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**EKT 441 MICROWAVE Communications**

CHAPTER 2: PLANAR TRANSMISSION LINES

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Transmission Lines A device used to transfer energy from one point to another point “efficiently” Efficiently = minimum loss, reflection and close to a perfect match as possible (VSWR = 1:1) Important to be efficient at RF and microwave frequencies, since freq used are higher than DC and low freq applications as freq gets higher, any energy loss is in transmission lines are more difficult and costly to be retrieved

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**Some Types of Transmission Lines**

Some types of transmission lines are listed as follow: Coaxial transmission line transmission line which a conductor completely surrounds the other Both shares the same axis, separated by a continuous solid dielectric or dielectric spacers Flexible – able to be bent without breaking

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**Some Types of Transmission Lines**

Waveguides Hollow-pipe structure, in which two distinct conductor are not present Open space of the waveguide is where electromagnetic energy finds the path of least resistance to propagate Do not need any dielectric medium as it uses air as medium of energy transfer Planar transmission lines Planar – looks like a 3D line that have been run over and flattened Usually made up of a layer of dielectric, and one or several ground (metallic planes) Four types of planar lines discussed in this chapter; (1) Stripline, (2) microstrip (3) dielectric waveguide (4) Slotline Our focus in this chapter

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**But Why Planar? Waveguides Coaxial lines**

High power handling capability low loss bulky expensive Coaxial lines high bandwidth, convenient for test applications difficult to fabricate complex microwave components in the medium

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**But Why Planar? Planar Transmission Lines Compact Low cost**

Capability for integration with active devices such as diodes, transistors, etc

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STRIPLINE Figure 3.1: Stripline transmission line (a) Geometry (b) Electric and magnetic field lines.

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STRIPLINE Also known as “sandwich line” – evolved from “flattened” coaxial transmission line The geometry of a stripline is shown in Figure 3.1. Consist of a; (1) top ground plane, (2) bottom ground plane and (3) a center conductor W is the width of thin conducting strip (centered between two wide conducting ground planes). b is the distance of ground planes separation. The region between the ground planes is filled with a dielectric. Practically, the centered conductor is constructed of thickness b/2.

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STRIPLINE Figure 3.2: Photograph of a stripline circuit assembly.

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**STRIPLINE The phase velocity is given by:**

[3.1] Thus, the propagation constant of the stripline is: [3.2]

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STRIPLINE From equation [3.1], c = 3 x 108 m/sec is the speed of light in free-space. The characteristic impedance of a transmission line is given by: [3.3] L and C are the inductance and capacitance per unit length of the line. There is a solution as explained in [M. Pozar’ book]. The resulting formula for the characteristic impedance is: [3.4]

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STRIPLINE Where We is the effective width of the center conductor given by: [3.5] These formulas assume a zero strip thickness, and are quoted as being accurate to about 1 % of the exact results. It is seen from equation [3.4] and [3.5] that the characteristic impedance decreases as the strip width W increase.

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STRIPLINE When designing stripline circuits, one usually needs to find the strip width, W. By given characteristic impedance (and height b and permittivity εr), the value of W can be find by the inverse of the formulas in equation [3.4] and [3.5]. The useful formulas is: [3.6] Where, [3.7]

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**STRIPLINE The attenuation due to dielectric loss is:**

[3.8] The attenuation due to the conductor loss: [3.9]

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STRIPLINE With: [3.10] [3.11] Where t is the thickness of the strip

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STRIPLINE [EXAMPLE 2.1] Find the width for a 50 Ω copper stripline conductor, with b = 0.32 cm and εr = If the dielectric loss tangent (tan δ) is and the operating frequency is 10 GHz, calculate the attenuation in dB/λ. Assume the conductor thickness of t = 0.01 mm and surface resistance, Rs of Ω

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**SOLUTION [EXAMPLE 2.1] Since and**

Eq [3.6] gives the width as W = bx = (0.32)(0.830) = cm. At 10 GHz, the wave number is

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**SOLUTION [EXAMPLE 2.1] since A = 4.74 The dielectric attenuation is**

Surface resistance of copper at 10 GHz is Rs = Ω. Then from eq [3.9] since A = 4.74 The total attenuation constant is

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**SOLUTION [EXAMPLE 2.1] In dB;**

At 10 GHz, the wavelength on the stripline is; So in terms of the wavelength the attenuation is

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SOLUTION [EXAMPLE 2.1] But why do we need to convert Np/m to dB/m using this way? loss to the transmission line is reflected by the attenuation constant. The amplitude of the signal decays as e-α. The natural units of the attenuation constant are Nepers/meter, but we often convert to dB/meter in microwave engineering. To get loss in dB/length, multiply Nepers/length by

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STRIPLINE [EXAMPLE 2.2] Find the width for 50 Ω copper stripline conductor with b = 0.5 cm and εr = 3.0. The loss tangent is and the operating frequency is 8 GHz. Calculate the attenuation in dB/λ. Assume a conductor thickness of t = cm. The surface resistance is 0.03 Ω.

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SOLUTION [EXAMPLE 2.2] The wave number at f = 8 GHz

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**SOLUTION (cont) [EXAMPLE 2.2]**

The dielectric attenuation is Np/m The conductor attenuation is

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**SOLUTION (cont) [EXAMPLE 2.2]**

Total loss: Np/m In dB: dB/m

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**SOLUTION (cont) [EXAMPLE 2.2]**

The guided wavelength: The attenuation in dB/λ:

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**STRIPLINE DISCONTINUITY**

Figure 3.3: geometry of enclosed stripline

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**STRIPLINE DISCONTINUITY**

By derivation found in M.Pozar’s book (page 141), the surface charge density on the strip at y = b/2 is: [3.12] The charge density on the strip line by uniform distribution: [3.13]

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**STRIPLINE DISCONTINUITY**

The capacitance per unit length of the stripline is: [3.14] The characteristic impedance is then found as: [3.15]

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MICROSTRIP Figure 3.3: Microstrip transmission line. (a) geometry. (b) Electric and magnetic field lines.

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MICROSTRIP Microstrip line is one of the most popular types of planar transmission line. Easy fabrication processes. Easily integrated with other passive and active microwave devices. The geometry of a microstrip line is shown in Figure 3.3 W is the width of printed thin conductor. d is the thickness of the substrate. εr is the relative permittivity of the substrate.

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MICROSTRIP The microstrip structure does not have dielectric above the strip (as in stripline). So, microstrip has some (usually most) of its field lines in the dielectric region, concentrated between the strip conductor and the ground plane. Some of the fraction in the air region above the substrate. In most practical applications, the dielectric substrate is electrically very thin (d << λ). The fields are quasi-TEM (the fields are essentially same as those of the static case.

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**MICROSTRIP The phase velocity and the propagation constant:**

[3.16] [3.17] Where εe is the effective dielectric constant of the microstrip line used to compensate difference between the top and bottom of the circuit line The effective dielectric constant satisfies the relation: and is dependent on the substrate thickness, d and conductor width, W

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MICROSTRIP The effective dielectric constant of a microstrip line is given by: [3.18] The effective dielectric constant can be interpreted as the dielectric constant of a homogeneous medium that replaces the air and dielectric regions of the microstrip, as shown in Figure 3.4. Figure 3.4: equivalent geometry of quasi-TEM microstrip line.

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**MICROSTRIP The characteristic impedance can be calculated as:**

[3.19] For a given characteristic impedance Z0 and the dielectric constant Єr, the W/d ratio can be found as: [3.20]

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MICROSTRIP Where: Considering microstrip as quasi-TEM line, the attenuation due to dielectric loss can be determined as [3.21] Where tan δ is the loss tangent of the dielectric.

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MICROSTRIP This result is derived from Equation [2.37] by multiplying by a “filling factor”: Which accounts for the fact that the fields around the microstrip line are partly in air (lossless) and partly in the dielectric. The attenuation due to conductor loss is given approximately by: [3.22] Where Rs = √(ωμ0/2σ) is the surface resistivity of the conductor.

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MICROSTRIP [EXAMPLE 2.3] Calculate the width and length of a microstrip line for a 50 Ω characteristic impedance and a 90° phase shift at 2.5 GHz. The substrate thickness is d = cm, with εr = 2.20. Solution: M.Pozar’ book (page: 146).

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**SOLUTION [EXAMPLE 2.3] εe = 1.87**

First we need to find W/d for Z0=50 Ω, and initially guess that W/d > 2. From eq [3.20]; B = and W/d = 3.081 Otherwise, we would use the expression for W/d<2 Then W = (3.081.d) = cm. From eq [3.18]; εe = 1.87

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SOLUTION [EXAMPLE 2.3] The line length, l, for a 90o phase shift is found as;

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MICROSTRIP [EXAMPLE 2.4] Design a microstrip transmission line for 70 Ω characteristic impedance. The substrate thickness is 1.0 cm, with εr = What is the guide wavelength on this transmission line if the frequency is 3.0 GHz?

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SOLUTION [EXAMPLE 2.4] Initially, it is guessed that W/d < 2

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**SOLUTION cont [EXAMPLE 2.4]**

Since the W/d < 2 assumption is valid; We proceed to calculate εe

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**SOLUTION cont [EXAMPLE 2.4]**

Thus the guided wavelength is given by;

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MICROSTRIP [EXAMPLE 2.5] Design a quarter wavelength microstrip impedance transformer to match a patch antenna of 80 Ω with a 50 Ω line. The system is fabricated on a 1.6 mm substrate thickness with εr = 2.3, that operates at 2 GHz.

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**MICROSTRIP [EXAMPLE 2.5] From the quarter wave transformer equation;**

Which is also Z0 of line ZS Z0 RL

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SOLUTION [EXAMPLE 2.5] Since W is not known, guess that W/d < 2

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**SOLUTION cont [EXAMPLE 2.5]**

From the calculation, the initial assumption of W/d < 2 is incorrect. The next formula (where W/d > 2) is used

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**SOLUTION cont [EXAMPLE 2.5]**

The next step is to find the effective dielectric constant (εe) To determine the quarter wavelength of the line, the guided wavelength λg need to be determined

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**SOLUTION cont [EXAMPLE 2.5]**

Thus the quarter wave length of line is determined by dividing the full wavelength by 4;

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MICROSTRIP Figure 3.5: Geometry of a microstrip line with conducting sidewalls.

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**MICROSTRIP DISCONTINUITY**

By derivation found in M.Pozar’s book (page 147), the surface charge density on the strip at y = d is: [3.23] The charge density on the microstrip line by uniform distribution: [3.24]

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**MICROSTRIP DISCONTINUITY**

The capacitance per unit length of the stripline is: [3.25] The characteristic impedance is then found as: [3.26]

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DIELECTRIC WAVEGUIDE Figure 3.6: Dielectric waveguide geometry.

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**DIELECTRIC WAVEGUIDE The dielectric waveguide is shown in Figure 3.6.**

εr2 is the dielectric constant of the ridge. εr1 is the dielectric constant of the substrate. Usually εr1 < εr2 The fields are thus mostly confined to the area around the dielectric ridge. Convenient for integration with active devices. Very lossy at bends or junctions in the ridge line. Many variations in basic geometry are possible.

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SLOTLINE Figure 3.7: Geometry of a printed slotline.

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**SLOTLINE The geometry of a slotline is shown in Figure 3.7.**

Consists of a thin slot in the ground plane on one side of a dielectric substrate. The characteristic impedance of the line can be change by changing the width of the slot.

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Chapter 2. Transmission Line Theory

Chapter 2. Transmission Line Theory

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