Transmission line characteristic impedance (Zo)

The transmission line is an RLC network (see Fig. 3-2), so it has a characteristic impedance Zo, also sometimes called a surge impedance. Network analysis will show that Zo is a function of the per unit of length parameters resistance R,con-ductance G,inductance L, and capacitance C, and is found from equation :

where Zo is the characteristic impedance, in ohms 
 R is the resistance per unit length, in ohms 
G is the conductance per unit length, in mhos  
L is the inductance per unit length, in henrys  
C is the capacitance per unit length, in farads 
ω is the angular frequency in radians per second (2πF)

 In microwave systems the resistances are typically very low compared with the reactances, so Eq. 3.1 can be reduced to the simplified form:

 Example 3-1A nearly lossless transmission line (Ris very small) has a unit length inductance of 3.75 nH and a unit length capacitance of 1.5 pF. Find the char-acteristic impedance Zo,
Solution:

The characteristic impedance for a specific type of line is a function of the con-ductor size, the conductor spacing, the conductor geometry (see again Fig. 3-1), andthe dielectric constant of the insulating material used between the conductors. Thedielectric constant eis equal to the reciprocal of the velocity (squared) of the wavewhen a specific medium is used

where e is the dielectric constant (for a perfect vacuum e= 1.000) v is the velocity of the wave in the medium

(a)Parallel line

where Zo is the characteristic impedance, in ohms e is the dielectric constant S is the center-to-center spacing of the conductors d is the diameter of the conductors

 (b) Coaxial line

where D is the diameter of the outer conductor d is the diameter of the inner conductor 

(c)Shielded parallel line

where A=s/d 

          B=s/D

 (d)Stripline

 where  

et ,is the relative dielectric constant of the printed wiring board (PWB) 

T is the thickness of the printed wiring board 

W is the width of the stripline conductor 

The relative dielectric constant e tused above differs from the normal dielectric constant of the material used in the PWB. The relative and normal dielectric con-stants move closer together for larger values of the ratio W/T.Example 3-2A stripline transmission line is built on a 4-mm-thick printed wiring board that has a relative dielectric constant of 5.5. Calculate the characteris-tic impedance if the width of the strip is 2 mm.

Solution :

 

 

In practical situations, we usually don’t need to calculate the characteristic im-pedance of a stripline, but rather design the line to fit a specific system impedance(e.g., 50 Ω). We can make some choices of printed circuit material (hence dielectricconstant) and thickness, but even these are usually limited in practice by the avail-ability of standardized boards. Thus, stripline widthis the variable parameter. Equa-tion 3.2 can be arranged to the form:

 The impedance of 50 Ωis accepted as standard for RF systems, except in the cable TV industry. The reason for this diversity is that power handling ability and lowloss operation don’t occur at the same characteristic impedance. For example, the maximum power handling ability for coaxial cables occurs at 30 Ω, while the lowest loss occurs at 77 Ω; 50 Ωis therefore a reasonable tradeoff between the two points.In the cable TV industry, however, the RF power levels are minuscule, but lines arelong. The tradeoff for TV is to use 75 Ωas the standard system impedance in orderto take advantage of the reduced attenuation factor.

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