ece 1100 introduction to electrical and computer engineeringcourses.egr.uh.edu/ece/ece6341/class...
TRANSCRIPT
![Page 1: ECE 1100 Introduction to Electrical and Computer Engineeringcourses.egr.uh.edu/ECE/ECE6341/Class Notes/Topic 3 Cylindrical/6341... · Prof. David R. Jackson . ECE Dept. Spring 2016](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a8933337f8b9ac96a8ece88/html5/thumbnails/1.jpg)
Prof. David R. Jackson ECE Dept.
Spring 2016
Notes 10
ECE 6341
1
![Page 2: ECE 1100 Introduction to Electrical and Computer Engineeringcourses.egr.uh.edu/ECE/ECE6341/Class Notes/Topic 3 Cylindrical/6341... · Prof. David R. Jackson . ECE Dept. Spring 2016](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a8933337f8b9ac96a8ece88/html5/thumbnails/2.jpg)
Dielectric Rod
a
z
,r rε µ
This serves as a model for a fiber-optic guide.
2
![Page 3: ECE 1100 Introduction to Electrical and Computer Engineeringcourses.egr.uh.edu/ECE/ECE6341/Class Notes/Topic 3 Cylindrical/6341... · Prof. David R. Jackson . ECE Dept. Spring 2016](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a8933337f8b9ac96a8ece88/html5/thumbnails/3.jpg)
Fiber Optic Guides
3
Two types of fiber-optic guides:
1) Single-mode fiber
2) Multi-mode fiber
This fiber carries a single mode (HE11). This requires the fiber diameter to be on the order of a wavelength. It has less loss, dispersion, and signal distortion. It is often used for long-distances (e.g., greater than 1 km). The diameter is typically 8 µm. It requires more expensive electro-optic equipment.
This fiber has a diameter that is large relative to a wavelength (e.g., 10 wavelengths). It operates on the principle of total internal reflection (critical-angle effect). It can handle more power than the single-mode fiber, but has more dispersion. The diameter is typically 50 µm. It is often used for LANs, etc.
![Page 4: ECE 1100 Introduction to Electrical and Computer Engineeringcourses.egr.uh.edu/ECE/ECE6341/Class Notes/Topic 3 Cylindrical/6341... · Prof. David R. Jackson . ECE Dept. Spring 2016](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a8933337f8b9ac96a8ece88/html5/thumbnails/4.jpg)
4
It usually has a yellow jacket.
Single-mode Fiber
![Page 5: ECE 1100 Introduction to Electrical and Computer Engineeringcourses.egr.uh.edu/ECE/ECE6341/Class Notes/Topic 3 Cylindrical/6341... · Prof. David R. Jackson . ECE Dept. Spring 2016](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a8933337f8b9ac96a8ece88/html5/thumbnails/5.jpg)
Multi-mode Fiber
5
http://en.wikipedia.org/wiki/Multi-mode_fiber
A 1.25 Gbit/s multi-mode fiber
![Page 6: ECE 1100 Introduction to Electrical and Computer Engineeringcourses.egr.uh.edu/ECE/ECE6341/Class Notes/Topic 3 Cylindrical/6341... · Prof. David R. Jackson . ECE Dept. Spring 2016](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a8933337f8b9ac96a8ece88/html5/thumbnails/6.jpg)
Fiber-optic Cable
6
Cable Types: (L to R): Zipcord, Distribution, Loose Tube, Breakout
http://www.thefoa.org/tech/ref/basic/cable.html
Fibers (single-mode or multi-mode) may be bundled together into a “fiber optic cable” that has one or more fibers.
Simplex: single fiber in a cable Duplex: two fibers in a cable
![Page 7: ECE 1100 Introduction to Electrical and Computer Engineeringcourses.egr.uh.edu/ECE/ECE6341/Class Notes/Topic 3 Cylindrical/6341... · Prof. David R. Jackson . ECE Dept. Spring 2016](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a8933337f8b9ac96a8ece88/html5/thumbnails/7.jpg)
7
Single-mode Fiber: Operation
HE11 mode
A single mode (HE11 mode) propagates on the dielectric rod. The mode is similar to the TM0 mode on a grounded slab. It has a zero cutoff frequency.
Fields decay away from the rod
![Page 8: ECE 1100 Introduction to Electrical and Computer Engineeringcourses.egr.uh.edu/ECE/ECE6341/Class Notes/Topic 3 Cylindrical/6341... · Prof. David R. Jackson . ECE Dept. Spring 2016](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a8933337f8b9ac96a8ece88/html5/thumbnails/8.jpg)
8
http://en.wikipedia.org/wiki/Optical_fiber
Higher index core region
A multimode fiber can be explained using geometrical optics and internal reflection. The “ray” of light is actually a superposition of many waveguide modes (hence the name “multimode”).
A laser bouncing down an acrylic rod, illustrating the
total internal reflection of light in a multi-mode optical fiber
Multi-mode Fiber: Operation
![Page 9: ECE 1100 Introduction to Electrical and Computer Engineeringcourses.egr.uh.edu/ECE/ECE6341/Class Notes/Topic 3 Cylindrical/6341... · Prof. David R. Jackson . ECE Dept. Spring 2016](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a8933337f8b9ac96a8ece88/html5/thumbnails/9.jpg)
Single-mode vs. Multi-mode Fibers
9
Graded index
![Page 10: ECE 1100 Introduction to Electrical and Computer Engineeringcourses.egr.uh.edu/ECE/ECE6341/Class Notes/Topic 3 Cylindrical/6341... · Prof. David R. Jackson . ECE Dept. Spring 2016](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a8933337f8b9ac96a8ece88/html5/thumbnails/10.jpg)
Dielectric Rod
Modes are hybrid unless 0 ( 0)nφ∂
= =∂
For example, assume TMz: zAψ =
1
1 z
H
jkEj z j
ρ
φ
ψµρ φ
ψ ψωµερ φ ωµερ φ
∂=
∂−∂ ∂
= =∂ ∂ ∂
a
z
1
0
,r rε µ
10
Note:
We can have TE0p, TM0p modes
![Page 11: ECE 1100 Introduction to Electrical and Computer Engineeringcourses.egr.uh.edu/ECE/ECE6341/Class Notes/Topic 3 Cylindrical/6341... · Prof. David R. Jackson . ECE Dept. Spring 2016](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a8933337f8b9ac96a8ece88/html5/thumbnails/11.jpg)
Dielectric Rod (cont.) At ρ = a :
so
1 1 0 0
1 0
H HE E
ρ ρ
φ φ
µ µ=
=
01
01
1 1 0 0
1 1
ψψφ φ
ψψµ ε φ µ ε φ
∂∂=
∂ ∂∂∂
=∂ ∂
Hence, for n > 0 we have 1 1 0 0µ ε µ ε= (Not true in general!) 11
![Page 12: ECE 1100 Introduction to Electrical and Computer Engineeringcourses.egr.uh.edu/ECE/ECE6341/Class Notes/Topic 3 Cylindrical/6341... · Prof. David R. Jackson . ECE Dept. Spring 2016](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a8933337f8b9ac96a8ece88/html5/thumbnails/12.jpg)
( ) ( )( ) ( )
1 1
1 1
sin
cos
z
z
jk zz n
jk zz n
A A J k n e
F B J k n eρ
ρ
ρ φ
ρ φ
−
−
=
=
where 2 2 21 1 zk k kρ = − (kz is unknown)
ρ < a:
Dielectric Rod (cont.)
12
Representation of potentials inside the rod:
![Page 13: ECE 1100 Introduction to Electrical and Computer Engineeringcourses.egr.uh.edu/ECE/ECE6341/Class Notes/Topic 3 Cylindrical/6341... · Prof. David R. Jackson . ECE Dept. Spring 2016](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a8933337f8b9ac96a8ece88/html5/thumbnails/13.jpg)
To see choice of sin/cos, examine the field components (for example Eρ):
1 z z zF k AEρ ερ φ ωµε ρ∂ − ∂
= − +∂ ∂
The field Eρ is assumed (arbitrarily) to vary as sin(nφ) for our choice of potentials.
We then have for the field components:
( ) ( ) ( )sin cos sinzE n E n E nρ φφ φ φ∝ ∝ ∝
Dielectric Rod (cont.)
13
( ) ( )cos sinz zF n A nφ φ∝ ∝
( ) ( ) ( )cos sin coszH n H n H nρ φφ φ φ∝ ∝ ∝
![Page 14: ECE 1100 Introduction to Electrical and Computer Engineeringcourses.egr.uh.edu/ECE/ECE6341/Class Notes/Topic 3 Cylindrical/6341... · Prof. David R. Jackson . ECE Dept. Spring 2016](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a8933337f8b9ac96a8ece88/html5/thumbnails/14.jpg)
where
( ) ( )2 20 0( ) ( )n nH k H jρ ρρ α ρ= −
ρ > a:
Note: αρ0 is interpreted as a positive real number in order to have decay radially in the air region, for a bound (non-leaky) mode.
2 20 0zk kρα = −
Use
Dielectric Rod (cont.)
( )1/22 20 0 0zk k k jρ ρα= − = −
14
Representation of potentials outside the rod:
![Page 15: ECE 1100 Introduction to Electrical and Computer Engineeringcourses.egr.uh.edu/ECE/ECE6341/Class Notes/Topic 3 Cylindrical/6341... · Prof. David R. Jackson . ECE Dept. Spring 2016](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a8933337f8b9ac96a8ece88/html5/thumbnails/15.jpg)
Kn (x) = Modified Bessel function of the second kind.
( )11( ) ( )2
nn nK x j H jxπ +≡
( ) ( ) ( )12 1( ) 1 ( )nn nH jx H jx+− = − +Useful identity:
Another useful identity:
Dielectric Rod (cont.)
15
![Page 16: ECE 1100 Introduction to Electrical and Computer Engineeringcourses.egr.uh.edu/ECE/ECE6341/Class Notes/Topic 3 Cylindrical/6341... · Prof. David R. Jackson . ECE Dept. Spring 2016](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a8933337f8b9ac96a8ece88/html5/thumbnails/16.jpg)
Modified Bessel function of the second kind
Dielectric Rod (cont.)
0 1 2 3 4 50
0.2
0.4
0.6
0.8
11
3.691 10 3−×
K0 x( )
K1 x( )
55 10 3−× x
( )0K x
( )1K x
x16
![Page 17: ECE 1100 Introduction to Electrical and Computer Engineeringcourses.egr.uh.edu/ECE/ECE6341/Class Notes/Topic 3 Cylindrical/6341... · Prof. David R. Jackson . ECE Dept. Spring 2016](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a8933337f8b9ac96a8ece88/html5/thumbnails/17.jpg)
The modified Bessel function of the first kind grows exponentially, so we don’t want this one.
Dielectric Rod (cont.)
17
( ) ( ) ( )nn nI x i J ix≡ −
x
I0
I1
I2
![Page 18: ECE 1100 Introduction to Electrical and Computer Engineeringcourses.egr.uh.edu/ECE/ECE6341/Class Notes/Topic 3 Cylindrical/6341... · Prof. David R. Jackson . ECE Dept. Spring 2016](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a8933337f8b9ac96a8ece88/html5/thumbnails/18.jpg)
Hence, we choose
0 0
0 0
( )sin( )
( )cos( )
z
z
jk zz n
jk zz n
A CK n e
F DK n eρ
ρ
α ρ φ
α ρ φ
−
−
=
=
Dielectric Rod (cont.)
18
![Page 19: ECE 1100 Introduction to Electrical and Computer Engineeringcourses.egr.uh.edu/ECE/ECE6341/Class Notes/Topic 3 Cylindrical/6341... · Prof. David R. Jackson . ECE Dept. Spring 2016](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a8933337f8b9ac96a8ece88/html5/thumbnails/19.jpg)
Summary of potentials:
0 0
0 0
( )sin( )
( )cos( )
z
z
jk zz n
jk zz n
A CK n e
F DK n eρ
ρ
α ρ φ
α ρ φ
−
−
=
=
Dielectric Rod (cont.)
19
( ) ( )( ) ( )
1 1
1 1
sin
cos
z
z
jk zz n
jk zz n
A A J k n e
F B J k n eρ
ρ
ρ φ
ρ φ
−
−
=
=
![Page 20: ECE 1100 Introduction to Electrical and Computer Engineeringcourses.egr.uh.edu/ECE/ECE6341/Class Notes/Topic 3 Cylindrical/6341... · Prof. David R. Jackson . ECE Dept. Spring 2016](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a8933337f8b9ac96a8ece88/html5/thumbnails/20.jpg)
Match Ez , Hz , Eφ , Hφ at ρ = a:
11 12 13 14
21 22 23 24
31 32 33 34
41 42 43 44
0000
M M M M AM M M M BM M M M CM M M M D
=
Dielectric Rod (cont.)
Example: 1 0z zE E=
( ) ( ) ( ) ( )2 2 2 21 1 0 0
1 1 0 0
1 1z n z nk k AJ k a k k CK a
j jρ ραωµ ε ωµ ε
− = −
( ) ( ) ( ) ( )2 2 2 211 1 1 13 0 0 12 14
1 1 0 0
1 1 0z n z nM k k J k a M k k K a M Mj jρ ραωµ ε ωµ ε
−= − = − = =
so
20
22
21
z zE k Aj zωµε
∂= + ∂
![Page 21: ECE 1100 Introduction to Electrical and Computer Engineeringcourses.egr.uh.edu/ECE/ECE6341/Class Notes/Topic 3 Cylindrical/6341... · Prof. David R. Jackson . ECE Dept. Spring 2016](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a8933337f8b9ac96a8ece88/html5/thumbnails/21.jpg)
To have a non-trivial solution, we require that
11 12 13 14
21 22 23 24
31 32 33 34
41 42 43 44
0000
M M M M AM M M M BM M M M CM M M M D
=
det , ) 0( zM k ω =
Dielectric Rod (cont.)
21
kz = unknown (for a given frequency ω)
![Page 22: ECE 1100 Introduction to Electrical and Computer Engineeringcourses.egr.uh.edu/ECE/ECE6341/Class Notes/Topic 3 Cylindrical/6341... · Prof. David R. Jackson . ECE Dept. Spring 2016](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a8933337f8b9ac96a8ece88/html5/thumbnails/22.jpg)
Set
0 0det , ) 0( c cM µ εω ω =
Dielectric Rod (cont.)
Cutoff frequency:
0 0 0z ck k ω µ ε= =
Then
22
Note: This is an open structure, so cutoff means the boundary
between proper and improper behavior (kz = k0).
The unknown is now ωc.
![Page 23: ECE 1100 Introduction to Electrical and Computer Engineeringcourses.egr.uh.edu/ECE/ECE6341/Class Notes/Topic 3 Cylindrical/6341... · Prof. David R. Jackson . ECE Dept. Spring 2016](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a8933337f8b9ac96a8ece88/html5/thumbnails/23.jpg)
Dominant mode (lowest cutoff frequency): HE11 (fc = 0)
The field shape is somewhat similar to the TE11 circular waveguide mode.
E
Dielectric Rod (cont.)
Note: The notation HE means that the mode is hybrid, and has both Ez and Hz, although Hz is stronger. (For an EH mode, Ez would be stronger.)
The physical properties of the fields are similar to those of the TM0 surface wave on a slab (For example, at low frequency the field is loosely bound to the rod.)
23
![Page 24: ECE 1100 Introduction to Electrical and Computer Engineeringcourses.egr.uh.edu/ECE/ECE6341/Class Notes/Topic 3 Cylindrical/6341... · Prof. David R. Jackson . ECE Dept. Spring 2016](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a8933337f8b9ac96a8ece88/html5/thumbnails/24.jpg)
When will the next mode be at cutoff? This determines the upper frequency limit for the single-mode fiber.
Dielectric Rod (cont.)
24
Cutoff: 0zk k=
0 0 ( )k kρ ρ= = in air region
21Ej zφ
ψωµερ φ
∂=
∂ ∂
22 2
2
1 1zE k k
j z j ρψ ψωµε ωµε
∂= + = ∂
The next mode (i.e., with the next lowest cutoff frequency) is the TM01 mode.
zAψ =
The tangential field in the air region at the boundary becomes zero at the cutoff frequency: the
rod acts like a circular waveguide with a PEC wall.
(Recall: For n = 0, the modes are TMz and TEz.)
![Page 25: ECE 1100 Introduction to Electrical and Computer Engineeringcourses.egr.uh.edu/ECE/ECE6341/Class Notes/Topic 3 Cylindrical/6341... · Prof. David R. Jackson . ECE Dept. Spring 2016](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a8933337f8b9ac96a8ece88/html5/thumbnails/25.jpg)
Dielectric Rod (cont.)
25
2 21 0 2.405k k a− =
TM01 mode at cutoff: 1 01 2.405k a xρ = =
20 1 1 2.405k a n − =
0 21
2.4051
k an
<−
or
Hence, to have only the HE11 mode, we have the following frequency restriction:
n1 = index of refraction of rod
![Page 26: ECE 1100 Introduction to Electrical and Computer Engineeringcourses.egr.uh.edu/ECE/ECE6341/Class Notes/Topic 3 Cylindrical/6341... · Prof. David R. Jackson . ECE Dept. Spring 2016](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a8933337f8b9ac96a8ece88/html5/thumbnails/26.jpg)
A round wire made of conducting material is examined.
The wire has a conductivity of σ.
Impedance of Wire
26
a
σ
z
We neglect the z variation of the fields inside the wire. (The wire is short compared with a wavelength.)
Goal: Determine the impedance per unit length (in the z direction).
![Page 27: ECE 1100 Introduction to Electrical and Computer Engineeringcourses.egr.uh.edu/ECE/ECE6341/Class Notes/Topic 3 Cylindrical/6341... · Prof. David R. Jackson . ECE Dept. Spring 2016](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a8933337f8b9ac96a8ece88/html5/thumbnails/27.jpg)
Inside the wire:
Impedance of Wire (cont.)
27
a
σ z
( )0zE AJ kρ=
( )
( )
/4
/4
1
22
c
j
j
k
j
j
j
e
e
π
π
ω µε
σω µεωε
σω µεωε
ωµσ
ωµσ
ωµσ
−
−
=
= −
≈ −
= −
=
=
(The field must be finite on the z axis.)
![Page 28: ECE 1100 Introduction to Electrical and Computer Engineeringcourses.egr.uh.edu/ECE/ECE6341/Class Notes/Topic 3 Cylindrical/6341... · Prof. David R. Jackson . ECE Dept. Spring 2016](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a8933337f8b9ac96a8ece88/html5/thumbnails/28.jpg)
Hence, we have
Impedance of Wire (cont.)
28
a
σ z
/40 2 j
zE AJ e πρδ
− =
2δωµσ
=
where
(skin depth)
We can also write the field as
3 /4 3 /40 02 2j j
zE AJ e AJ eπ πρ ρδ δ
= − =
![Page 29: ECE 1100 Introduction to Electrical and Computer Engineeringcourses.egr.uh.edu/ECE/ECE6341/Class Notes/Topic 3 Cylindrical/6341... · Prof. David R. Jackson . ECE Dept. Spring 2016](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a8933337f8b9ac96a8ece88/html5/thumbnails/29.jpg)
Impedance of Wire (cont.)
29
a
σ z
Therefore, we can write
3 /40 2 j
zE AJ e πρδ
=
( ) ( )( )( ) ( )( )
3 /4
3 /4
Ber Re
Bei Im
j
j
x J xe
x J xe
πυ υ
πυ υ
≡
≡
Definition of Kelvin functions:
0 0Ber 2 Bei 2zE A jρ ρδ δ
= +
![Page 30: ECE 1100 Introduction to Electrical and Computer Engineeringcourses.egr.uh.edu/ECE/ECE6341/Class Notes/Topic 3 Cylindrical/6341... · Prof. David R. Jackson . ECE Dept. Spring 2016](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a8933337f8b9ac96a8ece88/html5/thumbnails/30.jpg)
Impedance of Wire (cont.)
30
a
σ z
The current flowing in the wire is
2
0 0
0
0
2
2
zS
a
z
a
z
a
z
I J dS
J d d
J d
E d
π
ρ ρ φ
π ρ ρ
πσ ρ ρ
=
=
=
=
∫
∫ ∫
∫
∫
3 /40
0
2 2a
jI A J e dπρπσ ρ ρδ
= ∫Hence
3 /40 2 j
zE AJ e πρδ
=
![Page 31: ECE 1100 Introduction to Electrical and Computer Engineeringcourses.egr.uh.edu/ECE/ECE6341/Class Notes/Topic 3 Cylindrical/6341... · Prof. David R. Jackson . ECE Dept. Spring 2016](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a8933337f8b9ac96a8ece88/html5/thumbnails/31.jpg)
Impedance of Wire (cont.)
31
a
σ z
3 /40
3 /40
0
2
2 2
j
l aj
aAJ eZ
A J e d
π
π
δρπσ ρ ρδ
= ∫
The internal impedance per unit length is defined as:
( )zl
E aZ
I≡
Hence,
Note: The internal impedance accounts for the internal stored energy and power dissipation.
![Page 32: ECE 1100 Introduction to Electrical and Computer Engineeringcourses.egr.uh.edu/ECE/ECE6341/Class Notes/Topic 3 Cylindrical/6341... · Prof. David R. Jackson . ECE Dept. Spring 2016](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a8933337f8b9ac96a8ece88/html5/thumbnails/32.jpg)
Impedance of Wire (cont.)
32
a
σ z
( )
( )( )
23 /4 3 /4
0 00 0
1 0
1
22
a Lj j
L
J e d e J x xdx
xJ x
LJ L
π πρ δρ ρδ
− =
=
=
∫ ∫
We also have the following helpful integration identity:
( ) ( )0 1J x xdx xJ x=∫
3 /42 jaL e π
δ≡where
Hence
3 /4
3 /4
2
2
j
j
x e
dx d e
π
π
ρδ
ρδ
=
=
Use :
![Page 33: ECE 1100 Introduction to Electrical and Computer Engineeringcourses.egr.uh.edu/ECE/ECE6341/Class Notes/Topic 3 Cylindrical/6341... · Prof. David R. Jackson . ECE Dept. Spring 2016](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a8933337f8b9ac96a8ece88/html5/thumbnails/33.jpg)
Impedance of Wire (cont.)
33
a
σ z
Hence, we have
3 /40
23 /4 3 /4 3 /4
1
2
2 2 22
j
lj j j
aJ eZ
a ae e J e
π
π π π
δδπσ
δ δ−
=
3 /40 0 0
3 /41 1 1
Ber Bei
Ber Bei
j
j
a a aJ e j
a a aJ e j
π
π
δ δ δ
δ δ δ
= + = +
where
![Page 34: ECE 1100 Introduction to Electrical and Computer Engineeringcourses.egr.uh.edu/ECE/ECE6341/Class Notes/Topic 3 Cylindrical/6341... · Prof. David R. Jackson . ECE Dept. Spring 2016](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a8933337f8b9ac96a8ece88/html5/thumbnails/34.jpg)
Impedance of Wire (cont.)
34
a
σ z
At low frequency (a << δ):
( )0
2
18
rlZ j
aµ µω
πσ π ≈ +
where
At high frequency (a >> δ):
2s
lZZ
aπ≈
( )1
1 ( )2
s s
s
Z R j
R ωµσδ σ
= +
= = surface resistance of metal
![Page 35: ECE 1100 Introduction to Electrical and Computer Engineeringcourses.egr.uh.edu/ECE/ECE6341/Class Notes/Topic 3 Cylindrical/6341... · Prof. David R. Jackson . ECE Dept. Spring 2016](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a8933337f8b9ac96a8ece88/html5/thumbnails/35.jpg)
Impedance of Wire (cont.)
35
An extra impedance per unit length Zi is added to the TL model in order to account for the internal impedance of the conductors.
R ∆z
C ∆z G ∆z
L0 ∆z
Zi ∆z
Transmission Line Model
The external inductance per unit length L0 is calculated assuming perfectly conducting wires.
Note: The form of Zi will depend on the shape of the conductors: we have only analyzed the case of a round wire.
![Page 36: ECE 1100 Introduction to Electrical and Computer Engineeringcourses.egr.uh.edu/ECE/ECE6341/Class Notes/Topic 3 Cylindrical/6341... · Prof. David R. Jackson . ECE Dept. Spring 2016](https://reader031.vdocuments.site/reader031/viewer/2022030408/5a8933337f8b9ac96a8ece88/html5/thumbnails/36.jpg)
Impedance of Wire (cont.)
36
For a twin lead (two wires running parallel) we have:
Transmission Line Model
2i lZ Z≈
i
+ -
v z
We assume that the wires are far enough apart so that the current is uniform inside each one.