EKT 441 MICROWAVE COMMUNICATIONS CHAPTER 1: TRANSMISSION LINE THEORY

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

INTRODUCTION ELECTROMAGNETIC 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.

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 x105 3 x 106 3 x107 3x 108 3x109 3x1010 3x1011 3x1012 3x 1013 3x1014 103 102 101 1 10-1 10-2 10-3 10-4 10-5 10-6 Typical frequencies AM broadcast band 535-1605 kHz 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

MICROWAVE BAND DESIGNATION Frequency (GHz) Wavelength (cm) IEEE band 1 - 2 30 - 15 L 2 - 4 15 - 7.5 S 4 - 8 7.5 - 3.75 C 8 - 12 3.75 - 2.5 X 12 - 18 2.5 - 1.67 Ku 18 - 27 1.67 - 1.11 K 27 - 40 1.11 - 0.75 Ka 40 - 300 0.75 - 0.1 mm

APPLICATION OF MICROWAVE ENGINEERING Communication systems WLAN, GPS, GSM, DBS, UWB and etc. Radar system Environmental remote sensing Medical system Etc.

TYPICAL RECEIVER ARCHITECTURE Typical Dual conversion receiver

TYPICAL TRANSMITTER ARCHITECTURE

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

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

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.

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

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.

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.

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

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.

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]

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.

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]

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]

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)

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).

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

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.

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]

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]

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 (|V0+ |2 /2Z0), minus the reflected power (|V0-|2 |Γ|2 /2Z0). 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

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]

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

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 <

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)

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

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

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

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]

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.

SOLUTION TO EXAMPLE 1.4 It can be shown as:

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

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

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

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.

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

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.

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

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.