Transmission Lines

TRANSMISSION LINES  a metallic conductor system Behavior of Transmission Lines  used to transfer electrical energy from one point to  At low frequencies: another. o Resistive ...

TRANSMISSION LINES  a metallic conductor system Behavior of Transmission Lines  used to transfer electrical energy from one point to  At low frequencies: another. o Resistive  A material medium or structure that forms a path for o Ideal directing the transmission of energy from one place to o No loss another o No reactance  Impedance matching circuits  At high frequencies: Basic Kinds of Waves o Becomes complex o There are RLCG in the circuit a) Based on the direction of signal REMINDER: Source Load  The smaller the diameter of the wire, the higher the resistance  The longer the transmission line, the higher the  Incident Waves resistance 𝑷𝒊𝒏𝒄 +𝒅𝒃 = 𝟏𝟎𝒍𝒐𝒈 𝑷𝒓𝒆𝒇  Reflected Waves 𝑷𝒓𝒆𝒇 LOSSES IN TRANSMISSION LINES −𝒅𝒃 = 𝟏𝟎𝒍𝒐𝒈 𝑷𝒊𝒏𝒄 1. Conductor heating loss 𝑷𝒐𝒖𝒕 = 𝑷𝒊𝒏𝒄 − 𝑷𝒓𝒆𝒇  also called I2R loss and power loss  increases with frequency because of skin effect b) Based on the displacement of signal Solutions to Skin Effect  Longitudinal wave – the displacement of the o increase the diameter of the conductor medium is parallel to the propagation of the o Let the conductor be plated with silver wave. o Use multi-stranded wire  Transverse wave – the displacement of the 2. Dielectric heating medium is perpendicular to the direction of  It increases with frequency because of gradually propagation of the wave. worsening properties with increasing frequency for any given dielectric medium.  Observable in solid dielectric but remains TYPES OF TRANSMISSION LINES negligible for air dielectric BALANCED LINE/PARALLEL WIRE/DIFFERENTIAL LINE 3. Radiation loss  Occurs because transmission line may act as an  Common examples are open wire transmission line, antenna if the separation of the conductor is Twin Lead or Ribbon Cable, Twisted-Pair Cable. appreciable fraction of a wavelength.  made up of two parallel conductors spaced from one  The loss increases with frequency another by a distance of /12 inch up to several inches.  Can be reduced by properly shielding.  Two conductors carry current; one conductor carries 4. Coupling Loss the signal and the other is the return.  Occurs when two separate lines are o Metallic circuit currents – currents flow in connected/reconnected together. opposite directions  Mechanical connections are discontinuities o Longitudinal circuit currents – currents flow which tend to heat up, radiate energy and in the same directions dissipate heat.  UTP – Unshielded Twisted Pair  Types of Coupling Loss  STP – Shielded Twisted Pair o Lateral management UNBALANCED LINE/COAXIAL LINE/SINGLE-ENDED LINE o Gap displacement o Angular misalignment  Consist of a solid center conductor surrounded by a plastic 5. Corona or spark insulator.  A luminous discharge between two conductors of  Second conductor, a shield made of fine wires for ground transmission line potential.   Occurs when the difference potential between Where : u = permeability them exceeds the breakdown voltage of the u = uR x Uo dielectric uo = 1. 257 x 10-6 H/m or 4π x10-7 H/m  Permeability – measure of the superiority of GENERAL EQUIVALENT CIRCUIT OF TRANSMISSION LINES material over a vacuum as a path for magnetic lines of force Capcitance (C), F/m 𝜋𝜀 𝐶= 𝑠 𝑙𝑛 𝑟 Where : Ɛ = permittivity A. Primary Line Constants Ɛ=ƐRxƐo  Resistance  Inductance Ɛ o = 8.854 x 10-12 F/m or 1/36π x10-12 F/m  Capacitance  Permittivity – ability to concentrate magnetic lines  Conductance of force; pertains to the dielectric constant B. Secondary Line Constants  Characteristic impedance  Propagation constant Resistance (R), Ω/m o Attenuation Constant o Phase constant 1 𝑅= 𝜋𝑟𝛿𝜎𝑐 Where : δ = skin depth of the cable Telegrapher’s Equation/Telegraph Equation σo = conductivity of the conductor  Pairs of couples, linear differential equations that describe the voltage and current on an electrical transmission line with distance and time Conductance (G), mho/m 𝜋𝜎 𝐺= For Parallel Wire Line 𝑐𝑜𝑠ℎ−1 𝑠 2𝑟 Where : σ = conductivity of the dielectric For Coaxial Line Where : s = spacing between the conductor r = radius of the conductor Inductance (L), H/m Where : a = radius of the inner conductor 𝜇 𝑠 𝐿= 𝑙𝑛 b = radius of the outer conductor 𝜋 𝑟 Inductance (L), H/m  It is the impedance measured at the input of the 𝜇 line when its length is infinite. 𝐿= 𝑏 2𝜋 (𝑙𝑛 ) 𝑎 Capcitance (C), F/m Note: 2𝜋𝜀  A line terminated in its Zo is called non-resonant, 𝐶= resistive or flat line. 𝑏 𝑙𝑛 𝑎  The V and I of a lossless line are constant in phase. Resistance (R), Ω/m  The V and I of a line with loss are reduced exponentially. 1 𝑅= 1 1 2𝜋𝑟𝛿𝜎𝑐 ( + ) 𝑎 𝑏 Zo of Transmission Line Conductance (G), mho/m 2𝜋𝜎 𝑍 𝐺= 𝑍𝑜 = √ 𝑏 𝑌 𝑙𝑛 𝑎 𝑹 + 𝒋𝝎𝑳 𝒁𝒐 = √ 𝑮 + 𝒋𝝎𝑪 Velocity Factor (Vf) Where : Z = shunt impedance  the velocity reduction ratio of the electromagnetic waves that depends on the nature of the medium Y = shunt admittance through which they travel 𝑽𝒄 𝟏 At Radio Frequency 𝑽= ; 𝑽𝒇 = ; 𝑽 = 𝑽𝒄 𝑽𝒇 √𝒌 √𝒌 ωL >> R Where : V = velocity of the medium ωC >> G 8 Vc = velocity of light in a vacuum (3 x 10 m/s) 𝑳 𝒁𝒐 = √ 𝑪 k = dielectric constant of the medium At Audio Frequency Vf = velocity factor R >> ωL MATERIAL DIELECTRIC CONSTANT (k) G >> ωC Vacuum 1.0 Air 1.0006 Teflon 2.1 𝑹 𝒁𝒐 = √ Polyethylene (PE) 2.27 𝑮 Polystyrene 2.5 Paper, paraffined 2.5 Rubber 3.0 Nature of Transmission Line Polyvinyl chloride (PVC) 3.3 Mica 5.0 Glass 7.5 Characteristic Impedance (Zo)  also called surge impedance or intrinsic impedance  refers to the impedance that a cable presents to an electrical signal, determined by the cable’s physical structure—the arrangement of conductors, dielectric material, and dimensions.  Zo based on Physical Dimension 𝟏𝟑𝟖 𝑫 𝒁𝒐 = 𝒍𝒐𝒈 √𝒌 𝒅  Geometry of the conductor  Size of the conductor 138 𝐷 𝑍𝑜 = 𝑙𝑜𝑔  Spacing of the conductor √𝑘 2𝑟  Dielectric constant of the insulator 60 𝐷 𝑍𝑜 = 𝑙𝑛 √𝑘 𝑑 For Parallel Wire Line Note: if the outer diameter of the outer conductor (Do) is involved use the ff. formula: 𝑫𝒐 = 𝑫 + 𝟐𝒉 Where : Zo = character impedance k = dielectric constant D = inside diameter of the outer conductor d = diameter of the inside conductor 𝟐𝟕𝟔 𝟐𝒔 𝒁𝒐 = 𝒍𝒐𝒈 √𝒌 𝒅 Propagation Constant (δ) 276 𝑠 𝑍𝑜 = 𝑙𝑜𝑔  A secondary line constant √𝑘 𝑟  Determines the variation of current and voltage with 120 𝑠 distance along a transmission line and is found to 𝑍𝑜 = 𝑙𝑛 √𝑘 𝑟 vary exponentially. Note: if the diameter of the conductor is not equal use the ff. 𝐼 = 𝐼𝑠𝑒 −𝛿𝑥 formula: 𝑉 = 𝑉𝑠𝑒 −𝛿𝑥 𝒅 = √𝒅𝟏 + 𝒅𝟐 Where : δ = propagation constant Where : Zo = character impedance x = distance k = dielectric constant Is = magnitude of the current s = spacing of conductors from center to center Vs = magnitude of the voltage d = diameter of the conductor s = sending end or input r = radius of the conductor The propagation constant also depends on primary line constants and the angular velocity of the signal. Reminder: 𝜹 = √𝒁𝒀  Characteristic impedance is inversely proportional to dielectric constant 𝜹 = √(𝑹 + 𝒋𝝎𝑳)(𝑮 + 𝒋𝝎𝑪)  If Zo is in minimum value (Zomin), s = d 𝛿 =∝ +𝑗𝛽 Where : a = attenuation constant (dB/uL, Neper/uL) For Coaxial Line = determines how V and I decreases with distance along the line 1 db = 0.115 Neper 1 Neper = 8.686 dB β = phase shift/phase delay constant (°/uL, rad/uL)  = determines the phase angle of the V and I 𝟐𝝅 𝛌= variation with distance 𝜷 𝟑𝟔𝟎 𝛌= For Lossless Line: 𝜷 If : L/R = C/G Unit = uL Therefore : C = LG/R Two Types of Transmission Line Length 𝑮 𝜹 = (𝑹 + 𝒋𝝎𝑳)√ 𝑹 1. Physical Length (S)  Also called the mechanical length (unit: in, m, But if : G/R = C/L cm, etc.) 2. Electrical Length (°ƪ) 𝐺 𝐶 𝛿 = (𝑅 + 𝑗𝜔𝐿)√ → 𝛿 = (𝑅 + 𝑗𝜔𝐿)√ °ƪ = 𝜷𝑺 𝑅 𝐿 𝝎 °ƪ = 𝑺 𝑪 𝑽 𝜹 = 𝑹√ + 𝒋𝝎√𝑳𝑪 𝑳 𝟐𝝅𝒇 𝟑𝟔𝟎𝒇 °ƪ = 𝑺 𝒐𝒓 °ƪ = 𝑽 𝑽 Therefore: Unit = rad/uL or °/uL 𝐶 ∝= 𝑅√ 𝐿 Delay Lines 𝛽 = 𝜔√𝐿𝐶  Transmission lines designed to intentionally introduce a time delay in the path of an electromagnetic wave Speed of Propagation (Velocity, V) 𝑡𝑑 = 𝐿𝐶 At any condition: Where : td = time delay (s) 𝑫 𝑽= L = inductance (H) 𝑻 𝑫 C = capacitance (C) 𝑽= 𝑳𝑪 𝑡𝑑 = 1.016𝑘 𝑽 = 𝝀𝒇 𝝎 𝑽= ; 𝜷 = 𝜔√𝐿𝐶 𝜷 Things that may happen when the signal reaches the load: 𝟏 1. All signals that go to the load is totally absorbed by the 𝑽= √𝑳𝑪 load. (Zo = ZL) 2. Only portion or part of a signal is absorbed by the load. (mismatch) Wavelength (λ) 3. No signal is absorbed by the load (total reflection) the line is shorted (ZL = 0), or the line is open (ZL = ꚙ); or 𝑽 𝜔 𝛌= ; 𝑉= perfectly unmatched. 𝒇 𝛽 𝜔 λ= 𝛽𝑓 2𝜋𝑓 λ= 𝛽𝑓