Radio to Terahertz Receiver Systems
PHYS46611
Lecture 3
Transmission Lines, Impedance match and S-parameters
1 TEM transmission line From previous lecture Eq. to transverse x-­‐sec4on of 2 conductors TL ϕ electrosta4c poten4al We need + and – charges needs mul3-­‐conductor guides Conductors are equipoten4als Cst ET perpendicular to conductor  HT tangen3al to conductor surface Crea3on of a current (Ampère’s law) along z with the same dependence than the fields Characteris4c impedance of the line (per unit length) For lossless line Inductance (per unit length) Parameters of the dielectric Capacitance (per unit length) Electrical length 2 TEM transmission line and Power It can be shown that the surface charge per unit length is: General Transmission Line But also that the magne4c flux is: E
Poyn3ng vector is along z. For TEM mode: Pz = T
2!
2
Power transmiIed along the line (see deriva4on in Chap 10) PT =
1
2
V
2Z
Lecture 1: ohmic losses are arising from both the dielectric losses and losses in the conductor walls R’ being the resistance per unit length of the line, the power loss due to the conductor is: If the dielectric has a conduc3on σd then a current will flow between the conductors I’d shunt current It is shown that: and G’ conductance of the line power loss per unit length due to the shunt current P’d= and The aIenua4on coefficient Dielectric loss constant loss tangent 3 Parallel plate lines If w>>h this is equivalent to a capacitor Ground plane Losses due to conductor Where Even if unfortunately the nota4on is the same, do not mix it up with the loss tangent! 4 Microstrip lines Fringing  different fields / parallel plates Longitudinal – transverse decomposi4on Boundary condi3ons: are con3nuous at y=h Ey small at the interface but non-­‐zero Ez non-­‐zero  not a TEM wave but quasi-­‐TEM We replace the boundary air-­‐dielectric by an effec3ve dielectric 5 Microstrip lines (2) Duroid Al2O4 Impedance can be tuned by adap3ng the ra3o w/h Advantages: Compact and can be easily manufactured (printed circuit technology) 6 Disadvantages: generally lower power handling capacity, and higher losses. Microstrip is not enclosed -­‐ suscep3ble to cross-­‐talk and radia3on leakage. Striplines Conductor Dielectric Ground planes h\p://www.amanogawa.com/archive/Stripline/Stripline-­‐2.html Advantages: •  True TEM: nearly non-­‐dispersive and no cut frequency •  Very good isola3on Disadvantages: •  Difficult to manufacture •  For a given impedance, strip width much narrower than microstrip •  Limited power 7 Coaxial transmission line Poten4al follow Laplace’s equa4on TEM transmission line in cylindrical coordinates •  Boundary condi3ons: ϕ(b)=0 and ϕ(a)=V •  Axial symmetry  no dependence with φ h\p://www.amanogawa.com/archive/Coaxial/Coaxial-­‐2.html E field has only a radial component So H field as only an azimuthal component 8 Two wire lines h\p://www.amanogawa.com/archive/TwoWire/TwoWire-­‐2.html 9 Circuit model of transmission line L’ and C’ describe a lossless line Addi3onally R’ and G’ describe the losses of a TL series impedance per unit length We define: shunt admiIance per unit length Distributed parameter circuit For an element Δz the impedance is Z’Δz Solu4ons can be wriIen as sums of forward and backward waves Lossless line: R’=G’=0 10 Wave impedance and reflec4on We follow the same reasoning than for a uniform plane wave propaga3on in a medium. Wave impedance and reflec3on are defined as: For simplifica4on we will assume a lossless line: Z0 is real, or at least R’ and G’ very small Iden3cally for 2 points 1 and 2 on the transmission line we obtain: and but also the rela4on between Z and Γ
We can also use the admiIance Y=1/Z for which we will have
Using the propaga4on matrices 11 Wave impedance and reflec4on (2) We get an equivalent circuit Equivalent terminated line Again with similarity to the quarter and half wave slab thickness we get: and 12 Two port equivalent circuit Matching matrix Impedance matrix With Conven4on: + when currents come in the port 13 Terminated Transmission Line We had l: length of transmission line from the load Load We had We can now write Generator Length d of transmission line (with characteris4c impedance Z0) between the generator and the load Which gives With 14 Matched, open-­‐ and short-­‐circuited TL If the line and the load are matched (ZL = Z0) then ΓL=0  Γd=0 and Zd=Z0 for any distance l Meaning that: Only a forward wave à voltage and current stay the same along the line with just a phase varia4on VL= IL Open-­‐circuited line: Short-­‐circuited line: 15 VSWR (Voltage Standing Wave Ra4o) We have found that: For the forward wave: varies sinusoidally with l between the values Vmax and Vmin Wri4ng We define the voltage VSWR standing wave ra3o as: ! L = 0 " VSWR = 1
! L = 1 " VSWR = #
This the characteris4c of a matched line: At any point l 16 Z
By normalising the z=
= r + jx
impedance to the Z0
characteris3c impedance Z " Z 0 z "1
!=
=
Z + Z 0 z +1
Smith Chart r=0 17 Example x r More at: h\p://www.amanogawa.com/
archive/LossLessSmithChart/
LossLessSmithChartWide-­‐2.html 18 move of l=λ/2 corresponds to rota4on of 360deg angle=125.37deg l in wavelength A VSWR varies with l but circle of constant but also constant =0.726 for A VSWR 19 ScaIering parameters Let’s consider a 2 port network Transfer or ABCD matrix Impedance matrix ScaIering matrix S-­‐parameters Relates the outgoing waves with incoming ones S11 and S22: reflec3on coefficients S12 and S21: transmission coefficients Rela4ons between waves Z0 reference impedance (usually 50Ω) 20 VNA measurement equivalent to length of cables Measurement VNA will measure: a1!, a!2 , b1! and b2!
we want to know: a1, a2 , b1 and b2
propaga4on matrices S Phase lengths δ1 and δ2 will be determined through calibra3on 21 S21 Actual measurement S12 S11 S22 By connec3on to a matched load ZL = Z0, we can make a2=0 =Γ1 reflec3on coefficient =T1 transmission coefficient Similarly, both ports are transmi\er and receiver with matched load so S22=Γ2 and S12=T2 22 Prac4cal measurement 23 Some defini4ons We define the Return Loss or Reflec4on Loss (in dB) as: !P $
RL = 10 log10 # r & = 20 log10 ( ' ) = 20 log10 ( S11 ) or 20 log10 ( S22 )
" Pi %
Power reflected/Power incident Note: this is then a nega3ve value. You might find another defini3on for which the RL is posi3ve: !P $
RL = 10 log10 # i & = '20 log10 ( S11 )
" Pr %
We define the Inser4on Loss (in dB) as: IL = !20 log10 ( S21 )
Be careful ! When the VNA displays the S-­‐parameters in dB, it will give 20 log10 ( Sxy )
So it will be nega3ve, except if we have an ac3ve component for which S21 and/or S12 could be posi3ve 24 Example of measurement 20 log10 ( S11 )
25