EE-223 Microwave Circuits (Fall 2014) Lecture 2

Dr. Atif Shamim EE Program King Abdullah University of Science and Technology (KAUST) 1

Transmission Lines

• Transmission Lines, or T-Lines for short, are characterized by their ability to guide propagation of electromagnetic energy (EM) and their length, which is on the order of the wavelength

• Telephone lines, Cable TV lines and lines connecting circuits on a printed circuit board or integrated circuit are examples

• Important performance criteria is that the T-line transfers the EM energy from the source to the load in an efficient manner, meaning with minimum loss of signal

• Loss of signal can be due to multiple reasons like reflection due to mismatch, absorption of the signal in the medium and radiation from the T-line

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Transmission Lines

In (a), a sinusoidal voltage is dropped across a resistor. The supply and resistor are connected by an ideal (negligible length) conductor, and these are shown in (b) to be in phase. In (c) a quarter-wavelength long transmission line is added between the supply and the resistor and the voltage at the resistor in (d) is 90°out of phase with the supply voltage. 3

Typical Dual Conductor T-Lines

These structures are called TEM lines because propagation of EM waves occurs in the Transverse-Electro- Magnetic mode, i.e. E- fields and H-fields are perpendicular to the direction of propagation =>Just like Plane Waves

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Distributed Parameter Model

• To study the line, we must break it up into a series of identical segments, where each “differential segment” ∆z is taken to be much smaller than λ, and hence representable by pure circuit elements (R, L, G or C) • Interestingly, application of circuit theory to a cascade of such segments actually describes traveling wave and EM effects on the line.

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Distributed Parameter Model

The differential segment is modeled as: - • a series resistance R’ (ohms/m) representing power loss in the conductors • a series inductance L’ (henries/m) representing magnetic flux between conductors • a shunt capacitance C’ (farads/m) representing electric field between conductors • a shunt conductance G’ (siemens/m) representing power loss in dielectric medium between conductors where the primes indicate per-unit-length (or distributed) values. Notes: • “Ideal” T-line parameters have R’ and G’ = 0 but L’ and C’ > 0 • G’ is not supposed to be equal to 1/R’

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Coaxial Cable Distributed Parameters

G'=2πσ d

ln ba( )

C '= 2πε

ln ba( )

L'= µ2π

ln ba( )

R'= 12π

1a

+1b

πfµσ c

Where σd and σc are conductances for the dielectric and the conductor respectively ε is the permittivity of the dielectric and is given as ε = εo εr (εo=8.85 x 10-12F/m) μ is the permeability of the dielectric and is given as μ = μo μr (μo =4 x π x 10-7 H/m)

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Problem

• Find the distributed parameters of a coaxial cable at 1.0 GHz, if the radius of the inner conductor is 0.45 mm, and the outer conductor goes from a radius of 1.47 mm to 2.4 mm. Assume the dielectric to be Polyethylene with zero conductance (εr = 2.45) and the conductors to be copper with conductivity of 5.8 x 107 S/m)

Answer (R’=3.8Ω/m, L’=240nH/m, G’=0 S/m, C’= 110pF/m)

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Telegraphists Equations

Figure 6-4: The distributed-parameter model including instantaneous voltage and current.

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Telegraphists Equations

We will analyze the line with a differential elements of length ∆z

At a given point z along the line we have at time t:

Instantaneous voltage and currents at both ends of the segment

Voltage on the left side, given as v(z,t), indicates that it is function of both time and position

Voltage on the right side is ∆z further away so given as v(z+∆z , t)

Similar comments can be made for currents

Applying Kirchhof’s voltage law and noting that voltage across an inductor is given as v=Ldi/dt

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Telegraphists Equations

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Time Harmonic Waves on T- Lines

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Time Harmonic Waves on T- Lines

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Propagation constant

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Traveling Wave equations for T-Lines

Eqn (6.19) is the homogeneous second order differential equation and the general solution for that is given below in Eqn (6.21)

Eqn (6.21) till (6.24) are called the traveling wave equations for the transmission line 15

Characteristic Impedance

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Characteristic Impedance

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Problem

• Find γ, α, β, Zο for the transmission line with following distributed parameters at 1 GHz

R’= 3.8 Ω/m L’= 240 nH/m G’ = 0 S/m C’ = 110 pF/m Tip: The two square roots of a+bi are (x +yi) and -(x +yi) with y = sqrt((r - a)/2) and x = b/(2.y) Answers: γ=0.04 + j 32 m-1, α= 0.04 Np/m, β= 32 rads/m, Zo= 47- j0.06 Ω

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Lossless Line

If R’ << ωL’ and G’ << ωC’, we can assume that R’ =G’=0 and consider the line to be lossless. Evaluating the propagation constant of Eqn (6.20)

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Lossless Line

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Example 6.1

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Problem

• What outer radius of teflon dielectric is required in previous example (6.1) to give the line a 50 ohm characteristic impedance?

Answer= 1.7 mm

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Power Transmission

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Power Transmission

Power ratios expressed as Gain (in dB)

G = Pout / Pin

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Power Ratio

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Attenuation in Nepers

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Short Problems

1. The output of a 10 dB amplifier is measured at 10 mW. How much input power was applied? Answer= 1mW

2. Convert 1 mW and 10 mW into dBm. Answer= 0 dBm, 10 dBm

1. A 12 dB amplifier is in series with a 4-dB attenuator. What is the overall gain of the circuit? Answer= 8 dB

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