EC546 Microwave Technology - Impedance Matching Notes

EC546 Microwave Technology - Impedance Matching

Introduction to Microwaves

• Microwaves are electromagnetic waves with frequencies ranging from approximately 1 GHz to 30 GHz or more.

• Their wavelengths fall between 1 cm and 30 cm or less.

Advantages of Microwave Frequencies:

1. Large Bandwidth: Supports higher data rates.

2. Reliable Propagation: Due to line of sight propagation.

3. Ionosphere Penetration: Ability to penetrate the ionosphere layer.

4. Lower Power Requirements: Smaller power requirements for both transmitter (Tx) and receiver (Rx) compared to other frequency bands.

5. Smaller Antenna Size and Narrow Radiation Beam

Disadvantages of Microwave Frequencies:

1. Limited Communication: Primarily limited to point-to-point communication.

2. Atmospheric Dependence: High dependence on atmospheric interference.

Electromagnetic Spectrum and Band Designation

• The electromagnetic spectrum includes various bands such as Radio, Microwave, Infrared, Visible, Ultraviolet, X-ray, and Gamma Ray.

• Frequencies range from extremely low frequencies (ELF) to cosmic rays (covering a wide range from 10^2 Hz to 10^24 Hz).

• Wavelengths vary from 10^6 meters to 10^-16 meters.

Frequency Bands

*Various frequency bands include VLF, HF, VHF, UHF, SHF, and EHF.

• VLF: (10^3) Hz

• HF: (10^7) Hz

• VHF: (10^8) Hz

• UHF: (10^9) Hz

• SHF: (10^{10}) Hz

• EHF: (10^{11}) Hz

Size Reference and Wavelength

• Wavelengths are associated with physical sizes, ranging from the size of a football field to subatomic particles.

• Frequencies also span a wide range, with corresponding uses in various applications.

• Formulas:

  • $E = hf$ (Energy of a photon)

Applications of Different Frequency Bands

• AM Radio: 600KHz-1.6MHz

• FM Radio: 89-109 MHz

• TV Broadcast: 54-700 MHz

• Mobile Phones: 900MHz-2.4GHz

• Wireless Data: ~ 2.4 GHz

• Microwave Oven: 2.4 GHz

• Radar: 1-100 GHz

• Visible Light: 425-750THz (700-400nm)

• Medical X-rays: (10-0.1 \AA)

Frequency Bands for Transmission Lines

• Different frequency bands are used for various applications, including power transmission, radio communication, and microwave systems.

• Transmission lines like twisted pair, coaxial cable, and optical fiber are used in different frequency ranges.

Frequency Ranges

• ELF/VF/VLF: Power and telephone applications

• MF/HF/VHF: Radio and television

• UHF/SHF/EHF: Microwave and radar systems

Microwave Applications

• Communication: Cellular, satellite, and terrestrial links, wireless LANs

• Environmental: Remote sensing, mapping, water and mineral searching

• Military: Radar, missile guidance, jamming, HPMW

• Medical: Diagnostics, deep heat therapy

• Industrial: Heating, imaging

• Household: Cooking

Difference between RF and MW

• Both RF (Radio Frequency) and Microwave (MW) are used to represent frequency ranges in the electromagnetic spectrum.

• The EM spectrum is classified into eight regions based on radiation intensity, divided into radio and optical spectra.

• Radio spectrum includes radio waves, microwaves, and terahertz radiations.

• Optical spectrum includes infrared, visible, ultra violet, X-rays, and gamma radiations.

Definitions

• "Micro" means very small (millionth part of a unit).

• Microwave identifies EM waves above 1GHz due to the short physical wavelength.

• Microwaves are a subset of radio frequencies.

• Microwave range starts from 300MHz to 300GHz.

RF vs. Microwave Characteristics

• Microwave Example:

  • $f = 10 GHz$

  • $\lambda = \frac{c}{f} = \frac{3}{10} = 3 cm$

  • Electrical Length (EL) for 1 cm line: $\frac{1 cm}{3 cm} = 0.333 \lambda$

  • Phase Delay (PD): $2\pi \times EL = 2\pi \times 0.333 = 120^o$

• Radio Frequency (RF) Example:

  • $f = 100 KHz$

  • $\lambda = \frac{c}{f} = \frac{300}{100} = 3 Km$

  • Electrical Length (EL) for 1 cm line: $\frac{1 cm}{300000 cm} = 3.3 \mu\lambda$

  • Phase Delay (PD): $2\pi \times EL = 0.000006^o$

Implications

• In microwaves, voltage and current waves do not affect the entire circuit at the same time; transmission line treatment is necessary.

• In RF, the phase change is insignificant, and standard circuit theory can be applied.

RF and Microwave Components

• RF components are lumped elements; standard circuit theory applies.

• Microwave components are distributed elements; standard circuit theory is an approximation, and Maxwell's equations are more accurate.

Common Types of Transmission Lines

• Two-wire line

• Coaxial cable

• Microstrip

Types of Transmission Modes

• TEM (Transverse Electromagnetic): Electric and magnetic fields are orthogonal to each other and to the direction of propagation.

• Examples: Coaxial line, Two-wire line, Parallel-plate line, Strip line, Microstrip line, Coplanar waveguide

• Higher-Order Transmission Lines: Rectangular waveguide, Optical fiber

Why Impedance Matching?

• Maximum power is delivered when the load is matched to the line, minimizing power loss.

• Impedance matching improves the signal-to-noise ratio in sensitive receiver components.

• Reduces amplitude and phase errors in power distribution networks (e.g., antenna arrays).

Factors for Selecting a Matching Network

1. Complexity: Simpler networks are cheaper, more reliable, and have lower loss.

2. Bandwidth: Narrowband or broadband requirements.

3. Implementation: Depends on the technology used.

4. Adjustability: Needed for variable loads.

Quarter Wavelength Transformer

• It is a section of transmission line with length $\frac{\lambda}{4}$ and characteristic impedance (Z_1).

• Used as an intermediate matching section to connect two wave guiding systems of different characteristic impedances.

• Used to match real impedance to a real load.

Formula

• $Z<em>{in} = Z</em>1 \frac{R<em>L + j Z</em>1 tan \beta l}{Z<em>1 + j R</em>L tan \beta l}$

• For $\beta l = \frac{\pi}{2}$, $Z<em>1 = \sqrt{Z</em>0 R_L}$

Example

• Match a load resistance $R_L = 100 \Omega$ to a $50 \Omega$ line.

• $Z_1 = \sqrt{100 \times 50} = 70.71 \Omega$

• $\beta l = \frac{\pi f}{2 f_0}$

Imperfect Match

• $\Gamma = \frac{Z<em>{in} - Z</em>0}{Z<em>{in} + Z</em>0} = \frac{Z<em>L - Z</em>0}{Z<em>L + Z</em>0 + j 2 tan(\beta l) \sqrt{Z<em>0 Z</em>L}}$

• $|\Gamma| \approx \frac{|Z<em>L - Z</em>0|}{2 \sqrt{Z<em>0 Z</em>L}} |cos \theta|$

• $\Delta \theta = 2 (\frac{\pi}{2} - \theta_m)$

Bandwidth

• $\frac{\Delta f}{f<em>0} = 2 - \frac{4}{\pi} cos^{-1} [\frac{\Gamma</em>m 2 \sqrt{Z<em>0 Z</em>L}}{|Z<em>L - Z</em>0|}]$

Disadvantages of QWT

1. Narrow bandwidth

2. $Z<em>0$ and $Z</em>T$ are frequency-dependent in waveguides

3. Equivalent susceptance at junctions causes mismatching problems

Multi-section Quarter Wavelength Transformer

• Uses multiple sections of transmission line for broadband matching.

• Analysis is simplified using the theory of small reflections.

• Overall reflection coefficient can be written as

• $\Gamma (\theta) = \Gamma<em>0 + \Gamma</em>1 e^{-j2\theta} + \Gamma_2 e^{-j4\theta} + …$

• For symmetrical transformers:

• $\Gamma(\theta) = e^{-jN\theta} [\Gamma<em>0 (e^{jN\theta} + e^{-jN\theta}) + \Gamma</em>1 (e^{j(N-2)\theta} + e^{-j(N-2)\theta}) + …]$

• $\Gamma(\theta) = 2e^{-j N \theta} [\Gamma<em>0 cos N\theta + \Gamma</em>1 cos (N-2)\theta + …]$

Binomial Multi-section Transformers

*Coefficients are determined from the binomial expansion as

$(1+x)^{m-1} = 1 + (m-1)x + \frac{(m-1)(m-2)}{2!} x^2 + \frac{(m-1)(m-2)(m-3)}{3!} x^3 + …$

*Individual Section Calculation :
$\Gamma<em>n = 2^{-N} \frac{Z</em>L - Z<em>0}{Z</em>L + Z<em>0} C</em>n^N$

$Z<em>{n+1} = Z</em>n e^{2\Gamma_n}$

*Fractional Bandwidth :
$\frac{\Delta f}{f<em>0} = \frac{4}{\pi} cos^{-1} [ \frac{\Gamma</em>m}{\frac{1}{2} \frac{|Z<em>L - Z</em>0|}{\sqrt{Z<em>L Z</em>0}}} ]$

The response of the binomial multi-section transformer is optimum because for a given number of sections the response is as flat as possible near the design frequency

Example

*3 Sections Binomial Transformer :

$\Gamma<em>0 = \Gamma</em>3 = 2^{-3} \frac{100-50}{100+50} C_0^3$

$Z<em>1 = Z</em>0e^{2\Gamma_0} = 50e^{2(0.0416)} = 54.35 \Omega$

$\Gamma<em>1 = \Gamma</em>2 = 2^{-3} \frac{100-50}{100+50} C_1^3$

$Z<em>2 = Z</em>1e^{2\Gamma_1} = 54.35e^{2(0.125)} = 69.79 \Omega$

$Z<em>3 = Z</em>2e^{2\Gamma_2} = 69.79e^{2(0.125)} = 89.607 \Omega$

$Z<em>4 = Z</em>3e^{2\Gamma_3} = 89.607e^{2(0.0416)} = 97.3867 \Omega$

VSWR Calculation

$\Gamma_m = \frac{S-1}{S+1} = \frac{1.2-1}{1.2+1} = 0.091$

$\frac{\Delta f}{f<em>0} = \frac{4}{\pi} cos^{-1} [ 2^{-N} (\frac{\Gamma</em>m}{\frac{1}{2} \frac{|Z<em>L - Z</em>0|}{Z_0}} ) ] = 0.898$

Chebyshev Multisection Transformers

The 'nth'-order Chebyshev polynomial is polynomial of degree 'n', denoted by $T_n(x)$

First four Chebyshev polynomials:

$T_1(x) = x$

$T_2(x) = 2x^2 -1$

$T_3(x) = 4x^3 - 3x$

$T_4(x) = 8x^4 - 8x^2 + 1$

Higher order polynomials can be found using the recurrence formula
$T<em>n(x) = 2xT</em>{n-1}(x) - T_{n-2}(x)$

Design of Chebyshev Transformers

$cosh^{-1} (sec\theta<em>m) = \frac{1}{n} cosh^{-1}(\frac{1}{2\Gamma</em>m} ln(\frac{Z<em>L}{Z</em>0}))$
$T<em>N (sec\theta</em>m cos\theta ) = 2\Gamma<em>0 cos N\theta + 2\Gamma</em>1 cos(N-2)\theta + …$

Example

*3 section Tchebyshev transformer :
$sec\theta<em>m = cosh(\frac{1}{3}cosh^{-1} [\frac{ln(\frac{100}{50})}{2(0.05)}] = 1.4075$ $\theta</em>m = 44.72^\circ$
$cos(\frac{\pi}{4} sec^{-1}(1.4075))$

Calculate Values

Following similar calculations as above for reflection co-efficients and impedance values of sections can then be computed