quick check on the BC's met at low w and curl too REVIEWED
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Phil's working note in the Transmission Lines folder (Low Freq Aug/Sept), marked reviewed. It evaluates the low-ω Ez, Er and Eθ mode fields at r=a and confirms Eθ=0 and Er=(jω/σ)Nm using N0=CV/(2πa) and Rdc. It then checks the cylindrical curl components against several textbook and web sources and his own lines document, and concludes the curl expressions are correct.
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Extracted text (machine-read; may contain errors)
Quickie : Verify that the low-ω E fields satisfy the two BC's.
Quickie : Make sure curl components are correct 10.5.14
These are just a few "checks" on my theory, both OK. The more checks the better. Always looking for something wrong, it goes on forever.
Here are the E fields
Ez(r,m) = (1/2) ηm CV Rdc (ω/k) (r/a)|m| (|m|+1) (7.4.1)
Er(r,m) = (j/4) ηm CV Rdc (ωa) [(r/a)|m|+1 + (r/a)|m|-1]
Eθ(r,m) = (1/4) ηm CV Rdc (ωa) [(r/a)|m|+1 - (r/a)|m|-1]
Ez(r,0) = CV Rdc (ω/k)
Er(r,0) = (j/2) CV Rdc (ωr)
Eθ(r,0) = 0 // low ω E fields
Here are the E fields evaluated at r = a:
Ez(a,m) = (1/2) ηm CV Rdc (ω/k) (|m|+1) (7.4.1)
Er(a,m) = (j/4) ηm CV Rdc (ωa) (2)
Eθ(a,m) = 0
Ez(a,0) = CV Rdc (ω/k)
Er(a,0) = (j/2) CV Rdc (ωa)
Eθ(a,0) = 0 // low ω E fields
The Eθ BC is obviously OK.
The other BC says,
Er(r=a,m) = (jω/σ) Nm
But from (D.1.8) I know that
N0 = (1/2πa) q(0) = <n(θ)> // q(0) = CV (D.1.8)
or
N0 = CV /(2πa)
so
Nm = ηm CV /(2πa)
so this BC then requires that
Er(r=a,m) = (jω/σ) Nm = (jω/σ) ηm CV /(2πa) = jω ηm CV (1/2) a (1/σπa2)
= jω ηm CV (1/2) a Rdc = (j/2) ηm CV (ωa) Rdc
and this does in fact agree with the above box.
Quickie : Make sure my cylindrical curl equation is right.
In my diffops doc I state it this way
curl F = [ r-1∂θFz - ∂zFθ] + [∂zFr - ∂rFz] + [ r-1∂r(rFθ) - r-1∂θFr ]
Source #1: M&F state it this way which I already quote in lines doc
So this source agrees with me.
Source #2: M&S page 12 have it in matrix form. I read their thing as saying this
r curl E = [ ∂θEz- ∂z(Eφr)] - r [ ∂rEz - ∂zEr ] + [ ∂r(Eθr) - ∂θEr ]
so then
curl E = [ r-1∂θEz- ∂z(Eφ)] + [- ∂rEz + ∂zEr ] + [ r-1∂r(Eθr) - r-1∂θEr ]
OK OK OK
So this source agrees with me.
Source #3. Smythe page 60, also agrees with diffops above.
Source #4: A web pdf
which also agrees. That is enough.
Now using ejmθ and e-jkz I can say
curl F = [ r-1∂θFz - ∂zFθ] + [∂zFr - ∂rFz] + [ r-1∂r(rFθ) - r-1∂θFr ]
curl F = [ r-1jmFz+jkFθ] + [-jkFr - ∂rFz] + [ r-1∂r(rFθ) - r-1jmFr ]
Source #5 : lines doc
curl E = [ r-1∂θEz - ∂zEθ] + [∂zEr - ∂rEz] + [ r-1∂r(rEθ) - r-1∂θEr ] (D.4.2)
-jωB(r,m) = [ r-1jmEz +jkEθ] + [-jkEr - ∂rEz] + [ r-1∂r(rEθ) - r-1jmEr ] (D.4.6)
Source #6. My Chapter 7:
Conclusion: I think I can rule out any possible error I might have made with the correct expressions for the curl components.