equivalent capacitance of capacitors in series : 1 / C = 1 / C1 + 1 / C2 + 1 / C3 ...
2.4. Equivalent capacitance of capacitors in parallel: C = C1 + C2 + C3 +...

The Attempt at a Solution

I am trying to find the equivalent impedance of the first three elements first (R, r and B) and the for the six first... This should then converge, hopefully...

 

[SammyS]

Homework Statement

Find the equivalent impedance of the infinite series of resistors and capacitors as shown below


-R----R----R----R----...R----...
____B____B____B____B_______B
-r----r----r----r----...r----...

Homework Equations

2.1. Equivalent resistance of resistors in series : R = R1 + R2 + R3 + ..
2.2. Equivalent resistance of resistors in parallel: 1 / R = 1 / R1 + 1 / R2 + 1 / R3 +...

The Attempt at a Solution

I am trying to find the equivalent impedance of the first three elements first (R, r and B) and the for the six first... This should then converge, hopefully...

What is B ?

 

[gneill]

Is this what you're going for?

 

[kryzlof]

Yes, it is. Sorry for my misstake (it should be C, not B), I will correct my first post.

 

[gneill]

Rather than try to ferret out a converging series from increasingly complicated terms, consider what happens when you add one more "unit cell" at the front of an already infinite train of identical "cells"

 

[Curious3141]

Homework Statement

Find the equivalent impedance of the infinite series of resistors and capacitors as shown below


-R----R----R----R----...R----...
____C____C____C____B_______C
-r----r----r----r----...r----...

Homework Equations

2.1. Equivalent resistance of resistors in series : R = R1 + R2 + R3 + ..
2.2. Equivalent resistance of resistors in parallel: 1 / R = 1 / R1 + 1 / R2 + 1 / R3 +...
2.3. Equivalent capacitance of capacitors in series : 1 / C = 1 / C1 + 1 / C2 + 1 / C3 ...
2.4. Equivalent capacitance of capacitors in parallel: C = C1 + C2 + C3 +...

The Attempt at a Solution

I am trying to find the equivalent impedance of the first three elements first (R, r and B) and the for the six first... This should then converge, hopefully...

The "trick" to most of these problems is to imagine that the network is already infinitely extensive to the right. Adding one more identical unit of impedance to the left of the network will not change the overall impedance.

If you let the impedance of the network be the unknown Z, you should be able to derive a quadratic equation with complex coefficients that Z satisfies.