SLIDE 1
- P. Piot, PHYS 571 – Fall 2007
- P. Piot, PHYS 571 – Fall 2007
Boundary conditions I
- Electric field components
- Magnetic field components
Prescription for e.m. field calculations P. Piot, PHYS 571 Fall - - PowerPoint PPT Presentation
Prescription for e.m. field calculations P. Piot, PHYS 571 Fall - - PowerPoint PPT Presentation
Prescription for e.m. field calculations P. Piot, PHYS 571 Fall 2007 Boundary conditions I Electric field components z z S Magnetic field components end plates Proof: start with what we had last lesson P. Piot, PHYS 571
SLIDE 2
SLIDE 3
- P. Piot, PHYS 571 – Fall 2007
Boundary conditions II
- Which gives
- So our the boundary conditions are
But On Surface S This is the gradient of Bz projected on n
SLIDE 4
- P. Piot, PHYS 571 – Fall 2007
Resonant mode types
- Resonant mode categorization
The two aforementioned boundary conditions cannot be satisfied
- simultaneously. To each of this boundary condition corresponds a
mode type:
- In old books/literature this is referred to as H and E modes
SLIDE 5
- P. Piot, PHYS 571 – Fall 2007
Wave equation in (r,φ,z) and its solutions I
- We first want to find Ez or Bz.
- Boundary condition at end plates
- Defining the
wave equation takes the form
0 L z
SLIDE 6
- P. Piot, PHYS 571 – Fall 2007
Wave equation in (r,φ,z) and its solutions II
- Once Ψ(r,φ) is found the transverse field can be computed from
- In (r,φ,z) the wave equation is:
SLIDE 7
- P. Piot, PHYS 571 – Fall 2007
Wave equation in (r,φ,z) and its solutions III
- We now further assume an harmonic azimutal dependence
- The wave equation simplifies to the Bessel’s differential equation
[see Arken & Weber (6th Edition) p. 678]
- Which has solution of the form
since
SLIDE 8
- P. Piot, PHYS 571 – Fall 2007
Wave equation in (r,φ,z) – Bessel Functions
First kind Bessel Functions Second kind Bessel Functions Y ≡ N
SLIDE 9
- P. Piot, PHYS 571 – Fall 2007
Wave equation in (r,φ,z) and its solutions IV and resonant frequencies
- So finally
where – ......... – R is the cavity radius, – γnm is defined to insure Ψ→0 as r →R.
- From the definition
we note that only discrete frequencies are allowed:
SLIDE 10
- P. Piot, PHYS 571 – Fall 2007
TM mode
- For TM modes so that
- The transverse E-field is
using
SLIDE 11
- P. Piot, PHYS 571 – Fall 2007
TM mode: transverse E-field
SLIDE 12
- P. Piot, PHYS 571 – Fall 2007
TM mode: transverse B-field
SLIDE 13
- P. Piot, PHYS 571 – Fall 2007
TM mode: summary
SLIDE 14
- P. Piot, PHYS 571 – Fall 2007
TE mode: summary
SLIDE 15
- P. Piot, PHYS 571 – Fall 2007