4th Week Seminar Sunryul Kim 2018.08.01 Antennas & RF Devices Lab.

MICROWAVE ENGINEERING CONTENTS Lumped-Element Circuit Model Field Analysis of Transmission Lines Terminated Lossless Transmission Line Smith Chart Slotted Line Quarter-Wave Transformer Generator and Load Mismatches Lossy Transmission Lines MICROWAVE ENGINEERING David M . Pozar Antennas & RF Devices Lab.

Lumped-Element Circuit Model Transit Time Effect At some instant let the voltage at 𝐀𝐀′ be 𝑽 𝑷 . Then 𝑽 𝑷 will appear at 𝐁 𝐁 ′ only after 𝒕 𝒓 . However, during this time the voltage at 𝐀𝐀′ changes to 𝑽 𝑸 . Transit time 𝒕 𝒓 = 𝒍 𝒗 The transit time effect can be neglected if 𝑻≫ 𝒕 𝒓 → 𝛌≫𝒍 Antennas & RF Devices Lab.

𝑅=Ω/m , 𝐿=H/m , 𝐺=S/m , 𝐶=F/m Lumped-Element Circuit Model Transmission line Equations FIGURE 2.1 Voltage and current definitions and equivalent circuit for an incremental length of transmission line. (a) Voltage and current definitions. (b) Lumped-element equivalent circuit. KVL (Kirchhoff’s voltage law) KCL (Kirchhoff’s current law) Antennas & RF Devices Lab.

Lumped-Element Circuit Model Transmission line Equations Phasor General solution (2.3a) , (2.6a) Characteristic impedance Complex propagation constant Antennas & RF Devices Lab.

Lumped-Element Circuit Model Transmission line Equations (Lossless Line) General Lossless ‘Real’ Antennas & RF Devices Lab.

Field Analysis of Transmission Lines Parameters Chapter 1 Circuit theory FIGURE 2.2 Field lines on an arbitrary TEM transmission line. Magnetic energy Conductivity power loss Electric energy Dielectric power loss Antennas & RF Devices Lab.

Field Analysis of Transmission Lines Parameters TABLE 2.1 Transmission Line Parameters for Some Common Lines Antennas & RF Devices Lab.

Field Analysis of Transmission Lines Coaxial Line (TEM Wave) 𝜕/𝜕𝜙=0 𝐸 𝑧 = 𝐻 𝑧 =0 𝜕 𝜕𝜌 𝜌 𝐸 𝜙 =0 Independent of 𝜌 Antennas & RF Devices Lab.

Field Analysis of Transmission Lines Coaxial Line (TEM Wave) Voltage between the two conductors Eliminate 𝒉(𝒛) and 𝒈(𝒛) Current on the inner conductor (𝜌=𝑎) TABLE 2.1 Antennas & RF Devices Lab.

Terminated Lossless Transmission Line Total Voltage and Current FIGURE 2.4 A transmission line terminated in a load impedance 𝑍 𝐿 . Load impedance Rearrange Reflection coefficient Antennas & RF Devices Lab.

Terminated Lossless Transmission Line Return Loss Complex conjugate → zero Incident power Reflected power Return loss 1 2 𝑉 0 + 2 𝑍 0 1 2 𝑉 0 + 2 𝑍 0 𝛤 2 𝚪=𝟎 → RL =∞ 𝚪=𝟏 → RL =0 Second order! Antennas & RF Devices Lab.

Terminated Lossless Transmission Line Standing Wave Ratio 𝑉 𝑧 = 𝑉 0 + 𝑒 −𝑗𝛽𝑧 +𝛤 𝑉 0 + 𝑒 𝑗𝛽𝑧 Euler formula Standing Wave Ratio Antennas & RF Devices Lab.

Terminated Lossless Transmission Line Input Impedance 𝑒 𝑗𝜃 = cos 𝜃 +𝑗 sin 𝜃 Antennas & RF Devices Lab.

Terminated Lossless Transmission Line Short Circuit FIGURE 2.5 A transmission line terminated in a short circuit. =∞ Antennas & RF Devices Lab.

Terminated Lossless Transmission Line Open Circuit FIGURE 2.6 A transmission line terminated in a open circuit. =∞ Antennas & RF Devices Lab. Antennas & RF Devices Lab.

Terminated Lossless Transmission Line Half-Wavelength Line & Quarter-Wavelength Line Half-Wavelength Line 𝛽= 2𝜋 𝜆 Quarter-Wavelength Line “Quarter-wave transformer” =−𝛤(0) 180° phase shift ! Antennas & RF Devices Lab.

Terminated Lossless Transmission Line Junction of Two Transmission Lines FIGURE 2.9 Reflection and transmission at the junction of two transmission lines with different characteristic impedances. Equal when 𝑧=0 Return loss Transmission coefficient Insertion loss Antennas & RF Devices Lab.

Smith Chart Normalize Rearrange Rearrange Antennas & RF Devices Lab.

Smith Chart Antennas & RF Devices Lab.

Smith Chart Antennas & RF Devices Lab. Rotate the point clockwise an amount 2𝛽ℓ FIGURE 2.11 Smith chart for Example 2.2. Antennas & RF Devices Lab.

Slotted Line Antennas & RF Devices Lab. https://www.labvolt.com SWR, distance of the first voltage minimum from the load can be measured. Load impedance Antennas & RF Devices Lab.

Quarter-Wave Transformer The Impedance Viewpoint 𝑍 𝐿 , 𝑍 0 are both real FIGURE 2.16 The quarter-wave matching transformer. ℓ= 𝜆 4 Antennas & RF Devices Lab.

Quarter-Wave Transformer Multiple Reflection Viewpoint 180° phase shift ! numerator FIGURE 2.18 Multiple reflection analysis of the quarter-wave transformer. Antennas & RF Devices Lab.

Quarter-Wave Transformer Multiple Reflection Viewpoint numerator =0 numerator =0 This is the same result as (2.63) Antennas & RF Devices Lab.

Generator and Load Mismatches FIGURE 2.19 Transmission line circuit for mismatches load and generator. Voltage distribution Antennas & RF Devices Lab.

Generator and Load Mismatches FIGURE 2.19 Transmission line circuit for mismatches load and generator. Standing wave ratio Power delivered to the load Antennas & RF Devices Lab.

Generator and Load Mismatches Load Matched to Line Lossless line → 𝑍 0 is real → 𝑍 in is real Antennas & RF Devices Lab.

Generator and Load Mismatches Generator Matched to Loaded Line The power delivered to the load may be less than ‘Load Matched to Line’ Load Matched to Line Antennas & RF Devices Lab.

Generator and Load Mismatches Conjugate Matching *Find the maximum value through derivative. Antennas & RF Devices Lab.

Lossy Transmission Lines Low-Loss Line Taylor series expansion Low-loss line can be closely approximated by considering the line as lossless. Antennas & RF Devices Lab.

Lossy Transmission Lines Terminated Lossy Line FIGURE 2.20 A lossy transmission line terminated in the impedance 𝑍 𝐿 . Lossless tanh 𝑧 = 𝑒 𝑧 − 𝑒 −𝑧 𝑒 𝑧 + 𝑒 −𝑧 Antennas & RF Devices Lab.

Lossy Transmission Lines Terminated Lossy Line FIGURE 2.20 A lossy transmission line terminated in the impedance 𝑍 𝐿 . Both terms increase as 𝛼 increses. Antennas & RF Devices Lab.

Lossy Transmission Lines Perturbation Method for Calculating Attenuation *No use Transmission line parameters ( 𝐿, 𝐶, 𝑅, 𝐺 ). *Assume that the fields of the lossy line are not greatly different from the fields of the lossless line. Antennas & RF Devices Lab.

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