EKT 441 MICROWAVE COMMUNICATIONS CHAPTER 1 TRANSMISSION LINE

EKT 441 MICROWAVE COMMUNICATIONS CHAPTER 1: TRANSMISSION LINE THEORY 1

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

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. 3

INTRODUCTION Microwave refers to alternating current signals with frequencies between 300 MHz and 300 GHz. l Figure 1 shows the location of the microwave frequency l 3 x 105 3 x 106 wave radio 3 x 108 3 x 109 3 x 1010 3 x 1011 3 x 1012 3 x 1013 3 x 1014 FM Short AM Long wave broad radio Castin g radio 103 3 x 107 VHFbroad Far Microwaves infrared TV casting Visible light radio 102 101 1 10 -2 10 -3 10 -4 10 -5 10 -6 Typical frequencies AM broadcast band 535 -1605 k. Hz VHF TV (5 -6) 76 -88 MHz Shortwave radio 3 -30 MHz UHF TV (7 -13) 174 -216 MHz FM broadcast band 88 -108 MHz UHF TV (14 -83) 470 -890 MHz VHF TV (2 -4) 54 -72 MHz Microwave ovens 2. 45 GHz 4

MICROWAVE BAND DESIGNATION Frequency (GHz) Wavelength (cm) IEEE band 1 -2 2 -4 30 - 15 15 - 7. 5 L S 4 -8 8 - 12 12 - 18 18 - 27 7. 5 - 3. 75 - 2. 5 - 1. 67 - 1. 11 C X Ku K 27 - 40 40 - 300 1. 11 - 0. 75 - 0. 1 Ka mm 5

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

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

TYPICAL Rx ARCHITECTURE l l l 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 f. IF+fm OR f. IF-fm; f. IF<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 8

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

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

TRANSMISSION LINES + Low frequencies I l wavelengths >> wire length l current (I) travels down wires easily for efficient power transmission l measured voltage and current not dependent on position along wire High frequencies l wavelength » or << length of transmission medium l need transmission lines for efficient power transmission l matching to characteristic impedance (Z o) is very important for low reflection and maximum power transfer l measured envelope voltage dependent on position along line 11

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

TRANSMISSION LINE EQUATIONS l 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. 13

TRANSMISSION LINE EQUATIONS l 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 14

TRANSMISSION LINE EQUATIONS l l l 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. 15

TRANSMISSION LINE EQUATIONS l 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. 16

TRANSMISSION LINE EQUATIONS l The traveling wave solution to the equation [2] and [3] before can be found as: [4] [5] The characteristic impedance, Z 0 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 17

TRANSMISSION LINE EQUATIONS l l 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) 1. 5 attenuation is lowest at 77 ohms 1. 4 1. 3 1. 2 normalized values l 50 ohm standard 1. 1 1. 0 0. 9 0. 8 power handling capacity peaks at 30 ohms 0. 7 0. 6 0. 5 10 20 30 40 50 60 70 80 90 100 characteristic impedance for coaxial airlines (ohms) 18

TRANSMISSION LINE EQUATIONS l Voltage waveform can be expressed in time domain as: [7] The factors V 0+ and V 0 - represent the complex quantities. The Φ± is the phase angle of V 0±. 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] 19

EXAMPLE 1. 1 l A transmission line has the following parameters: R = 2 Ω/m G = 0. 5 m. S/m f = 1 GHz L = 8 n. H/m C = 0. 23 p. F Calculate: 1. The characteristic impedance. 2. The propagation constant. 20

THE LOSSLESS LINE The general transmission line are including loss effect, while the propagation constant and characteristic impedance are complex. l 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 l In many practical cases, the loss of the line is very small and so can be neglected. R = G = 0 [10] l So, the propagation constant is: l [10 a] [10 b] 23

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

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 p. F. Calculate the characteristic impedance and propagation constant of this line at 500 MHz. Recalculate these quantities in the absence of loss (R=G=0) 25

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. Incident Reflected Transmitted Lightwave DUT RF 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 29 written V- (voltage) I- (current) or b (normalised power amplitude).

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

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

TERMINATED LOSSLESS TRANSMISSION LINE l 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] 32

TERMINATED LOSSLESS TRANSMISSION LINE l Solving for V 0+ 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] 33

TERMINATED LOSSLESS TRANSMISSION LINE [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) l timethere average alongto thethe line at the pointcase) z: • If. The |Γ|=1, is nopowerflow delivered load. (worst • So reflection coefficient will only have values between 0 < |Γ| < 1 34

STANDING WAVE RATIO (SWR) l 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] 35

STANDING WAVE RATIO (SWR) l 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 36

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

SUMMARY l Three parameters to measure the ‘goodness’ or ‘perfectness’ of the termination of a transmission line are: 1. 2. 3. Reflection coefficient (Γ) Standing Wave Ratio (SWR) Return loss (RL) 38

EXAMPLE 1. 3 SWR 1. 00 1. 01 |Γ| 0. 00 RL (d. B) Calculate the SWR, reflection coefficient magnitude, |Γ| and 0. 01 return loss values to complete the entries in the following 30. 0 table: 2. 50 39

EXAMPLE 1. 3 The formulas that should be used in this calculation are as follow: [20] [24] mod from [20] mod from [24] 40

EXAMPLE 1. 3 SWR |Γ| RL (d. B) 1. 00 0. 00 1. 01 0. 005 46. 0 Calculate the SWR, reflection coefficient magnitude, 1. 02 0. 01 40. 0 |Γ| and return loss values to complete the entries in the following 0. 0316 30. 0 table: 1. 07 2. 50 0. 429 7. 4 41

TERMINATED LOSSLESS TRANSMISSION LINE l 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; 42

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

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

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 – j 25 . Calculate: (i) The reflection coefficient, ГL (ii) VSWR (iii) The input impedance seen by the source. 45

SOLUTION TO EXAMPLE 1. 4 It can be shown as: 46

SOLUTION TO EXAMPLE 1. 4 (Cont’d) (i) The reflection coefficient, (ii) VSWR 47

SOLUTION TO EXAMPLE 1. 4 (Cont’d) (iii) The input impedance seen by the source, Zin Need to calculate Therefore, 48

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

SPECIAL CASE OF LOSSLESS TRANSMISSION LINES l Referred to Figure 4, equation [17] and [18] the voltage and current on the line are: [27] [28] From [26 c], 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. 50

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

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

SPECIAL CASE OF LOSSLESS TRANSMISSION LINES For an open circuit I = 0, while the voltage is a maximum. l The input impedance is: l [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 [26 c] is given by: [32] Note: also known as quarter wave transformer. 53

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

SPECIAL CASE OF LOSSLESS TRANSMISSION LINES 55