Electromagnetic problem

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UNIVERSITY OF CALIFORNIA, DAVIS Department of Electrical and Computer Engineering

EEC130A Introductory Electromagnetics I Winter 2021

OPTIONAL - BONUS PROBLEMS

Adds an extra 12.5% on TOP of your overall EEC130A grade

Deadline: February 22nd at midnight

BACKGROUND

In our lectures, we learned about the single-stub matching method where the two degrees of freedom are the stub length ℓ and the distance to the load 𝑑𝑑. In some applications, moving the location where the stub is connected is difficult. In such cases, a double-stub matching approach can be used. In double-stub matching, two stubs are used and both are connected at fixed locations. By changing the length of the stubs, ℓ1 and ℓ2, one can effective achieve impedance matching. The drawback of double stub tuning is that a certain range of load impedances cannot be matched once the stub locations are fixed – the so- called forbidden impedance region.

The working principle of the double-stub matching circuit is similar to the single-stub matching in the sense that we want to match the real part first, i.e., move onto the constant 𝑟𝑟 = 1 circle, and then cancel the reactive part by a shunt stub (ℓ2). Due to the existence of the 𝑑𝑑0 transmission line, however, we cannot move to the constant 𝑟𝑟 = 1 circle directly. Instead, we first rotate the constant 𝑟𝑟 = 1 circle toward to the load by an angle of 2𝛽𝛽𝑑𝑑0. With the help of the first stub, we move along the constant 𝑟𝑟 circle to intersect with the rotated constant 𝑟𝑟 = 1 circle (point Γ𝐴𝐴, Fig. 2). Then we rotate 2𝛽𝛽𝑑𝑑0 onto the actual constant 𝑟𝑟 = 1 circle (point Γ𝐵𝐵, Fig. 2). Finally, we use the second stub to cancel out the remaining reactance at point Γ𝐵𝐵.

You can find additional information about Double stub-matching in Canvas->Files->Bonus and in the course bibliography.

Fig. 1. Schematic of a double-stub circuit for impedance matching Fig. 1. Schematic of a double-stub circuit for impedance matching Fig. 1. Schematic of a double-stub circuit for impedance matching

Fig. 2. Illustration of the double-stub matching procedures in the Smith Chart.

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Problem 1: Impedance matching with a double-stub configuration (5 pts).

Let us consider the circuit shown in Fig. 1. The load impedance is 𝑍𝑍𝐿𝐿 = 100 − 𝑗𝑗50Ω. The stubs are terminated with a short circuit and have a characteristic impedance of 50 Ω. The first stub is located at a distance

𝑑𝑑 𝜆𝜆

= 0.028 from the load. The phase provided by the intermediate transmission line is 𝛽𝛽𝑑𝑑0 = 3𝜋𝜋 4

. Determine the length of the stubs.

Problem 2: Manipulating the forbidden impedance region (7.5 pts).

Let us consider the circuit shown in Fig. 3. Determine the length of the transmission lines (𝑑𝑑1 and 𝑑𝑑2) as well as the length (ℓ1 and ℓ2) and termination (short or open) of the stubs to enforce that the circuit exhibits the forbidden impedance region shown in the Smith Chart (Fig. 4). The operation frequency is

5GHz, 𝑑𝑑3 = 𝜆𝜆 2 , 𝑍𝑍0 = 50Ω, �̅�𝑍𝐹𝐹𝐹𝐹 = 0.2, and the radius of the forbidden impedance region is �̅�𝑟=0.2.

Fig. 4. Desired forbidden impedance region for the circuit shown in Fig. 3

Fig. 3. Schematic of a double-stub circuit for impedance matching.