Nonlinear Dynamics and Chaos
Nonlinear Dynamics and Chaos
2nd Edition
ISBN: 9780429972195
Author: Steven H. Strogatz
Publisher: Taylor & Francis
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Chapter 4.6, Problem 1E
Interpretation Introduction

Interpretation:

To sketch the super-current Ic sin ϕ (t) as a function of t assuming that IIc is slightly greater than 1 and IIc >> 1, for a Josephson junction in the over-damped limit β = 0. To sketch the instantaneous voltage V(t) for both the cases

Concept Introduction:

The Josephson relation in the over-damped limit β = 0 is given by ϕ˙ = IIc - sin ϕ.

Josephson voltage-phase relation is given by V = 2eϕ˙

Expert Solution & Answer
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Answer to Problem 1E

Solution:

a) The super-current Ic sin ϕ (t) as a function of t assuming that IIc is slightly greater than 1 and IIc >> 1 for a Josephson junction in the over-damped limit β = 0 is sketched.

b) The instantaneous voltage V(t) for both the cases is sketched.

Explanation of Solution

Josephson junctions are superconducting devices that are capable of generating voltage oscillations of extraordinarily high frequency. It consists of two closely spaced superconductors separated by a weak connection.

The Josephson relation in the over-damped limit β = 0 is given by

ϕ˙ = IIc - sin ϕ

Here, Ic is the critical current.

Assuming that IIc is slightly greater than 1

Consider IIc = 1.1

ϕ˙ = 1.1 - sin ϕ

The graph of ϕ vs. t is shown below

Nonlinear Dynamics and Chaos, Chapter 4.6, Problem 1E , additional homework tip  1

Assuming that IIc >> 1.

Consider IIc = 10,

ϕ = 10 - sin ϕ

The graph of ϕ vs. t is shown below.

Nonlinear Dynamics and Chaos, Chapter 4.6, Problem 1E , additional homework tip  2

b) Josephson voltage-phase relation is given by

V = 2eϕ˙

Substituting ϕ˙ = IIc - sin ϕ, we get

V = 2e(IIc - sin ϕ)

Consider IIc = 1.1

The graph of voltage vs. ϕ is shown below.

Nonlinear Dynamics and Chaos, Chapter 4.6, Problem 1E , additional homework tip  3

Nonlinear Dynamics and Chaos, Chapter 4.6, Problem 1E , additional homework tip  4

V = 2e(IIc - sin ϕ)

Consider IIc = 10

The graph of voltage vs. ϕ is shown below.

Nonlinear Dynamics and Chaos, Chapter 4.6, Problem 1E , additional homework tip  5

Nonlinear Dynamics and Chaos, Chapter 4.6, Problem 1E , additional homework tip  6

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