Transmission Lines. Ranga Rodrigo. January 13, Antennas and Propagation: Transmission Lines 1/46

Transmission Lines Ranga Rodrigo January 13, 2009 Antennas and Propagation: Transmission Lines 1/46

1 Basic Transmission Line Properties 2 Standing Waves Antennas and Propagation: Transmission Lines Outline 2/46

Outline 1 Basic Transmission Line Properties 2 Standing Waves Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 3/46

Introduction A transmission line is a circuit element that transfers energy in the form of electromagnetic waves from one place to the other. Transmission Lines Balanced E.g., flat twin Unbalanced E.g., coaxial cable Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 4/46

The construction of the line will vary depending on its end use: Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 5/46

The construction of the line will vary depending on its end use: Copper wires for low-frequency audio applications. Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 5/46

The construction of the line will vary depending on its end use: Copper wires for low-frequency audio applications. Copper dielectric mixtures for VHF, UHF, and microwave. Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 5/46

The construction of the line will vary depending on its end use: Copper wires for low-frequency audio applications. Copper dielectric mixtures for VHF, UHF, and microwave. Solid dielectric such as plastic or glass for optical use. Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 5/46

The construction of the line will vary depending on its end use: Copper wires for low-frequency audio applications. Copper dielectric mixtures for VHF, UHF, and microwave. Solid dielectric such as plastic or glass for optical use. By carefully exploiting the properties of a given line configuration, useful circuit components such as filters and impedance matching networks can be built. Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 5/46

Distributed Nature of Trans. Lines 10 MHz and Below Wavelengths are long (> 30 m). Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 6/46

Distributed Nature of Trans. Lines 10 MHz and Below Wavelengths are long (> 30 m). Standard components, capacitors, inductors, etc., appear very short. Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 6/46

Distributed Nature of Trans. Lines 10 MHz and Below Wavelengths are long (> 30 m). Standard components, capacitors, inductors, etc., appear very short. Gives rise to the notion of lumped circuits. Higher Frequencies Wavelength is comparable with that of components. Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 6/46

Distributed Nature of Trans. Lines 10 MHz and Below Wavelengths are long (> 30 m). Standard components, capacitors, inductors, etc., appear very short. Gives rise to the notion of lumped circuits. Higher Frequencies Wavelength is comparable with that of components. Distributed circuit techniques must be used. Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 6/46

Distributed Nature of Trans. Lines 10 MHz and Below Wavelengths are long (> 30 m). Standard components, capacitors, inductors, etc., appear very short. Gives rise to the notion of lumped circuits. Higher Frequencies Wavelength is comparable with that of components. Distributed circuit techniques must be used. Division occurs when the component dimension is not greater than 1/20 of the wavelength. Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 6/46

Z 0 R L R L R L G C G C G C Z 0 R: series loss resistance (copper losses). G: shunt loss conductance (losses in dielectric). L: series inductance representing energy storage within the line. C: shunt capacitance representing energy storage within the line. Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 7/46

I 1 I 2 I 3 V 1 V 2 V 3 l l Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 8/46

I 1 I 2 I 3 V 1 V 2 V 3 l l If an infinitely long pair of wires were considered and the voltage and current were somehow measured at uniform spaced points along the line, then V 1 I 1 = V 2 I 2 = = V k I k = constant = Z 0 Ω. (1) Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 8/46

I 1 I 2 I 3 V 1 V 2 V 3 l l If an infinitely long pair of wires were considered and the voltage and current were somehow measured at uniform spaced points along the line, then V 1 I 1 = V 2 I 2 = = V k I k = constant = Z 0 Ω. (1) This is termed the characteristic impedance of the line and is denoted by Z 0. Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 8/46

Z Y V Z 0 Z in = Z 0 Figure: Characteristic line impedance. The input impedance Z in for the L-section shown above is Z in = Z 0 = (Z 0 + Z )/ Y Z 0 + Z + 1/ Y = Z 0 + Z Y, Z 0. 1 + Z 0 Y (2)

Z Y V Z 0 Z in = Z 0 Figure: Characteristic line impedance. Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 9/46

Z Y V Z 0 Z in = Z 0 Figure: Characteristic line impedance. The input impedance Z in for the L-section shown above is Z in = Z 0 = (Z 0 + Z )/ Y Z 0 + Z + 1/ Y = Z 0 + Z Y, Z 0. 1 + Z 0 Y (2) Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 9/46

Solving Equation 2 for Z 0 gives ( ) Z 1/2 Z 0 =, (3) Y where Z = R + j ωl and Y = G j ωc hence, ( ) R + j ωl 1/2 Z 0 = Ω. (4) G + j ωc This expression is important as it relates the lumped circuit model for the transmission line to one of the primary line constants, characteristic impedance. At very low frequencies: ( ) R 1/2 Z 0 = Ω. (5) G Antennas and For Propagation: high Transmission frequencies Lines (ωl RBasic and Transmission ωcline Properties G): 10/46

Let s consider Equation 4, ( ) R + j ωl 1/2 Z 0 = Ω. G + j ωc At very low frequencies: ( ) R 1/2 Z 0 = Ω. (7) G For high frequencies (ωl R and ωc G): ( ) L 1/2 Z 0 = Ω (8) C Since transmission line circuit designs are done at high frequencies, Equation 8 is often used. If losses cannot be neglected, then the line is said to be lossy. Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 11/46

Example A lossless transmission line has a characteristic impedance of 50 Ω and s self-inductance of 0.08 µh/m. Calculate the capacitance of a 4-meter length of line. Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 12/46

Example A lossless transmission line has a characteristic impedance of 50 Ω and s self-inductance of 0.08 µh/m. Calculate the capacitance of a 4-meter length of line. Solution The line is lossless. Therefore, R = 0, G =. ( ) L 1/2 Z 0 = C = L 0.08 10 6 = = 32 pf/m. C 50 50 Z 2 o The total capacitance for a 4-meter length is 4 32 pf/m = 128 pf/m. This capacitance will limit the high frequency response of the cable. Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 12/46

Propagation Constant Consider the lumped equivalent circuit again. Z 0 R L R L R L G C G C G C Z 0 The voltage drop V across one lumped section is V = I (R + j ωl) x. (9) here x is the incremental length of the line represented by the lumped section. Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 13/46

Dividing Equation 9 by x and since x 0, dv = (R + j ωl)i. (10) dx Similarly we can obtain d I = (G + j ωc)v. (11) dx By differentiating and substitution d 2 V = (R + j ωl)(g + j ωc)v. (12) dx2 Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 14/46

Equation 12 can be written as d 2 V dx 2 = γ2 V, (13) where γ = (R + j ωl)(g + j ωc) is termed the propagation constant. This is usually expressed as γ = α + j β, (14) where α represents the attenuation per unit length and β the phase shift per unit length. Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 15/46

Equation 13 has a solution of the form V (x) = Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 16/46

Equation 13 has a solution of the form V (x) = Ae γx + Be γx. (15) This suggests that the line will contain two waves, one traveling in the positive x-direction (e γx ) and the other traveling in the negative x-direction (e γx ). To evaluate A, let s consider an infinite-length line excited by sine wave of amplitude V in. If the line contains resistive elements, then at x =, any voltage would have decayed to zero. Therefore, we have 0 = Ae γ + Be γ B = 0. (16) V (x) = Ae γx. (17) Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 16/46

When we consider that at the driving end of the line (x = 0), the voltage is V in, we get A = V in. Therefore, Since γ = α + j β, V (x) = V in e γx. (18) V (x) = V in e αx e j βx. (19) Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 17/46

e αx Term Let s consider the attenuation term, e αx. For a line containing uniform loss per unit length, V 2 = kv 1, V 3 = kv 2,. =. V n+1 = kv n. (20) where k = e αx is a constant less than one (k = 1 for lossless). Therefore, V n+1 = V 1 k n. (21) Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 18/46

Taking the natural logarithm, we have ln ( Vn+1 V 1 ) = nαx. (22) The term nαx is defined as the total line attenuation and is measured in units called nepers (Np). It can be shown that 1 Np is equal to 8.686 db. Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 19/46

e j βx Term Now let s consider the term e j βx. Taking the current I as a reference, we find that the voltage drop across a single incremental inductance element L is j ωli x, while the voltage across the element Z 0 is I Z 0. β I Z 0 ωli x ( ) ωli x β = tan 1. (23) I Z 0 Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 20/46

For small angles, tanθ θ, so For a lossless line, β = ωl x Z 0. (24) β = ωl x (L/C) 1/2 = ω(lc)1/2 x. (25) The quantity β/ x is called the phase-shift-change per unit length or the wave number β. We have β = ω(lc) 1/2. (26) v p Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 21/46

For small angles, tanθ θ, so For a lossless line, β = ωl x Z 0. (24) β = ωl x (L/C) 1/2 = ω(lc)1/2 x. (25) The quantity β/ x is called the phase-shift-change per unit length or the wave number β. We have β = ω(lc) 1/2. (26) v p = f λ = ω 2π λ, β Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 21/46

For small angles, tanθ θ, so For a lossless line, β = ωl x Z 0. (24) β = ωl x (L/C) 1/2 = ω(lc)1/2 x. (25) The quantity β/ x is called the phase-shift-change per unit length or the wave number β. We have β = ω(lc) 1/2. (26) v p = f λ = ω 2π λ, β = 2π λ, v p Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 21/46

For small angles, tanθ θ, so For a lossless line, β = ωl x Z 0. (24) β = ωl x (L/C) 1/2 = ω(lc)1/2 x. (25) The quantity β/ x is called the phase-shift-change per unit length or the wave number β. We have β = ω(lc) 1/2. (26) v p = f λ = ω 2π λ, β = 2π λ, v p = ω β, Z 0 Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 21/46

For small angles, tanθ θ, so For a lossless line, β = ωl x Z 0. (24) β = ωl x (L/C) 1/2 = ω(lc)1/2 x. (25) The quantity β/ x is called the phase-shift-change per unit length or the wave number β. We have β = ω(lc) 1/2. (26) v p = f λ = ω 2π λ, β = 2π λ, v p = ω β, Z 0 = v p L = 1 v p C. (27) Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 21/46

Sending End Impedance When a load is placed at the end of a section of transmission line the line will produce a transformation effect. The load impedance will appear different when viewed from the sending end of the line. Using this effect, the impedance transformation caused by inserting a section of transmission line between a given impedance and a measurement point can be carefully controlled to allow impedance matching. Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 22/46

Sending End Impedance When a load is placed at the end of a section of transmission line the line will produce a transformation effect. The load impedance will appear different when viewed from the sending end of the line. Using this effect, the impedance transformation caused by inserting a section of transmission line between a given impedance and a measurement point can be carefully controlled to allow impedance matching. If measurements are to be made on an unknown load impedance, the impedance transformation caused by inserting a connecting section of Antennas and transmission Propagation: Transmission Lines line needs to bebasic taken Transmission into Line Properties account. 22/46

Let s consider Equation 15, V (x) = Ae γx + Be +γx. I inc I ref Z 0 Z T Z S x = 0 l Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 23/46

Let s consider Equation 15, V (x) = Ae γx + Be +γx. I inc I ref Z 0 Z T Z S x = 0 l The total current flowing in the line I T is the sum of current traveling in the forward direction from x = 0 to x = l, I inc, and the amount of current that is reflected from the load termination Z T, I ref. Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 23/46

This reflected current is the portion of the incident (forward) current that is not absorbed by the load. I T = I inc I ref, (28) I T = A Z 0 e γx B Z 0 e +γx. (29) At position x = l the impedance is Z T. Z T = V T I T = Ae γl + Be +γl Ae γl Be +γl Z 0. (30) Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 24/46

This reflected current is the portion of the incident (forward) current that is not absorbed by the load. I T = I inc I ref, (28) I T = A Z 0 e γx B Z 0 e +γx. (29) At position x = l the impedance is Z T. Z T = V T I T = Ae γl + Be +γl Ae γl Be +γl Z 0. (30) Therefore, B A = Z e 2γl T Z 0. (31) Z T + Z 0 The term e 2γl is the loss encountered by the signal traversing from source to termination and back. Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 24/46

Considering the situation at x = 0, we get, Z S = V S I S = A + B A B Z 0. (32) Z S = 1 + B/A Z 0 1 B/A. (33) Here Z S is the sending end impedance, i.e., the impedance seen looking into the line toward the load. Substituting, Z S = 1 + e 2γl ZT Z0 Z T +Z 0 Z 0 1 e 2γl Z T Z 0 Z T +Z 0 = Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 25/46

Considering the situation at x = 0, we get, Z S = V S I S = A + B A B Z 0. (32) Z S = 1 + B/A Z 0 1 B/A. (33) Here Z S is the sending end impedance, i.e., the impedance seen looking into the line toward the load. Substituting, Z S = 1 + e 2γl ZT Z0 Z T +Z 0 Z 0 1 e 2γl Z T Z 0 Z T +Z 0 = Z T (1 + e 2γl ) + Z 0 (1 e 2γl ) Z T (1 e 2γl ) + Z 0 (1 + e 2γl ). (34) Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 25/46

sinhγl = 1 2 [ e γl e γl], 1 e 2γl = e γl [ e γl e γl], = 2sinh(γl)e γl. 1 + e 2γl = 2cosh(γl)e γl. (35) Z S Z 0 = Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 26/46

sinhγl = 1 2 [ e γl e γl], 1 e 2γl = e γl [ e γl e γl], = 2sinh(γl)e γl. 1 + e 2γl = 2cosh(γl)e γl. (35) Z S Z 0 = Z T cosh(γl) + Z 0 sinh(γl) Z T sinh(γl) + Z 0 cosh(γl). (36) Equation 36 is the desired result. It enables us to evaluate the sending end impedance Z S, in terms of the termination impedance Z T and characteristic impedance Z 0. Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 26/46

Sometimes, it is convenient to use the following equation as the governing equation (be dividing each term by Z 0 cosh(γl)): Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 27/46

Sometimes, it is convenient to use the following equation as the governing equation (be dividing each term by Z 0 cosh(γl)): Z S = Z T cosh(γl) + Z 0 sinh(γl) Z 0 Z T sinh(γl) + Z 0 cosh(γl) Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 27/46

Sometimes, it is convenient to use the following equation as the governing equation (be dividing each term by Z 0 cosh(γl)): Z S = Z T cosh(γl) + Z 0 sinh(γl) Z 0 Z T sinh(γl) + Z 0 cosh(γl) Z S Z 0 = Z T Z 0 + tanh(γl) 1 + Z T Z 0 tanh(γl). (37) Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 27/46

From Equation 36, we see that if the load termination Z T = Z 0, then the sending end impedance Z S will equal to Z 0. In this case, no reflection will occur from the load, and the line is said to be matched. Example A transmission line of length 100 m is operated at 10.0 MHz, and has an attenuation per unit length of 0.002 nepers/m. The phase velocity of the line is 2.7 10 8 m/s, and the line has a characteristic impedance of 50 Ω. What is the value of the terminal load impedance, if the input impedance looking into the line is measured to be 30 j 10 Ω? Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 28/46

Solution Following relations are useful: tanh(j βl) = j tan(βl). tanh(γl) = tanh(αl + j βl), tanh(αl) + j tan(βl) = 1 tanh(αl)tan(βl) sinh(2αl) ± j sin(2βl) tanh(α ± j βl) = cosh(2αl) + cos(2βl). (38) Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 29/46

We need to find Z T, given Z S and line parameters. From Equation 36, Z S = Z T cosh(γl) + Z 0 sinh(γl) Z 0 Z T sinh(γl) + Z 0 cosh(γl), we obtain Rearranging, Z S Z 0 = Z T Z 0 + tanh(γl) 1 + Z T Z 0 tanh(γl). (39) Z T Z S = Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 30/46

We need to find Z T, given Z S and line parameters. From Equation 36, Z S = Z T cosh(γl) + Z 0 sinh(γl) Z 0 Z T sinh(γl) + Z 0 cosh(γl), we obtain Rearranging, Z S Z 0 = Z T Z 0 + tanh(γl) 1 + Z T Z 0 tanh(γl). (39) Z T Z S = Z S Z 0 tanh(γl) 1 Z S Z 0 tanh(γl). (40) Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 30/46

Equation 40 is written for traveling from load to generator. However, as we need to find Z T, we need to travel toward the load. Therefore, we need to use l in place of +l. We note that tanh( γl) = tanh(γl). Z T = quantity to be found, Z S = 30 j 10 Ω, Z 0 = 50 + j 0 Ω, l = 100 m, α = 0.002 nepers/m, v p = 2.7 10 8 m/s. Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 31/46

Z S Z 0 = 0.6 0.2j, αl = 0.002 100 = 0.2 nepers, 2αl = 0.4 nepers, βl = ωl v p = 2π 10 106 100 2.7 10 8 = 23.27 radians, 2βl = 46.54 radians, tanh(γl) = tanh(αl + j βl), = tanh(0.2 + j 23.27). Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 32/46

Z S tanh(γl) = tanh(0.2 + j 23.27), = 0.6 0.2j, Z 0 sinh0.4 + j sin46.54 αl = 0.002 100 = 0.2 nepers, = cosh0.4 + cos46.54. 2αl = 0.4 nepers, βl = ωl = 2π 10 sinh0.4 = e0.4 e 0.4 = 0.411. 106 100 2 = 23.27 radians, v p 2.7 cosh0.4 10 8 = 1.081. 2βl = 46.54 radians, sin 46.54 = 0.5513. tanh(γl) = tanh(αl + j cos βl), 46.54 = 0.8343. 0.411 + j 0.5513 = tanh(0.2 + jtanh(γl) 23.27). = 1.081 0.8343. Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 32/46

Z S Z 0 = 0.6 0.2j, αl = 0.002 100 = 0.2 nepers, 2αl = 0.4 nepers, βl = ωl v p = 2π 10 106 100 2.7 10 8 = 23.27 radians, 2βl = 46.54 radians, tanh(γl) = tanh(αl + j βl), = tanh(0.2 + j 23.27). 0.411 + j 0.5513 tanh(γl) = 1.081 0.8343, = 1.666 + j 2.2347. Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 32/46

Substituting this into the governing equation gives Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 33/46

Substituting this into the governing equation gives Z S Z 0 = Z T Z 0 + tanh(γl) 1 + Z T Z 0 tanh(γl).

Substituting this into the governing equation gives Z T 0.6 j 0.2 (1.666 + j 2.2347) = Z 0 1 (0.6 j 0.2)(1.666 + j 2.2347), 4.55 j 0.7063 =. 3.0 and finally, the desired result Z T = 76 j 12 Ω. Z S Z 0 = Z T Z 0 + tanh(γl) 1 + Z T Z 0 tanh(γl). Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 33/46

Short Circuit Termination If a short circuit is used as the load termination, then the sending-end impedance becomes, Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 34/46

Short Circuit Termination Z S Z 0 = Z T Z 0 + tanh(γl) 1 + Z T Z 0 tanh(γl). If a short circuit is used as the load termination, then the sending-end impedance becomes, Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 34/46

Short Circuit Termination If a short circuit is used as the load termination, then the sending-end impedance becomes, Z S SC = Z 0 tanh(γl), (41) and, neglecting line losses for short lengths we get Z S SC = Z 0 tanh(j βl) = j Z 0 tan(βl). (42) Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 34/46

Open Circuit Termination When the termination is open circuit, then the sending-end impedance becomes, Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 35/46

Open Circuit Termination Z T Z S Z = 0 + tanh(γl) Z When the termination is open circuit, 0 1 + then the Z T Z 0 tanh(γl). sending-end impedance becomes, Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 35/46

Open Circuit Termination When the termination is open circuit, then the sending-end impedance becomes, Z S OC = Z 0 tanh(γl), (43) and, neglecting line losses for short lengths we get Z S OC = j Z 0 tanh(j βl) = j Z 0 cot(βl). (44) Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 35/46

Remarks Equations 42 and 44 indicate that, by correctly choosing l, we can make the line (stub) to have the sending end impedance equivalent to a lumped capacitor or an inductor. Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 36/46

Remarks Equations 42 and 44 indicate that, by correctly choosing l, we can make the line (stub) to have the sending end impedance equivalent to a lumped capacitor or an inductor. When a transmission line is terminated in a perfect open or a short circuit, the incident energy is reflected back into the line. Product of Equations 42 and 44 gives Z S OC Z S SC = Z 2 0 (45) This is a useful method of measuring the characteristic impedance of a transmission line. For this the line length selected should be a value close to an odd number of λ/8. Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 36/46

tan(βl) 6 4 2 0 π 2 π 3π 2 2π 5π 2 βl = 2πl λ 2 4 6 Normalized input resistance versus βl for a Antennas and Propagation: short-circuited, Transmission Lines lossless, transmission Basic Transmission Line Properties line. 37/46

cot(βl) 6 4 2 0 π 2 π 3π 2 2π 5π 2 βl = 2πl λ 2 4 6 Normalized input resistance versus βl for a Antennas and Propagation: open-circuited, Transmission Lines lossless, transmission Basic Transmission Line Properties line. 38/46

Example The design of a microwave amplifier requires an inductor to series resonate a capacitive reactance of 15,000 Ω. Calculate the residual resistance if the inductor consists of a short-circuited line with Z 0 = 75 Ω, α = 0.002 db/cm, and λ = 10 cm. Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 39/46

Example The design of a microwave amplifier requires an inductor to series resonate a capacitive reactance of 15,000 Ω. Calculate the residual resistance if the inductor consists of a short-circuited line with Z 0 = 75 Ω, α = 0.002 db/cm, and λ = 10 cm. Solution To series resonate the j 15000 Ω, the input impedance of the shorted line must be +j 15000 Ω. If the line were lossless, Z in = j 15000 = j 75tan(βl), and therefore βl = Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 39/46

Example The design of a microwave amplifier requires an inductor to series resonate a capacitive reactance of 15,000 Ω. Calculate the residual resistance if the inductor consists of a short-circuited line with Z 0 = 75 Ω, α = 0.002 db/cm, and λ = 10 cm. To series resonate the j 15000 Ω, the input impedance of the shorted line must be +j 15000 Ω. If the line were lossless, Z in = j 15000 = j 75tan(βl), and therefore βl = tan 1 200 π 2 1 200 rad. Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 39/46 Solution

Since β = 2π/λ, l = Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 40/46

Since β = 2π/λ, ( π l = 2.5 2 1 ) cm. 100π Thus a shorted lossless line about 0.3% less than a quarter wavelength (2.5 cm) would provide the desired reactance without adding any resistance. However, in this line is not lossless and αl = 0.005 db or 5.76 10 4 Np (1 Np = 8.686 db). Since tanh(αl) αl, substituting Z 0 = 75 Ω and tan(βl) = 200 in to Equation 37 yields Z in = Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 40/46

Since β = 2π/λ, ( π l = 2.5 2 1 ) cm. 100π Thus a shorted lossless line about 0.3% less than a quarter wavelength (2.5 cm) would provide the desired reactance without adding any resistance. However, in this line is not lossless and αl = 0.005 db or 5.76 10 4 Np (1 Np = 8.686 db). Since tanh(αl) αl, substituting Z 0 = 75 Ω and tan(βl) = 200 in to Equation 37 yields Z in = 75 5.76 10 4 + j 200 1 + j 0.115 = 1700 + j 14800 Ω. Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 40/46

Note the effect of line loss. The slight loss in the reactive portion can be adjusted by slightly increasing the line length. The resistive component is significant. If we adjust the length, the residual resistance will actually be 1748 Ω. Antennas and Propagation: Transmission Lines Basic Transmission Line Properties 41/46

Outline 1 Basic Transmission Line Properties 2 Standing Waves Antennas and Propagation: Transmission Lines Standing Waves 42/46

Standing Waves For any transmission line, a sinusoidal signal of appropriate frequency introduced at the sending end by a generator will propagate along the length of the transmission line. If the line has infinite length, then the signal never reaches the end of the line. If the signal is viewed at some distance down the line away from the generator, then it will appear to have the same frequency, but will exhibit smaller peak to peak voltage swing than at the generator. Antennas and Propagation: Transmission Lines Standing Waves 43/46

Standing Waves For any transmission line, a sinusoidal signal of appropriate frequency introduced at the sending end by a generator will propagate along the length of the transmission line. If the line has infinite length, then the signal never reaches the end of the line. If the signal is viewed at some distance down the line away from the generator, then it will appear to have the same frequency, but will exhibit smaller peak to peak voltage swing than at the generator. The signal, therefore, has been attenuated by the line losses due to the conductor resistance Antennas and and Propagation: dielectric Transmission Linesimperfections. Standing Waves 43/46

Reflection If the line is lossless, then the signal viewed at some remote point will be identical to that at the generator but time delayed by an amount dependant on the position. Antennas and Propagation: Transmission Lines Standing Waves 44/46

Reflection If the line is lossless, then the signal viewed at some remote point will be identical to that at the generator but time delayed by an amount dependant on the position. When a sinusoidal signal reaches the open end of a section of a lossless transmission line, it can dissipate no energy. This means that all the energy propagating along the line in the forward direction (incident) will be reflected completely, on reaching the open circuit termination. Antennas and Propagation: Transmission Lines Standing Waves 44/46

Reflection If the line is lossless, then the signal viewed at some remote point will be identical to that at the generator but time delayed by an amount dependant on the position. When a sinusoidal signal reaches the open end of a section of a lossless transmission line, it can dissipate no energy. This means that all the energy propagating along the line in the forward direction (incident) will be reflected completely, on reaching the open circuit termination. The reflected wave (backward wave) must be such that the total current at the open circuit is Antennas and zero. Propagation: Transmission Lines Standing Waves 44/46

Reflection As the reflected signal travels back along the line toward the generator, it reinforces the incident waveform at certain points forming maxima (nodes). Antennas and Propagation: Transmission Lines Standing Waves 45/46

Reflection As the reflected signal travels back along the line toward the generator, it reinforces the incident waveform at certain points forming maxima (nodes). Similarly, it can cancel the incident waveform at certain other points producing minima (antinodes). In an open circuit line, node voltage points will occur at the same position as the antinode current points. Antennas and Propagation: Transmission Lines Standing Waves 45/46

Reflection As the reflected signal travels back along the line toward the generator, it reinforces the incident waveform at certain points forming maxima (nodes). Similarly, it can cancel the incident waveform at certain other points producing minima (antinodes). In an open circuit line, node voltage points will occur at the same position as the antinode current points. These waves do not represent traveling waves. They are standing waves, implying that there is no net power flow from generator to the Antennas and (open-circuit) Propagation: Transmission Lines load. Standing Waves 45/46

A point one-quarter wavelength away from a short circuit will have voltage and current magnitudes equivalent to those obtained for an open circuit. Antennas and Propagation: Transmission Lines Standing Waves 46/46

A point one-quarter wavelength away from a short circuit will have voltage and current magnitudes equivalent to those obtained for an open circuit. For a lossless line the peak value of the standing wave envelope is twice that of the incident wave. There are nulls as well due to the complete cancelation. Antennas and Propagation: Transmission Lines Standing Waves 46/46

A point one-quarter wavelength away from a short circuit will have voltage and current magnitudes equivalent to those obtained for an open circuit. For a lossless line the peak value of the standing wave envelope is twice that of the incident wave. There are nulls as well due to the complete cancelation. For lossy (practical) line, the peak will be less than twice that of the incident wave, and complete cancelation rarely results. Antennas and Propagation: Transmission Lines Standing Waves 46/46