TELECOMMUNICATIONS

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TELECOMMUNICATIONS

• Dr. Hugh Blanton
• ENTC 4307/ENTC 5307

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Complex Numbers
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Complex numbers
ARGAND diagram

• M A jB
• Where j2 -1 or j v-1
• M v(A2 B2)
• and tan ? B/A

Imaginary
B
M
?
A
Real
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j notation

• Refers to the expression
• Z R jX
• X is not imaginary
• Physically the j term refers to
• j 90o lead and -j 90o lag

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Complex Number Definitions

• Rectangular Coordinate System
• Real (x) and Imaginary (y) components, A x jy
• Complex Conjugate (A?A) refers to the same real
  part but the negative of the imaginary part.
• If A x jy, then A x ? jy.

jy
2
1
?x
x
1
2
?1
?2
?jy
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Complex Number Definitions

• Polar Coordinates Magnitude and Angle
• Complex conjugate has the same magnitude but the
  negative of the angle.
• If A M?90?, then AM?-90?

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• Rectangular to Polar Conversion
• By trigonometry, the phase angle q is,
• Polar to Rectangular Conversion
• y imaginary part M(sin q)
• x real part M(cos q)

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jy
2
1
x
1
2
?1
?2
?jy
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Vector Addition Subtraction

• Vector addition and subtraction of complex
  numbers are conveniently done in the rectangular
  coordinate system, by adding or subtracting their
  corresponding real and imaginary parts.
• If A 2 j1 and B 1 j2
• Then their sum is
• A B (21) j(1 2) 3 j1
• and the difference is
• A - B (2 ? 1) j(1 ?( 2)) 1 j3

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• For vector multiplication use polar form.
• The magnitudes (MA,MB) are multiplied together
  while the angles (q) are added.
• MuItiplying A and B
• AB (2.24 ?26.60?)(2.24 ??63.40?)
• 5 ??36.8 ?

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• Vector division requires the ratio of magnitudes
  and the differences of the angles

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jy
A-B
2
1
x
?1
?2
1
2
?1
AB
?2
?jy
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Complex Impedance System
inductive

• RF components are frequently defined by their
  terminal impedances or admittances in the complex
  rectangular coordinate system.
• Complex impedance is the vector sum of resistance
  and reactance.
• Impedance Resistance j Reactance

jX
?R
R
?jX
capacitive
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• Series connections are handled most conveniently
  in the impedance system.

?
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Complex Admittance System

• Parallel circuit descriptions may be viewed in
  the complex admittance system
• Complex impedance is the vector sum of
  conductance and susceptance.
• Admittance Conductance j Susceptance
• where and

capacitive
jB
?G
G
?jB
inductive
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• Parallel connections are handled most
  conveniently in the admittance system.

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Z dependence on ??(RCL )
Impedance
1000
500
series
100
50
parallel
?o
10
5
frequency
1
1
2
5
10.
20.
50.
100.
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Currrent dependence on ??
Current (ma)
1000
parallel
??o
500
Imaxxv2
series
Imin
100
frequency
1
2
5
10.
20.
50.
100.
?o
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• At RF, particularly at high power levels, it is
  very important to maximize power transfer through
  careful impedance matching.
• Improperly matched component connection leads to
  mismatch loss.

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RF Components Related Issues

• Unique component problems at RF
• Parasitics change behavior
• Primary and secondary resonances
• Distributed vs. lumped models
• Limited range of practical values
• Tolerance effects
• Measurements and test fixtures
• Grounding and coupling effects
• PC-board effects

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V and I Phase relationships
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R, XC and Z relationships
XL
Z
XL-XC
?
I
R
XC
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Example 1

• Consider this circuit with ? 105 rad s-1

1 k?
0.01 ?F
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Example 2
20 ?
10 ?
10 ?
5 ?
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Example 2 Contd
200 V
20 ?
10 ?
10 ?
5 ?

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Example 2 Contd
VL
XL
VR
R
?
VL-VC
?
XL-XC
I
I
VS
Z
VC
XC
Z 15 - 10j
I 8/13(15 10j)
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General procedures

• convert all reactances to ohms
• express impedance in j notation
• determine Z using absolute value
• determine I using complex conjugate
• draw phasor diagram
• Note j -1/j so
• R (1/j?C) R - j/?C

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Example 3
10 ?
20 ?
15 ?
5 ?
- express the impedance in j notation - determine
Z (in ?s) and ? - determine I for a voltage of 24V
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Example 3 Contd
10 ?
20 ?
5 ?
5 ?
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Example 3 Contd
24V
20 ?
10 ?
10 ?
5 ?

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Example 4

• Construct a circuit which contains at least one L
  and one C components which could be represented
  by
• Z 10 - 30j

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Parallel circuits

10 - 30j
20 - 10j

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Parallel circuits

20 - 30j
20 - 30j