Transmission line and Impedance matching.

the ojective is to design transmission line work at 2.45 GHz and bandwidth 100 kHz and match it with antenna in small transmiting circuit.

1.1 Transmission line type.
The main purposes of this section are discussing the type of transmission line which is more suitable for small system design and then match it with transmitter and antenna impedance. Transmission lines are designed to transfer electromagnetic power from transmitter to antenna without loss or radiation (Kaiser, 2006). It is clear that the mismatch between transmission line and transmitter or antenna will increase the level of reflections which decreases the radiated power by the antenna and standing waves will occur (Collin, 2001).
In this assignment, the transmitter impedance is 50 ohm which means the transmission line should be designed with characteristic impedance equal to 50 ohm. Moreover, the size of the system is very small which requires very small dimensions to design the transmission line and should be suitable for the system bandwidth 100 kHz at operating frequency 2.45 GHz. In our design, a microstrip transmission line is used. It has been stated by Chang (2005) that a microstrip line has wide bandwidth and can be used in high frequency range. Furthermore, it is a planar transmission line which allows controlling its characteristics by the dimensions in one plane (Chen et al., 2004).
1.2 Transmission line design.
As shown in Figure 1, a microstrip line is a conductor and ground plane separated by dielectric substrate where W is the width of conductor, h is the width of the dielectric substrate and ε_r is a dielectric constant (Chen et al., 2004).

Figure 1: Geometry of a microstrip line (Chen et al., 2004).
It has been introduced by Chen et al (2004) that the concept of effective dielectric permittivity ε_eff to find the parameters of the transmission line such as wavelength, phase velocity v_p and characteristic impedance Z_c:
λ_g=λ_0/(√ε_eff ) (1)
v_p=c/(√ε_eff ) (2)
Z_c=(Z_c^0)/(√ε_eff ) (3)
ε_eff=(ε_r+1)/2+ε_(r-1)/2 [1/(√(1+12⁄u) )+0.041(1-u)^2 ] (for u≤1) (4)
ε_eff=(ε_r+1)/2+(ε_r-1)/2 1/(√(1+12/u) ) (for u>1) (5)
Where c is the light speed, Z_c^0 is the characteristic impedance of the microstrip line when the dielectric is air and u=W/h is relative microstrip width.
To calculate Z_c of microstrip, these two expressions are given by Kaiser (2006):
Z_c=60/(√ε_eff ) ln(8h/W+0.25 W/h) if W/h ≤1 (6)
Z_c=377/(√ε_eff [W/h+1.393+0.667 ln(W/h+1.444)] ) if W/h ≥1 (7)
In our design, the characteristic impedance should be equal to 50 ohm to match the transmitter impedance. It has been stated by Steer (2010) that when W/h=2.056 and the dielectric is S_i O_2 or FR-4, the dielectric constant ε_r=4, the effective dielectric permittivity ε_eff=3.077 and Z_c=50 Ω so that the microstip transmission line is designed depending on these specifications.
The final design of the microstrip transmission line is:
W= 2.056 mm, h= 1 mm, ε_r=4 (The dielectric substrate is S_i O_2 or FR-4), ε_eff=3.077 and Z_c=50 Ω.
1.3 Transmission line impedance matching with antenna impedance.
It has been shown by Collin (2001) that the transmission line can be matched with load by using lumped element or stub. In addition, in lumped element technique, parallel and series elements will be connected to the matching circuit. However, stub circuit could be parallel or series and either open or short circuit. Moreover, a parallel and open-circuited stub is preferred to be used with microstrip. Therefore, to match the dipole antenna in our design, shunt stub is used and the calculations are done by using smith chart as below:
Normalized antenna impedance:
z_l=Z_L/Z_c =(20-j20)/50=0.4-j0.4
Locate z_l on smith chart.
Draw line from z_l passed through the Centre and draw a circle to intersect with the line to find the value of y_l.
y_l=1.25+j1.25
To find the location of stub, rotate y_l toward the generator until the first intersection with r=1 as shown on figure 2.
y_l^’=1-j1.13=1-jB
d= (0.25-0.182)+(0.334-0.25)=0.152 λ.
The imaginary part of y_l^’ is B=-1.13 and to match the load, the imaginary part should be equal to zero and to do that B=+1.13 will be added.
Locate this imaginary value on smith chart.
Now, to find the length of stub, rotate toward the load from +jB.
As shown on figure 2, the distance to y_sc is longer than that to y_oc and the shortest distance is preferred (to y_sc) because the size of the system is small.
l=(0.5-.365)=0.135 λ.
From equation (2) v_p=(3×〖10〗^8)/(√3.077)=1.71×〖10〗^8 m/s.
Find the wavelength: λ=v_p/f=(1.71×〖10〗^8)/(2.45×〖10〗^9 )=69.8×〖10〗^(-3) m/s.
Use this value to calculate the real values of the length of stub and its location from the load:
l=0.135×69.8×〖10〗^(-3)=9.423 mm.
d=0.152×69.8×〖10〗^(-3)=10.61 mm .
The final design of stub as shown in figure 3 is:
Open-circuited, length=9.423 mm and at distance from the antenna d=10.61 mm.

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