1. EKT 441 MICROWAVE COMMUNICATIONS
CHAPTER 1:
TRANSMISSION LINE THEORY

2. OUR MENU (PART 1) Introduction to Microwaves
Transmission Line Equations
The Lossless Line
Terminated Transmission Lines
Reflection Coefficient
VSWR
Return Loss
Transmission Lines Impedance Equations
Special Cases of Terminated Transmission Lines

3. SPECTRUM & WAVELENGTHS
Wavelength of a wave is the distance we have to move along the transmission line for the sinusoidal voltage to repeat its pattern
Waves in the electromagnetic spectrum vary in size from very long radio waves the size of buildings, to very short gamma-rays smaller than the size of the nucleus of an atom.

4. INTRODUCTION
Microwave refers to alternating current signals with frequencies between 300 MHz and 300 GHz.
Figure 1 shows the location of the microwave frequency
Long wave radio
AM broad
Casting radio
Short
wave
radio
VHF TV FM
broad
casting
Microwaves
Far
infrared
Visible
light
3 x x x x x x x x x x1014
Typical frequencies
AM broadcast band kHz VHF TV (5-6) MHz
Shortwave radio 3-30 MHz UHF TV (7-13) MHz
FM broadcast band MHz UHF TV (14-83) MHz
VHF TV (2-4) MHz Microwave ovens 2.45 GHz

5. MICROWAVE BAND DESIGNATION
Frequency (GHz)
Wavelength (cm)
IEEE band
1 - 2 L
2 - 4 S
4 - 8 C
8 - 12 X Ku K Ka mm

6. APPLICATION OF MICROWAVE ENGINEERING
Communication systems
UHF TV
Microwave Relay
Satellite Communication
Mobile Radio
Telemetry
Radar system
Search & rescue
Airport Traffic Control
Navigation
Tracking
Fire control
Velocity Measurement
Microwave Heating
Industrial Heating
Home microwave ovens
Environmental remote sensing
Medical system
Test equipment

7. TYPICAL Rx ARCHITECTURE
Typical receiver (from RF & Microwave Wireless Systems, Wiley)

8. TYPICAL Rx ARCHITECTURE
When signal arrives at Rx, normally it is amplified by a Low Noise Amplifier (LNA)
Mixer then produce a down-converted signal at freq of fIF+fm OR fIF-fm; fIF<fm
Signal is then filtered to remove undesired harmonics & spurious products from mixing process
Signal is then amplified by an intermediate freq (IF) amplifier
Output signal of amplifier goes to detector for the recovery of the original message

9. TYPICAL Tx ARCHITECTURE
Typical transmitter architecture (from RF & Microwave Wireless System, Wiley)

10. TYPICAL Tx ARCHITECTURE
Input baseband signals (video, data, or voice) is assumed to be bandlimited to a freq fm
Signal is filtered to remove any components beyond passband
Message signal is then mixed with a local oscillator (LO) to produce modulated carrier (up-conversion) (fLO+fm OR fLO-fm), fm<fLO
Modulated carrier can be amplified & transmitted by the antenna

11. TRANSMISSION LINES - I Low frequencies+ -
Low frequencies
wavelengths >> wire length
current (I) travels down wires easily for efficient power transmission
measured voltage and current not dependent on position along wire
High frequencies
wavelength » or << length of transmission medium
need transmission lines for efficient power transmission
matching to characteristic impedance (Zo) is very important for low
reflection and maximum power transfer
measured envelope voltage dependent on position along line
The need for efficient transfer of RF power is one of the main reasons behind the use of transmission lines. At low frequencies where the wavelength of the signals are much larger than the length of the circuit conductors, a simple wire is very useful for carrying power. Current travels down the wire easily, and voltage and current are the same no matter where we measure along the wire.
At high frequencies however, the wavelength of signals of interest are comparable to or much smaller than the length of conductors. In this case, power transmission can best be thought of in terms of traveling waves.
Of critical importance is that a lossless transmission line takes on a characteristic impedance (Zo). In fact, an infinitely long transmission line appears to be a resistive load! When the transmission line is terminated in its characteristic impedance, maximum power is transferred to the load. When the termination is not Zo, the portion of the signal which is not absorbed by the load is reflected back toward the source. This creates a condition where the envelope voltage along the transmission line varies with position. We will examine the incident and reflected waves on transmission lines with different load conditions in following slides

12. TRANSMISSION LINE EQUATIONS
Complex amplitude of a wave may be defined in 3 ways:
Voltage amplitude
Current amplitude
Normalized amplitude whose squared modulus equals the power conveyed by the wave
Wave amplitude is represented by a complex phasor:
length is proportional to the size of the wave
phase angle tells us the relative phase with respect to the origin or zero of the time variable

13. TRANSMISSION LINE EQUATIONS
Transmission line is often schematically represented as a two-wire line.
Figure 1: Voltage and current definitions.
The transmission line always have at least two conductors.
Figure 1 can be modeled as a lumped-element circuit, as shown in Figure 2.

14. TRANSMISSION LINE EQUATIONS
The parameters are expressed in their respective name per unit length.
Figure 2: Lumped-element equivalent circuit
R = series resistant per unit length, for both conductors, in Ω/m
L = series inductance per unit length, for both conductors, in H/m
G = shunt conductance per unit length, in S/m
C = shunt capacitance per unit length, in F/m

15. TRANSMISSION LINE EQUATIONS
The series L represents the total self-inductance of the two conductors.
The shunt capacitance C is due to close proximity of the two conductors.
The series resistance R represents the resistance due to the finite conductivity of the conductors.
The shunt conductance G is due to dielectric loss in the material between the conductors.
NOTE: R and G, represent loss.

16. TRANSMISSION LINE EQUATIONS
By using the Kirchoff’s voltage law, the wave equation for V(z) and I(z) can be written as:
[1]
[2]
where
[3]
γ is the complex propagation constant, which is function of frequency.
α is the attenuation constant in nepers per unit length, β is the phase constant in radians per unit length.

17. TRANSMISSION LINE EQUATIONS
The traveling wave solution to the equation [2] and [3] before can be found as:
[4]
[5]
The characteristic impedance, Z0 can be defined as:
[6]
Note: characteristic impedance (Zo) is the ratio of voltage to current in a forward travelling wave, assuming there is no backward wave

18. TRANSMISSION LINE EQUATIONS
Zo determines relationship between voltage and current waves
Zo is a function of physical dimensions and r
Zo is usually a real impedance (e.g. 50 or 75 ohms)
characteristic impedance
for coaxial airlines (ohms) 10 20 30 40 50 60 70 80 90
100
1.0
0.8
0.7
0.6
0.5
0.9
1.5
1.4
1.3
1.2
1.1
normalized values
50 ohm standard
attenuation is lowest at 77 ohms
power handling capacity peaks at 30 ohms
RF transmission lines can be made in a variety of transmission media. Common examples are coaxial, waveguide, twisted pair, coplanar, stripline and microstrip. RF circuit design on printed-circuit boards (PCB) often use coplanar or microstrip transmission lines. The fundamental parameter of a transmission line is its characteristic impedance Zo. Zo describes the relationship between the voltage and current traveling waves, and is a function of the various dimensions of the transmission line and the dielectric constant (er) of the non-conducting material in the transmission line. For most RF systems, Zo is either 50 or 75 ohms.
For low-power situations (cable TV, for example) coaxial transmission lines are optimized for low loss, which works out to about 75 ohms (for coaxial transmission lines with air dielectric). For RF and microwave communication and radar applications, where high power is often encountered, coaxial transmission lines are designed to have a characteristic impedance of 50 ohms, a compromise between maximum power handling (occurring at 30 ohms) and minimum loss.

19. TRANSMISSION LINE EQUATIONS
Voltage waveform can be expressed in time domain as:
[7]
The factors V0+ and V0- represent the complex quantities. The Φ± is the phase angle of V0±. The quantity βz is called the electrical length of line and is measured in radians.
Then, the wavelength of the line is:
[8]
and the phase velocity is:
[9]

20. EXAMPLE 1.1 A transmission line has the following parameters:
R = 2 Ω/m G = 0.5 mS/m f = 1 GHz
L = 8 nH/m C = 0.23 pF
Calculate:
The characteristic impedance.
The propagation constant.

21. SOLUTION (EXAMPLE 1.1) 1. The characteristic impedance of the line is:
[6]

22. SOLUTION (EXAMPLE 1.1)
2. The propagation constant is:
[3]

23. THE LOSSLESS LINE
The general transmission line are including loss effect, while the propagation constant and characteristic impedance are complex.
On a lossless transmission line the modulus or size of the wave complex amplitude is independent of position along the line; the wave is neither growing not attenuating with distance and time
In many practical cases, the loss of the line is very small and so can be neglected. R = G = 0
So, the propagation constant is:
[10]
[10a]
[10b]

24. THE LOSSLESS LINE The wavelength is: and the phase velocity is:
For the lossless case, the attenuation constant α is zero.
Thus, the characteristic impedance of [6] reduces to:
[11]
The wavelength is:
[11a]
and the phase velocity is:
[11b]

25. EXAMPLE 1.2
A transmission line has the following per unit length parameters: R = 5 Ω/m, G = 0.01 S/m, L = 0.2 μH/m and C = 300 pF. Calculate the characteristic impedance and propagation constant of this line at 500 MHz. Recalculate these quantities in the absence of loss (R=G=0)

26. SOLUTION (EXAMPLE 1.2) 1. The characteristic impedance of the line is:
[6]

27. SOLUTION (EXAMPLE 1.2) 2. The propagation constant is: Np/m or rad/m
[3]
Np/m or rad/m

28. SOLUTION (EXAMPLE 1.2)
At R = G = 0; for a lossless line, we know that attenuation constant, α is;
From eq [10a], β is given as
And finally Z0 from eq [11]

29. TERMINATED TRANSMISSION LINES
Network analysis is concerned with the accurate measurement
of the ratios of the reflected signal to the incident signal, and
the transmitted signal to the incident signal. RF
Incident
Reflected
Transmitted
Lightwave
DUT
One of the most fundamental concepts of high-frequency network analysis involves incident, reflected and transmitted waves traveling along transmission lines. It is helpful to think of traveling waves along a transmission line in terms of a lightwave analogy. We can imagine incident light striking some optical component like a clear lens. Some of the light is reflected off the surface of the lens, but most of the light continues on through the lens. If the lens were made of some lossy material, then a portion of the light could be absorbed within the lens. If the lens had mirrored surfaces, then most of the light would be reflected and little or none would be transmitted through the lens. This concept is valid for RF signals as well, except the electromagnetic energy is in the RF range instead of the optical range, and our components and circuits are electrical devices and networks instead of lenses and mirrors.
Network analysis is concerned with the accurate measurement of the ratios of the reflected signal to the incident signal, and the transmitted signal to the incident signal.
Waves travelling from generator to load have complex amplitudes usually written V+ (voltage) I+ (current) or a (normalised power amplitude).
Waves travelling from load to generator have complex amplitudes usually written V- (voltage) I- (current) or b (normalised power amplitude).

30. TERMINATED LOSSLESS TRANSMISSION LINE
Most of practical problems involving transmission lines relate to what happens when the line is terminated
Figure 3 shows a lossless transmission line terminated with an arbitrary load impedance ZL
This will cause the wave reflection on transmission lines.
Figure 3: A transmission line terminated in an arbitrary load ZL

31. TERMINATED LOSSLESS TRANSMISSION LINE
Assume that an incident wave of the form V0+e-jβz is generated from the source at z < 0.
The ratio of voltage to current for such a traveling wave is Z0, the characteristic impedance [6].
If the line is terminated with an arbitrary load ZL= Z0 , the ratio of voltage to current at the load must be ZL.
The reflected wave must be excited with the appropriate amplitude to satisfy this condition.

32. TERMINATED LOSSLESS TRANSMISSION LINE
The total voltage on the line is the sum of incident and reflected waves:
[12]
The total current on the line is describe by:
[13]
The total voltage and current at the load are related by the load impedance, so at z = 0 must have:
[14]

33. TERMINATED LOSSLESS TRANSMISSION LINE
Solving for V0+ from [14] gives:
[15]
The amplitude of the reflected wave normalized to the amplitude of the incident wave is defined as the voltage reflection coefficient, Γ:
[16]
The total voltage and current waves on the line can then be written as:
[17]
[18]

34. TERMINATED LOSSLESS TRANSMISSION LINE
The time average power flow along the line at the point z:
[19]
[19] shows that the average power flow is constant at any point of the line.
The total power delivered to the load (Pav) is equal to the incident power minus the reflected power
If |Γ|=0, maximum power is delivered to the load. (ideal case)
If |Γ|=1, there is no power delivered to the load. (worst case)
So reflection coefficient will only have values between 0 < |Γ| < 1

35. STANDING WAVE RATIO (SWR)
When the load is mismatched, the presence of a reflected wave leads to the standing waves where the magnitude of the voltage on the line is not constant.
[21]
The maximum value occurs when the phase term ej(θ-2βl) =1.
[22]
The minimum value occurs when the phase term ej(θ-2βl) = -1.
[23]

36. STANDING WAVE RATIO (SWR)
As |Γ| increases, the ratio of Vmax to Vmin increases, so the measure of the mismatch of a line is called standing wave ratio (SWR) can be define as:
[24]
This quantity is also known as the voltage standing wave ratio, and sometimes identified as VSWR.
SWR is a real number such that 1 ≤ SWR ≤
SWR=1 implies a matched load

37. RETURN LOSS
When the load is mismatched, not all the of the available power from the generator is delivered to the load.
This “loss” is called return loss (RL), and is defined (in dB) as:
[20]
If there is a matched load |Γ|=0, the return loss is dB (no reflected power).
If the total reflection |Γ|=1, the return loss is 0 dB (all incident power is reflected).
So return loss will have only values between 0 < RL <

38. SUMMARY
Three parameters to measure the ‘goodness’ or ‘perfectness’ of the termination of a transmission line are:
Reflection coefficient (Γ)
Standing Wave Ratio (SWR)
Return loss (RL)

39. EXAMPLE 1.3
Calculate the SWR, reflection coefficient magnitude, |Γ| and return loss values to complete the entries in the following table:
SWR
|Γ|
RL (dB)
1.00
0.00
1.01
0.01
30.0
2.50

40. EXAMPLE 1.3
The formulas that should be used in this calculation are as follow:
[20]
[24]
mod from [20]
mod from [24]

41. EXAMPLE 1.3
Calculate the SWR, reflection coefficient magnitude, |Γ| and return loss values to complete the entries in the following table:
SWR
|Γ|
RL (dB)
1.00
0.00
1.01
0.005
46.0
1.02
0.01
40.0
1.07
0.0316
30.0
2.50
0.429
7.4

42. TERMINATED LOSSLESS TRANSMISSION LINE
Since we know that total voltage on the line is
[12]
And the reflection coefficient along the line is defined as Γ(z):
Defining or ΓL as the reflection coefficient at the load;

43. TRANSMISSION LINE IMPEDANCE EQUATION
Substituting ΓL into eq [14 and 15], the impedance along the line is given as:
At x=0, Z(x) = ZL. Therefore;

44. TRANSMISSION LINE IMPEDANCE EQUATION
At a distance l = -z from the load, the input impedance seen looking towards the load is:
[25a]
[25b]
When Γ in [16] is used:
[26a]
[26b]
[26c]

45. EXAMPLE 1.4
A source with 50  source impedance drives a 50  transmission line that is 1/8 of wavelength long, terminated in a load ZL = 50 – j25 . Calculate:
(i) The reflection coefficient, ГL
(ii) VSWR
(iii) The input impedance seen by the source.

46. SOLUTION TO EXAMPLE 1.4
It can be shown as:

47. SOLUTION TO EXAMPLE 1.4 (Cont’d)
(i) The reflection coefficient,
(ii) VSWR

48. SOLUTION TO EXAMPLE 1.4 (Cont’d)
(iii) The input impedance seen by the source, Zin
Need to calculate
Therefore,

49. SPECIAL CASE OF LOSSLESS TRANSMISSION LINES
For the transmission line shown in Figure 4, a line is terminated with a short circuit, ZL=0.
From [16] it can be seen that the reflection coefficient Γ= -1.
Then, from [24], the standing wave ratio is infinite.
Figure 4: A transmission line terminated with short circuit

50. SPECIAL CASE OF LOSSLESS TRANSMISSION LINES
Referred to Figure 4, equation [17] and [18] the voltage and current on the line are:
[27]
[28]
From [26c], the ratio V(-l) / I(-l), the input impedance is:
[29]
When l = 0 we have Zin=0, but for l = λ/4 we have Zin = ∞ (open circuit)
Equation [29] also shows that the impedance is periodic in l.

51. SPECIAL CASE OF LOSSLESS TRANSMISSION LINES
Figure 5: (a) Voltage (b) Current (c) impedance (Rin=0 or ∞) variation along a short circuited transmission line

52. SPECIAL CASE OF LOSSLESS TRANSMISSION LINES
For the open circuit as shown in Figure 6, ZL=∞
The reflection coefficient is Γ=1.
The standing wave is infinite.
Figure 6: A transmission line terminated in an open circuit.

53. SPECIAL CASE OF LOSSLESS TRANSMISSION LINES
For an open circuit I = 0, while the voltage is a maximum.
The input impedance is:
[30]
When the transmission line are terminated with some special lengths such as l = λ/2,
[31]
For l = λ/4 + nλ/2, and n = 1, 2, 3, … The input impedance [26c] is given by:
[32]
Note: also known as quarter wave transformer.

54. SPECIAL CASE OF LOSSLESS TRANSMISSION LINES
Figure 7: (a) Voltage (b) Current (c) impedance (R = 0 or ∞) variation along an open circuit transmission line.

55. SPECIAL CASE OF LOSSLESS TRANSMISSION LINES