Complex Numbers
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Complex Numbers - Telecommunications - Lecture Slides, Slides of Telecommunication electronics
This is the Lecture Slides of Telecommunications which includes Phase Lock Loop, Feedback System, Selected Input Signal, Frequency Changes, Phase Detector, Loop Filter, Voltage Controlled Oscillator, Periodic Input Signal etc. Key important points are: Complex Numbers, Not Imaginary, Complex Number Definitions, Rectangular Coordinate System, Complex Conjugate, Real Part, Negative of the Imaginary Part, Polar Coordinates, Magnitude and Angle, Negative of the Angle
Typology: Slides
2012/2013
1 / 32
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Complex Numbers
Complex numbers
M = A + jB
Where j 2 = -1 or j = √-
|M | = √(A^2 + B^2 )
and tan φ = B/A
B
A
M
Real
I m a g i n a r y
ARGAND diagram
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.
−x
+jy
−jy
+x
A = x + jy = 2 + j 1 = 2. 24 ∠ 26. 6^
B = x − jy = 1 − j 2 = 2. 24 ∠− 63. 4^
1
2
− 1 − 2
1 2
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 A*=M∠-90°
+jy
−jy
+x
A = x + jy = 2 + j 1 = 2. 24 ∠ 26. 6^
B = x − jy = 1 − j 2 = 2. 24 ∠− 63. 4^
1
2
− 1 − 2
1 2
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 = (2+1) + j(1 – 2) = 3 – j
- and the difference is:
- A - B = (2 − 1) + j(1 −(– 2)) = 1 + j
- Then their sum is:
• Vector division requires the ratio of
magnitudes and the differences of the angles:
B
A
+jy
−jy
+x
A = x + jy = 2 + j 1 = 2. 24 ∠ 26. 6^
B = x − jy = 1 − j 2 = 2. 24 ∠− 63. 4
1
2
− 1 − 2
− 2 −^11
A-B
A+B
- Series connections are handled most
conveniently in the impedance system.
( ) 1 2 (^ 1 2 )
1 2 1 1 2 2 R R jX jX
Z (^) T Z Z R jX R jX = + + ± ±
= + = ± + ± ( )
( ) ⇐
Z 1 Z 2
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
G jB Mhos or Siemens Z
Y = = ( ± ) ,
1 −G
+jB
−jB
+G
inductive
Y = G 1 + jB 1
G 1
− jB 1
jB 1
capacitive
Y = G 1 − jB 1 ∗ R
G
1 = jB^ jX
1 ± =
Z dependence on ω (RCL )
1 frequency
Impedance
ωo
parallel
series
Currrent dependence on ω
frequency
Current (ma)
∆ωo
ωo
Imin
Imax x√
parallel
series
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
V and I Phase relationships
V (^) S = V (^) R
- (^) ( V (^) L − VC )
and
tan ϕ =
( V^ L −^ VC )
V (^) R
Io = V (^) o / Z
VL
V^ I
R