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ECE 710
Review of Transmission Line Theory
Chapter 2
Patrick Roblin $
Department of Electrical & Computer Engineering
The Ohio State University Columbus, OH 43210 & OHIO S ATE T
UNIVERSITY T.H.E Patrick Roblin 0 The Ohio State University % '
Lecture 2 $ Types of Electromagnetic waves Tranmission lines and waveguides Voltage and current waves Introduction to transmission line theory & OHIO S ATE T
UNIVERSITY T.H.E Patrick Roblin 1 The Ohio State University % '
Introduction $ In this course we will study basic passive devices used in microwave circuit systems. Passive devices are devices which do not amplify the input power, i.e. they do not provide any power gain. These include impedance matching networks, couplers, lters, attenuators, phase shifters, mixers, isolator, circulator, etc. Microwave circuits including active devices are covered in ECE 620, ECE 723 and ECE 832. & OHIO S ATE T
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inductors resistors capacitors Distributed Circuit Theory $ relies on basic lumped elements derived from Maxwell's equations such as: At low frequencies circuits are designed using circuit theory. Circuit theory Similarly at microwave frequencies we will design microwave circuits using distributed circuit theory. Distributed circuit theory relies on basic elements also derived from Maxwell's equations. such as:
transmission lines (telegraphist's equations) shorted stubs (generalized inductor) open stubs (generalized capacitor) coupled lines tapered lines & beside the lumpedelements of regular circuit theory. 3 OHIO S ATE T
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Attributes of Distributed Circuit Theory $ Like circuit theory, distributed circuit theory is a simpler model to use than Maxwell Equations. Circuit theory is a subset of distributed circuit theory. New design techniques and synthesis theorems exit in distributed circuit theory without equivalent in circuit theory. & OHIO S ATE T
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Guided Waves $ Guides waves system can be classi ed into 3 major groups: Transmission line systems featuring 2 or more conductors guiding the EM waves along the z axis Closed metalic waveguides consisting of hollow conductive pipes guiding the EM waves along the z axis Dielectric waveguides typically consisting of a material with a high dielectric constant (slab of rod) sandwitched by materials with low dielectric constants.
1111111 0000000 1111111 0000000 1111111 0000000 111 000
Magic Tee Waveguide Microstrip Stripline & 1 0 1 0 11 00 2 wires Coaxial cable Low Pass Filter OHIO S ATE T
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Electromagnetic Waves $
D= E H=B Let us start with Maxwell's equations: r D= r E = ; @B @t r H = J + @D r B=0 @t Using: and assuming that and are scalar, we can rewrite Maxwell Equations in free space ( = J = 0) as:
rD =0 =0 r r E = ; @B & rB H= @D @t @t OHIO S ATE T
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Derivation of the wave equation in free space
r r E $
=
; & @ 2E : r =; @t2 Wave equation in free space (Pozar p. 16): 2 2 E ; @ E = 0: r @t2 Similarly we can obtain: 2 2 H ; @ H = 0: r @t2
r E ; r2 E @ @t r B OHIO S ATE T
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Phasor Notation $
y z t) = Re E (x y h i j!t H(x y z t) = Re H (x y z )e
E (x
h z)ej!t i Wave equation in free space (Pozar p. 16): r2 E + ! 2 E = 0: r 2 H + ! 2 H = 0: & OHIO S ATE T
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Modes in Waveguide Systems $ Various modes of electromagnetic waves can propagate in transmission lines and waveguide systems: TEM (tranverse electromagnetic) waves: Ez = Hz = 0 TE (tranverse electric) or H waves: Hz 6= 0 and Ez = 0 TM (tranverse magnetic) or E waves: Ez 6= 0 and Hz = 0 where E and H are respectively the electric and magnetic elds. For example TEM waves are of the following form: E (x y z t) = E t+ (x y )e(!t; z) + E t; (x y )e(!t+ z) H(x y z t) = Ht+ (x y )e(!t; z) + Ht; (x y )e(!t+ z) with = !p . 9 & OHIO S ATE T
UNIVERSITY T.H.E Patrick Roblin The Ohio State University % ' TEM Waves $ TEM waves require at least two conductors.
Other propagation modes (TE and TM) are possible in systems supporting TEM mode but are usually not desired. Example of the coaxial line (Pozar, Sec. 2.2, p.55): The TEM mode will be the only mode propagating in the frequency range c 0 < f < fc (TE11) = (1) (a + b)p r r where fc (TE11) is the cuto frequency at which the TE11 mode (transverse electric mode) starts to propagate in the coaxial cable. & b a OHIO S ATE T
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2conductor waveguiding systems supporting the TEM mode $
I I I I Example of systems support TEM modes: 11111111 00000000 11111111 00000000
E H & 2 wires Coaxial cable Stripline Microstrip OHIO S ATE T
UNIVERSITY T.H.E Patrick Roblin 11 The Ohio State University % ' TEM Mode in a Coaxial System $ As an example consider the frequency ranges for some common coaxial cables connectors (p.130): BNC f < 1 or 4 GHz APC 7mm (sex less) f < 18 GHz b APC 3.5 mm f < 34 GHz a SMA f < 24 GHz SSMA f < 38 GHz & BNC APC 7mm 12 APC 7mm SMA OHIO S ATE T
UNIVERSITY T.H.E Patrick Roblin The Ohio State University % ' Modes in Hollow Waveguides $ rectangular waveguide (Pozar, Sec. 3.3, p. 107). The dominant mode is TE10 . This mode only propagates for frequencies verifying: < c 10 = 2a 1 f > fc 10 = 2ap No TEM mode can propagate in waveguides (only one conductor). Consider the λ E H E & H Distribution of the electric and magnetic elds for the TE10 mode. OHIO S ATE T
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Rectangular Waveguide Fundamental Mode $
p q2 = ! ; !c2
s The propagation constant for TE10 is given in terms of the frequency ! by = with !c = 2 fc , ! = 0 cd = ω
2 0; p 0= a . 2 vp ωc vg ω= β0 c & β Dispersion curves for a waveguide (full line) and a TEM mode (dashed line).OHIO
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ωc Phase and Group Velocity $
0 =β The phase velocity (cordal slope) and the group velocity (di erential slope) are respectively larger and smaller than the speed of light in the dielectric material considered: vp = ! > cd = p1 > vg = @! @ ω
vp vg
c ω & β No information is carried by the wave with the phase velocity vp Do you know an example of phenomena faster than light OHIO S ATE T
UNIVERSITY T.H.E Patrick Roblin 15 The Ohio State University % ' Group Velocity Theory $ A general solution f (t z ) of the waveequation is obtained by using a wavepacket F (!) superposition of the various propagating eigen waves Z1 f (t z ) = F (!) exp fi !t ; (!)z]g d! ;1 Let us assume that we have a signal f (t z ) of su ciently narrow bandwidth !0 ; ! !0 + !] such that we can linearize the propagation wave vector (!) in that frequency range: & vg @ where vg is called the group velocity. Using the change of variable !0 = ! ; !0 we can rewrite f (t z ) as (" #) Z! z f (t z) = exp fi !0t ; (!0)z]g F (! ; !0) exp i!0 t ; v d!0 ;! g ( " #) ! = exp i!0 t ; (!0 ) z g t ; z !0 vg where g() is identi ed as the bandpass limited signal modulating OHIO S ATE T the carrier of frequency !0 .
T.H.E UNIVERSITY (!) ' (!0 ) + ! ; !0 with vg = @! Patrick Roblin 16 The Ohio State University % '
Group Velocity: Interpretation $
0 0 So the modulated waveform is at any position z given by: ( " #) ! f (t z) = exp i! t ; (!0) z g t ; z the modulation signal g(t ; vzg ) is moving at the socalled group velocity !0 vg It is clearly seen that although the carrier of frequency !0 is moving with the cordal phase velocity : !0 v p = (! ) vg = @! @ & Note that only the modulation carries any information.
OHIO S ATE T
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Group Velocity and EM Modes $ We have seen that the group velocity vg gives the velocity of the information transmission The group velocity can be frequency dependent (e.g., waveguide). As a result, signals at di erent frequencies will be transmitted at di erent speeds leading to the distortion of the wavepackets transmitted. Such a communication medium is said to be dispersive. On the contrary TEM modes are not dispersive as they verify & vp = ! = cd = p1 = vg = @! @
OHIO S ATE T
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Lecture 3 & 4 $ Introduction to Transmission Line Characteristic Impedance Concepts Lossy Lines Slow Waves and Skin E ect Modes & OHIO S ATE T
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Distributed Circuit Theory $
Z C2 The development of a distributed circuit theory is based on the introduction of a local voltage V (z t) and local current I (z t) from the electric eld E (x y z t) and magnetic elds H (x y z t). In a transmission line system a voltage between the conductors C 1 and C 2 can be obtained from the following path integration (Pozar, Sec. 4.1, p.162): & V (z t) = ; E (x y z t) d` C1 Similarly the current current owing in conductor C 1 or C 2 is obtained from the closed path integration: I I (z t) = H (x y z t) d` C 1 or C2 For TEM modes these integrations are path independent and the voltage and current so de ned assume a unique value. OHIO S ATE T
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Wave Equation for Transmission Lines Using Maxwell Equations $
a V0+ e(!t; z) ; V0; e(!t+ z) I (z t) = Z Z0 0 p where = ! is the wave number and where Z0 is the characteristic impedance The solution of the waveequation derived from Maxwell's Equation leads to a voltage wave of the form: V (z t) = V0+e(!t; z) + V0;e(!t+ z) and a current wave of the form: b 1 =p : Z0 = 0 0 & vpC C C 0 is the capacitance per unit length between conductor C 1 and C 2. For example in a coaxial cable we have: 2 1 = 1 r ln b C 0 = b and Z0 = C0v 2 a ln a p
21 OHIO S ATE T
UNIVERSITY T.H.E Patrick Roblin The Ohio State University % ' Proof Wave equation in free space (Pozar p. 16):
r2 E ; $
Z C2 @ 2E = 0: and r2H ; @ 2H = 0: @t2 @t2 p TEM waves are of the following form ( = ! ): E (x y z t) = E t+(x y)e(!t; z) + E t;(x y)e(!t+ z) H (x y z t) = H t+(x y)e(!t; z) + H t;(x y)e(!t+ z) Voltage wave between conductors C 1 and C 2: Z C2 V (z t) = ; E (x y z t) d`
C1 Z C2 & =; E t+(x y) d` C1 = V0+ e(!t; z) + V0;e(!t+ z) Current wave in I conductor C 1 or C 2: H (x y z t) d` I (z t) = = H t+(x y) d` C 1 or C2 = I0+ e(!t; z) + I0; e(!t+
IC 1 or C2 e(!t; z) ; C1 E t;(x y) d` e(!t+ z) e(!t; z) +
z) I C 1 or C2 H t;(x y) d` e(!t+ OHIO S ATE T
UNIVERSITY T.H.E z) Patrick Roblin 22 The Ohio State University % '
Ht =
s Characteristic Impedance From Maxwell Equation $
s Using (see Equ. 3.18 p.108 in Pozar or Equ. 3.17c in Collin) z Et ^
s and H t d` = z Et ^ d` = s d` z E t = ^ n E d` with n a normalized vector normal to C 2. The characteristic Impedance Z0 is de ned from the voltageto currentwave amplitude ratios: s s R C2 V0 = RHC12 E t (x y) d` = V0 E t (x y)d` = C Z0 I H C1 S 0 C 2 n E t (x y ) d` C 2 H t (x y )d` where S is the surface charge on conductor C 2. De ning the capacitance per unit length as: C 0 = & The characteristic impedance Z0 is then V0+ = ; V0; = r = 10 Z0 = 0 V0 S = H C2 n Et RC ; C 12 E t d` d` I0+ I0; C vpC with vp = 1=p the phase velocity. OHIO S ATE T
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QuasiTEM mode in Microstrip Line $
ε0
1 This TEM derivation holds for 2 conductor systems with a uniform dielectric between the two conductors. In the case of a microstrip systems, two di erent dielectric materials with dielectric constant 0 (air) and 1 are used. In such a case an e ective dielectric constant can be de ned for the quasiTEM mode such that if it were the only dielectric material present in the system, the same capacitance per unit length would result
ε ε 11 00 11 00 11 00 ε 111111111 111111111 111111111 000000000 000000000 000000000 C’ C’ C’
0 eff 0 & (a) (b) (c) OHIO S ATE T
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E ective Dielectric Constant
ε ε 11 00 11 00 11 00 ε 111111111 111111111 111111111 000000000 000000000 000000000 C’ C’ C’
1 0 eff 0 $
ε0 (a) (b) (c) The e ective dielectric constant can be obtained from the capacitance per unit 0 length C 0 and C0 from the above systems. 0 From C 0= eff = C0= 0 we have: & C0 eff = 0 0 C0 For such a quasiTEM mode the propagation constant is then now given by = !p eff and the characteristic impedance by: 1 = p eff : Z0 = v C 0 C0 p OHIO
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Approximate Formula For Microstrip
strip conductor dielectric εr h t W
11111111 00000000 11111111 00000000 11111111 00000000 $
ground plane
eff = r+1 2 + r ;1q 2 1 h 1 + 12 W
! & for w=h 1 Zo = p60 ln 8h + W W 4h eff 120 1 Zo = p W + 1:393 + 0:667 ln W + 1:444 eff h h 26 for w=h 1
OHIO S ATE T
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Z0 TE10 = Zv
s Characteristic Impedance of Waveguides $ The de nition of a unique voltage is not possible in a waveguide. Depending of the choice selected the characteristic impedance for the TE10 mode in a rectangular waveguide is found to be
r q b=a r 1 ; (fc =f )2 with = 377 < Zv < 754 : Clearly the characteristic impedance is not uniquely de ned in waveguides. However once a de nition has been selected the voltage and current waves can be productively used to design waveguide circuits. λ E H & E H OHIO S ATE T
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r Parallel Plate Approximation $ TEM characteristic impedance of a parallel plate line (neglecting fringe e ect) is b Z0 = a This is a rst order approximation the TE10 characteristic impedance s Z0 TE10 = Zv r q b=a 2 with = 377 < Zv < 754 : r 1 ; (fc =f ) A parallel plate is also a rstorder approximate model for a microstrip line. However the fringe capacitance in a microstrip typically accounts for about half the capacitance per unit length.
1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000
a (a) & b a (b) OHIO S ATE T
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I(z) V(z) Transmission Line Equation $
V(z+dz)=V(z)+dV/dz dz Having obtained a solution of Maxwell's equation in terms of voltage and current, we introduce now a transmission line (telegraphist) model which can yield the same result.
I(z+dz)=I(z)+dI/dz dz L’ dz Z’ dz C’ dz Y’ dz For the section dz of the above distributed circuit we can write: V (z + dz) = V (z) + dV dz = V (z) ; I (z)Z 0dz & dz I (z + dz) = I (z) + dI dz = I (z) ; V (z + dz)Y 0dz dz This means that dV = ;I (z )Z 0 and dI = ;V(z)Y0 dz dz Combining these two equations we obtain d2V (z) = Z 0Y 0 V (z) and d2I(z) = Z0Y0 I(z) dz2 dz2
29 OHIO S ATE T
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Transmission Line Equation Solution $ The transmission line equations d2V (z) = Z 0Y 0V (z) and & Z0 q p 0 Y 0 and with Z0 = Z 00 . with = Z Y We used the identity I (z ) = ; 10 dV Z dz d2I(z) = Z0 Y0 I(z) dz2 dz2 are of ?? order and therefore admits ?? solutions: V (z) = V +e; z + V ;e z I (z) = 1 V +e; z ; V ;e z to relate I (z ) to V (z )
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I(z) L’ dz Z’ dz V(z) Loss Less Case
I(z+dz)=I(z)+dI/dz dz $
V(z+dz)=V(z)+dV/dz dz C’ dz Y’ dz For a lossless circuit (R0 = G0 = 0) we have Z 0 = j!L0 and Y 0 = j!C 0 , so p p p 0 Y 0 = ;! 2 L0 C 0 = j ! L0 C 0 =Z and
s & L0 =0 Z0 = C The wave solution obtained obtained from Maxwell's equations (ME) and the transmission line (TL) model can be made equivalent: to obtain C 0 and L0: 9 8 p q0 L = Z (TL) > = < C 0 (TL) = C0 (ME) Z0(ME) = C0(ME) = C0(TL) 0 ): p p 0 C0 = (TL) > L0(TL) = C0(ME) OHIO (ME) = ! =! L
T.H.E Z0 Y0 s S ATE T
UNIVERSITY Patrick Roblin 31 The Ohio State University % ' Inductance per Unit Length $ & C0 C0 c2 C00 c2C00 since usually r = 1. One notices that the e ective inductance per unit length is independent of the dielectric constant . This result can be extended to multiline transmission system L0 = c12 C0;1 0 where the inductance and capacitance per unit length L0 and C00 are now matrices. In practice microwave engineers rely on CAD tools as an expert system for evaluating the characteristic impedance Z0 and propagation constant in terms of the frequency and the physical parameters of the OHIO transmission lines. S ATE T
T.H.E UNIVERSITY Note that the inductance can be expressed in terms of the capacitance of the 0 transmission line C0 when the dielectric is replaced by air: L0 = = r 0 r 0 = r : ' 1 : Patrick Roblin 32 The Ohio State University % ' Loss in Transmission Lines $
c) d) What are the source of loss in transmission lines? In physical transmission lines the propagation of the voltage and current waves is attenuated by the nonin nite conductivity of the conductors (wave attenuation constant: lossy dielectric between the conductors (wave attenuation constant: radiation loss (large microstrips act as antennas) (wave attenuation constant: r) It results from the loss that the propagation constant becomes complex =j + and the voltage waves are accordingly attenuated: V (z) = V +e;j z e; z + V ;e z e z 33 & OHIO S ATE T
UNIVERSITY T.H.E Patrick Roblin The Ohio State University % ' Calculation of Loss with Maxwell's Equation
r $ Rewriting the 3rd Maxwell's Equation D H = J + @@t in the frequency domain we get using j = E and D = E r H = j + j! D = ( + j! ) E = j! eE where we introduced a complex dielectric constant: e = ; j ! The derivation of the wave equation in the same as for the loss less case except that now is replaced by e. As a result the propagation constant is now: q 1 r ' j + 1 tan = j e = j! e ' j + 2 2
where we have introduced the tangent loss de ned as tan = It results that the dielectric attenuation constant is for small loss: d = 1 tan 2 In microstrip lines, the \material lling the line" is usually an excellent dielectric like Duroid (te on) which features a dielectric constant of approximately 2 OHIO S ATE T and a loss tangent of 9 10;4 .
T.H.E UNIVERSITY The calculation of the dielectric loss is straight forward. ! & Patrick Roblin 34 The Ohio State University % '
Conductive Loss $
c=; The calculation of the conductive loss c and R0 parameter is a little more di cult as the wave is no longer purely TEM. Since the power transmitted varies as P (z ) = P (0) exp(;2 cz ) along the z axis of the waveguide, the attenuation constant can be simply evaluated from the rate of variation of the power dissipated using: 1 dP 2P (z ) dz It is then just su cient to evaluate the power loss per unit length P` = ;dP=dz: & An example of calculation for the rectangular waveguide is given in Chapter 3 of Pozar (Sec. 3.3, p.141). 35 OHIO S ATE T
UNIVERSITY T.H.E Patrick Roblin The Ohio State University % ' Wheeler Incremental Inductance Rule $ For TEM and quasiTEM lines a simple theory: the "Wheeler incremental inductance rule" can be developed to calculate c . The penetration of the magnetic eld inside the conductor up to the skin depth s creates an increase L0 of the line inductance per unit length L0 and therefore the characteristic impedance Z0 = L0vp . It results (see Pozar, Sec. 2.7, p.85) that the attenuation constant is given by: = ! L = Rs dZ0 c 2Z0 2Z0 d` q the with Rs = 1=( s ) the surface resistivity of the conductor and = intinsic impedance of the dielectric and ` the distance into the conductors. Applied to the coaxial line the attenuation constant for conductor loos is found to be (see Pozar, Sec. 2.7, p. 86). Rs 1 + 1 c= 4 Z0 b a OHIO
b
T.H.E & a S ATE T
UNIVERSITY Patrick Roblin 36 The Ohio State University % '
I(z) V(z) Lossy Transmission Lines Model $ Let us see how we can include these loss e ects by modifying our transmission line circuit model. This is done by including a series resistance R0 (conductor loss) and shunt conductance (dielectric loss) G0 per unit length:
I(z+dz)=I(z)+dI/dz dz R’ dz L’ dz Z’ dz C’ dz G’ dz Y’ dz V(z+dz)=V(z)+dV/dz dz & The transmission line equations remain the same except that we have now:q p Z Z 0 = R0 + j!L0 and Y 0 = G0 + j!C 0. Substituting in = Z 0Y 0 and Z0 = Y 00 we obtain: s q R + j!L0 0 + j!L0 )(G0 + j!C 0 ) = + j = (R and Z0 = 0 G + j!C0 and Z0 are complex, indicating respectively power loss during OHIO propagation and a phase shift between voltage and current waves. S ATE T
T.H.E UNIVERSITY Patrick Roblin 37 The Ohio State University % '
q LowLoss Lines
s s 0 C 0 G 0 L0 p j ! {z 0C}0 + R L0 + 2 C 0 L  2 {z }  {z } j
c d $
s For lowloss lines (the desired case), we can approximate the propagation constant as = (R0 + j!L0 )(G0 + j!C 0 ) R 0 = G0 L0 C 0 we obain
=
q Of particular interest is the case of the \distortionless line" discovered by Heaviside. By selecting (R0 + j!L0 )(G0 + j!C 0 ) = j! p L0C 0 + R0 & L0 : Z0 = =0 C This is a nondispersive line, even though it is lossy!
38 s R + j!L0 G0 + j!C 0 s C0 L0
OHIO S ATE T
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Strongly Lossy Lines
For larger loss in the dielectric, additional modes of propagation such as the slowwave mode or skine ect mode can take place. Consider the case of a Silicon substrate (RFIC)J. Depending on the conductivity of the silicon layer and the wave frequency the silicon layer can behaves like a shunt resistance, a capacitance or a ground plane.
Condutor Si Condutor Si Condutor Condutor SiO 2 $ & (a) (b) OHIO S ATE T
UNIVERSITY T.H.E Patrick Roblin 39 The Ohio State University % '
I(z) V(z) Solution for Strong Loss $
V(z+dz)=V(z)+dV/dz dz Consider a system the following model applies:
I(z+dz)=I(z)+dI/dz dz R’ dz L’ dz Z’ dz C’ dz G’ dz Y’ dz When the loss in the dielectric region is important, an exact solution must be obtained for the propagation constant and the attenuation factor . q Starting from = + i = (G + i!C )(R + i!L) one can verify that is given by: where we use the following de nitions: = f1 + f2 2
0 s p & RC c= 2L 2 f1 = 0 ; 4 s c G L and and d= 2C 2 2 and f2 = ( 0 ; 4 c )( d
40 s = ! LC p 2 2 0 ; 4 d) OHIO S ATE T
UNIVERSITY T.H.E Patrick Roblin The Ohio State University % '
ω ωc Slow Wave Dispersion $
Condutor Si Condutor ω
vg vp 0 =β c β The lower slope observed at low frequency is associated with a lower phase vp and group vg velocities. & This originates from the penetration of the magnetic eld in the lossy silicon layer whereas the electric eld remains mostly con ned in the p layer. As a SiO2 result both C 0 and L0 are large and the phase velocity vp = 1= L0C 0 becomes very large. OHIO
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UNIVERSITY Patrick Roblin 41 The Ohio State University % ' Various Modes of Propagation for 20 m line on a Lossy Substrate $ For high enough frequencies the propagation mode switches to either: the dielectric quasiTEM mode at high silicon resistivity the skin e ect mode at low silicon resistivity when the skin depth becomes comparable or smaller than the silicon layer thickness.
10 Frequency (GHz) Skin effect mode Dielectric quasi−TEM mode Slow wave mode 1 & Condutor
10
−3 10 −2 10 −1 10 0 10 1 10 2 Si Condutor
T.H.E OHIO T Di erent modes of propagation for a 20 m line on a lossy substrate. S ATE
UNIVERSITY Substrate Resistivity ( Ω . cm) Patrick Roblin 42 The Ohio State University % ' Power Transmitted in Lossy lines $ & The time averaged power transmitted down a line, is obtained from Maxwell's equations (see Pozar, Sec. 2.3, p.57) from the integration of the complex Poynting power density over the transmission line cross section STL : 1Z PC = 2 S E H dS TL = 1 V (z)I (z) 2 TL The complex power can be expressed in terms of its real part PL and imaginary part as: PC = PL + j 2!(Wm ; We) PL is the power dissipated by the load (z > 0). According to Poynthing's theorem (power balance equation: see Pozar, Sec. 1.6 p. 2627) the load power PL is given by: 1 Re fV (z )I (z )g = Z jE j2dv = jV + j2 ; jV ; j2 PL = 2 2v 2 Z0 2Z0 using the incident & re ected waves V = V + + V ; and I = (V + ; V ; )=Z0 . OHIO
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UNIVERSITY Patrick Roblin 43 The Ohio State University % ' Energy Stored
jV + j2 jV ; j2 $ Consider the complex power delivered to the load (z > 0): & V + V ; ; V +V ; PC = PL + j 2!(Wm ; We) = V (z)I (z) = 2Z ; 2Z + 2Z 2Z0 0 0 0 The imaginary part of the complex in the load (z > 0) is 2! times the total energy stored in the load. According to Poynthing's theorem (see Pozar, Sec. 1.6 p. 2627) it is simply given by: Im 1 VI = j2!(Wm ; We ) 2 where Wm and We are respectively the magnetic and electric energy stored in the load (right side: z > 0). 1 Z jEj2 dv 1 Z jH j2 dv and We = Wm = 4 4v v In terms of the incident and re ected waves using V = V + + V ; and I = (V + ; V ;)=Z0 the stored energy is given by: V + V ; ; V +V ; j 2!(Wm ; We) = 2Z 2 Z0 0
44
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UNIVERSITY T.H.E Patrick Roblin The Ohio State University % '
Discontinuities in Transmission Lines
The transmission line model we have derived is only rigorously valid for in nite lines. Corrections to the transmission line model for lines of nite length is needed at the discontinuities or junctions. At a junction or discontinuity evanescent modes are launched. Although those modes do not propagate they do store either electrical or magnetic energy. Corrections to the transmission line model can be implemented with the use of an equivalent circuit. $ & Inductors or capacitors can be used to account for magnetic and electrical energy stored at the discontinuities.
OHIO S ATE T
UNIVERSITY T.H.E Patrick Roblin 45 The Ohio State University % '
11 00 11 00 1 0 11 00 1 0 11 00
C Example of Discontinuities
Discontinuity Open line $
11 00 1 0 11 00 11 00 111 000 111 000 1 0 11 00 11 00
Cf
Fringing capacitance 1111111 0000000
Z1 Z2 111111111111 000000000000
Z0 (a) (b) For example the junction between two transmission lines of di erent impedances or the fringe capacitance at the end of an open stub store electrical energy. & In a rst cut design we will usually neglect the e ect of discontinuities. Once a rst cut design has been obtained we can account for these e ects with the help of a CAD simulator to obtain a more realistic simulation and OHIO optimize the design. S ATE T
T.H.E UNIVERSITY Patrick Roblin 46 The Ohio State University % '
Lecture 5 $ Impedance of Transmission Lines Quarter Wave Transformer Re ection Coe cient Power Dissipation & OHIO S ATE T
UNIVERSITY T.H.E Patrick Roblin 47 The Ohio State University % '
Recall: Solution of the Telegraphist WaveEquation $
0 d2V (x) = Z 0Y 0 V (x) dx2 admits for solution V (x) = V +e; x + V ;e x 1 I (x) = Z V + e ; x ; V ; e x & with the propagation constant given by: q p 0 Y 0 = (R0 + j!L0 )(G0 + j!C 0 ) = j + =Z and with Z0 the characteristic impedance given by: s s 0 Z = R0 + j!L0 Z0 = Y 0 G0 + j!C
OHIO S ATE T
UNIVERSITY T.H.E Patrick Roblin 48 The Ohio State University % '
Low less case: $
Z0
p = ;!2 L0C 0
s For a lossless circuit we have Z 0 = j!L0 and Y 0 = j!C 0 , so that the voltage and current waves are: V (x) = V +e;j x + V ; ej x I (x) = 1 V +e;j x ; V ; ej x with and = j ! L0 C 0 = j
p & L0 ( ) (pure resistive ... for loss less case !) Z0 = C 0 Typical value selected for Z0 (to reduce high voltage breakdown) is 50 .
OHIO S ATE T
UNIVERSITY T.H.E Patrick Roblin 49 The Ohio State University % ' Impedance of Loaded Transmission lines
V+ V_ $
z 11 00 11 00
Z0 γ= j β+ α ZL Z(d) 11 00
0 d l 0 l & The impedance along a transmission line at a position x is given by x Z (x) = V ((x)) I where the complex voltage V (x) and current I (x) are: V (x) = V + e ; x + V ; e x V + e; x ; V ; e x I (x) = Z Z0 0 The reference plane for V + and V ; is located at x = 0.
50 OHIO S ATE T
UNIVERSITY T.H.E Patrick Roblin The Ohio State University % ' Impedance Calculation $
(2) (3) & V + e; ` ; V ; e ` I L = I ( `) = Z Z0 0 Solving for the incident wave amplitudes V + and V ; we obtain 1 V + = 2 (ZL + Z0) ILe ` 1 V ; = 2 (ZL ; Z0) ILe; `: Substituting the incident wave V + and V ; amplitudes is Equ. 2 and 3 x = 0) +Z Z (;`) = V ((x = 0) = Z0 ZL + Z 0 tanh ` : I Z0 L tanh `
51 The impedance at the position x = ` is the load impedance ZL Z (`) = V ((``)) = VL = ZL: I IL Now from the voltage and current wave solutions we have V L = V (`) = Z L I L = V + e ; ` + V ; e ` (4)
T.H.E OHIO S ATE T
UNIVERSITY Patrick Roblin The Ohio State University % '
Loss Less Case $ For a loss free line we have = j and the impedance reduces to: + jZ Z (d) = Z0 ZL + jZ 0 tan d Z0 L tan d The impedance Z is then a periodic function of frequency and position: In terms of the electrical angle = d the impedance Z repeats every period In terms of position d it repeats every half wavelength =2 since we have d= 2 d & In terms of radial frequency ! = 2 f it repeats every period 2!c with !c v ! de ned as the frequency at which d = c =4 = 2 !pc since d = v!p d = 2 !c
OHIO S ATE T
UNIVERSITY T.H.E Patrick Roblin 52 The Ohio State University % '
Transmission Line with Various Terminations
V+ V_ $
111 000 111 000
Z0 Z(d) γ= j β ZL 111 000
d & + jZ Z (d) = Z0 ZL + jZ 0 tan d Z0 L tan d For a short circuited line, ZL = 0 and we have Z (d) = jZ0 tan d. For an open circuited line, ZL = 1 and we have Z (d) = ;jZ0 cot d. For a matched load, ZL = Z0 , and we have Z (d) = Z0 for all values or d. l 0 OHIO S ATE T
UNIVERSITY T.H.E Patrick Roblin 53 The Ohio State University % '
Inductive Impedance for a Shorted Transmission Line $
d The input impedance alternates between shorts (Z = 0) and opens (Z = 1) The short is transformed into an open for d = =4.
Im[ Z(d) ] open 0 Capacitive short λ/4 λ/2 3/4 λ λ & OHIO S ATE T
UNIVERSITY T.H.E Patrick Roblin 54 The Ohio State University % '
Matched Line
Z0 Z 0 Z 0 $ Equivalence between a matched transmission line and a match load in steady state. This is referred as impedance matching: ZL = Z0 . & OHIO S ATE T
UNIVERSITY T.H.E Patrick Roblin 55 The Ohio State University % '
Impedance Matching and Conjugate Impedance Matching
Z0 $
Z G ZL
Z0 Z 0 & Impedance matching is to be distinguished from conjugate impedance matching ZG = ZL used for maximum power transfer between a generator of impedance ZG and a load ZL . Both impedance and conjugate impedance matching can be achieved for resistive loads: ZG = ZL = ZL = Z0 ! For example: ZG = ZL = Z0 = 50
OHIO S ATE T
UNIVERSITY T.H.E Patrick Roblin 56 The Ohio State University % ' Quarter Wave Transformer for a Resistive Load RL
4 $
= 2 and
RL 2 Z0 The line input is then matched to the generator: Zin = RG = RL q R if we use Z0 = RG RL :
G For a line of one quarter of a wavelength (d = =4) we have d = 2 since tan( =2) = 1 the line impedance is: RL + jZ0 tan d = Z02 Zin = Z (d = =4) = Z0 Z + jR tan d R 0 L L RG Z 0 =( R G R L ) 1/2 & β d= π /2 d= λ/4 The impedance matching is only realized at the frequency where the transmission line length is a quarter wavelength. 57 OHIO S ATE T
UNIVERSITY T.H.E Patrick Roblin The Ohio State University % '
Quarter Wave Transformer for a Complex Load ZL : Option 1
RG
1/2 $
1.0 Zin= R G RL Z’0 =( R G R L ) Z0 Z L =50+j50 1+j1 111 000 111 000
β d= π /2 d= λ/4 βl 111 000 1.0 RL 2.6 R’L 111 000 & 2.0 2. 2βl 0 OHIO S ATE T
UNIVERSITY T.H.E Patrick Roblin 58 The Ohio State University % 1.0 ' Quarter Wave Transformer for a Complex Load ZL : Option 2
Zin = R G RG RL YL SHORTED STUB Z0 $ Z’0 =( R G R L ) 1/2 β d= π /2 d= λ/4 Zin = R G RG RL YL βl OPEN STUB Z’0 =( R G R L ) 1/2 Z0 & β d= π /2 d= λ/4 βl Cancel the admittance with an open or shorted stub 59 OHIO S ATE T
UNIVERSITY T.H.E Patrick Roblin The Ohio State University % '
Re ection Coe cient $
Γ(x) line V (x) V (x)
− + We de ne the re ection coe cient at a position x as the ratio of the re ected wave to the incident wave: V ;e x = V ; e2 x = ; e2 x ;(x) = + ; x L Ve V+ where ;L = ;(x = 0) = V ; V+ ΓL load & x=−d d 0 OHIO S ATE T
UNIVERSITY T.H.E Patrick Roblin 60 The Ohio State University % '
Re ection Coe cient for a Loss Less Line $
d For a loss less line we have: V ;ej x = V ; ej2 x = ; e;j2 ;(x = ;d) = + ;j x L Ve V+ where ;L = ;(x = 0) = V ; V+
Γ(x) line V (x) V (x)
− + for loss line
ΓL load & x=−d d 0 OHIO S ATE T
UNIVERSITY T.H.E Patrick Roblin 61 The Ohio State University % ' Re ection Coe cient Along a Line $ For a loss less line the re ection coe cient can be written: ;(d) = ;L e;2j d = j;L j ej ( ;2 d) if we de ne ;L = j;L j ej
Im[ Γ]
toward the load 111 000 Γ 111 000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 α 111111111 000000000 111111111 000000000 111111111 000000000 L = ΓL e jα toward the generator − 2 βd Re[ Γ] Γ (d) for d >0 & As we move along the line, ;(d) moves along a circle of radius j;L j
T.H.E Γ plane The re ection coe cient ;(d) rotates clock wise as d increases and we OHIO move toward the measurement point usually a generator.
S ATE T
UNIVERSITY Patrick Roblin 62 The Ohio State University % '
Measurement Point De nition
Γ(d ) Measurement point ΓL Load ZL Z0 d Toward the measurement point (Toward the generator) Toward the load $ & The notation toward the measurement point is more rigorous than toward the generator as the load is sometimes the impedance of a generator!
OHIO S ATE T
UNIVERSITY T.H.E Patrick Roblin 63 The Ohio State University % '
PL = RefV I
= Re
( Power Dissipated at the Load
ΓL ZL
Z0 γ= j β $
11 00 11 00
ZL Power dissipated by the load:
gjrms 11 00
gjamplitude ;! ) = 1 RefV I 2 V++V;
; jV ; j2 & Z0 = P + ; P ; = P + 1 ; j;L j2
64 = j V + j2 V+ ; V Z0 Z0 Z0 (incident minus re ected power) OHIO S ATE T
UNIVERSITY T.H.E Patrick Roblin The Ohio State University % '
Lecture 6 $ Smith Chart Voltage Standing Wave Ratio Design of a Single Stub Tuner ADS tutorial & OHIO S ATE T
UNIVERSITY T.H.E Patrick Roblin 65 The Ohio State University % '
Relation between Re ection Coe cient and Impedance
Γ(x) Z(x)
γ= j β $
ΓL ZL
Z0 111 000 111 000
ZL 111 000 111 000 & V (x) = V +e; x + V ;e x = Z 1 + ;(x) Z (x) = I (x) V + ; x V ; x 0 1 ; ;(x) e ; Z0 e Z0 Inverting: ;(x) = Z (x) ; Z0 and particularly at the load: ;L = ZL ; Z0 Z (x) + Z0 ZL + Z0 Note: for a matched load ZL = Z0 and we have ;L = 0 (no re ection)
66 x 0 OHIO S ATE T
UNIVERSITY T.H.E Patrick Roblin The Ohio State University % '
x 1 0.5 50 ohm 0 −0.5 −1 0 0.5 1 The Smith Chart $
j 1 0.5 0 0.5 0 −0.5 −1 −j 1 1 Open Bilateral transform connecting the re ection coe cient ; and the impedance Z : Z Z ; Z0 = Z0 ; 1 = z ; 1 with z = Z = r + jx ;= Z Z + Z0 Z0 + 1 z + 1 Z0 r −1 0 Short & z plane Γ plane The Smith Chart is a graph mapping the zplane inside the ; plane. 67 OHIO S ATE T
UNIVERSITY T.H.E Patrick Roblin The Ohio State University % '
Extended Smith Chart $
x
1 −0.5 0.5 j 0 Short 0 0.5 1 0 −1 Short 0.5 1 1 Open For negative resistance r < 0 we have ;j > 1
1 0.5 0 −0.5 −0.5 r r>0 r<0 −1 −0.5 −j −1 r<0 r>0 & z plane Γ plane Note: for r = ;1 (RefZg = ;50 ) we have ; = 1. 68 OHIO S ATE T
UNIVERSITY T.H.E Patrick Roblin The Ohio State University % '
1 Active & Passive Load
Im[ΓL ] 1 ΓL >1
Active devices $
Re[ΓL ] 0 ΓL <1
Passive devices 1 & OHIO S ATE T
UNIVERSITY T.H.E Patrick Roblin 69 The Ohio State University % '
Inductance and Capacitance on Smith Chart $
ω=0 Short ω= Open ω= Short ω=0 Open Γ plane Locus of the re ection coe cient for an inductor and a capacitor in a Z Smith chart Z Smith chart & OHIO S ATE T
UNIVERSITY T.H.E Patrick Roblin 70 The Ohio State University % '
Short YSmith Charts $
Short The YSmith Chart can be obtained by expressin ; in terms of Y : 1 ZL ; 1 Z0 z ; 1 = y ; 1 = ; y ; 1 = ; ZL ; 1 = ; YL ; 1 Z0 Y0 ;L = ZL = 1 +1 Z0 + 1 YL + 1 y+1 y ZL Z0 + 1 z + 1 Y0 Open Open Normal Γ plane Rotated − Γ plane & The Y Smith chart is obtained by inverting (;1 ) the Z Smith chart. The Y Smith chart is therefore the Z Smith chart rotated by half a turn. OHIO T In the rotated (;; ) YSmith Chart the Short and Open are exchanged. S ATE
T.H.E UNIVERSITY Patrick Roblin 71 The Ohio State University % '
2.3 Impedance and Admittance Smith Charts
Z Smith chart ZY Smith chart
Short Open Γ plane Short Open Γ plane $ Open Short −Γ plane Short Open Γ plane & Y Smith chart Y Smith chart OHIO S ATE T
UNIVERSITY T.H.E Patrick Roblin 72 The Ohio State University % '
Voltage Standing Wave Ratio $ The voltage wave inside a transmission line can be written using ; = V ; =V + : V (z) = V +e;j z + V ;ej z = V +e;j z 1 + ; ej2 z : The voltage varies between: jV jmax = jV + j (1 + j;j) jV jmin = jV + j (1 ; j;j) The voltage standing wave ratio (VSWR) is de ned as: 1+ V SWR = jjV jjmax = 1 ; jj;jj : V min ; A perfect matching (j;j = 0) corresponds to a VSWR of 1. & A VSWR of 1.2 would usually still be a very good matching. A VSWR larger than 2 would usually not be very good.
OHIO S ATE T
UNIVERSITY T.H.E Patrick Roblin 73 The Ohio State University % ' Calculation of the VSWR with the Smith chart $ The re ection coe cient ; = j;jej can be rotated toward the re ection coe cient j;j with zero angle to read the VSWR like a normalized impedance.
1.0 1+j1
2. 0 11 00 111 000 11111111111111111111111111111111111111111111111111111111111111111111111111111111 00000000000000000000000000000000000000000000000000000000000000000000000000000000 111 000
1.0 2.6 Read VSWR & 2.0 Example for an impedance of Z = 1 + j . 74 OHIO S ATE T
UNIVERSITY T.H.E Patrick Roblin The Ohio State University % 1.0 '
l2 β RG Z0
Short Design of a Stub Tuner
1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111 0000 1111111111 0000000000 1111 0000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000
Chip resistor $
Short Z0 β Z L =100+j100 Microstrip implementation l1 ystub = − j 1.6 RG
yload = 1 β Z0 z L= 2 + j 2 RG RG Z0 β ZL y line =1 + j 1.6 Plane A Z* L Plane A ystub = − j 1.6 RG Z0 β Short Z* L z* = 2 − j 2
L & RG ZL Z0 β ytotal =1 − j 1.6 Plane B yG = 1 Plane B OHIO S ATE T
UNIVERSITY T.H.E Patrick Roblin 75 The Ohio State University % ' Design Example with Plane A $
Short Consider the load: ZL = 100 + j 100. Using Z0 = 50 we have zL = 2:0 + j 2:0
ystub = − j 1.6
yload = 1 β Z0 z L= 2 + j 2 RG l2 β l1 Z0 ZL Plane A y line =1 + j 1.6 Load
T.H.E & Design Solution: Line length: `1 = 0:2193 Stub length: `1 = 0:089 Measurement Moving toward the the measurement point (generator) OHIO S ATE T
UNIVERSITY Patrick Roblin 76 The Ohio State University % '
tor era gen
the Plane A Design: as easy as 1, 2, 3
= l1
1.6 $
2. 0 1 0.2 93 λ 1.0 3
yline = 1 + j 1.6 111 000 Towa 1
0.25 11 00 11 00 rd z L=2 + j 2 Open Circuit 111111111111111111111111111111111111111111 000000000000000000000000000000000000000000 111111111111111111111111111111111111111111 000000000000000000000000000000000000000000 111111111111111111111111111111111111111111 000000000000000000000000000000000000000000 111111111111111111111111111111111111111111 000000000000000000000000000000000000000000 111111111111111111111111111111111111111111 000000000000000000000000000000000000000000 111111111111111111111111111111111111111111 000000000000000000000000000000000000000000 111111111111111111111111111111111111111111 000000000000000000000000000000000000000000 2 111111111111111111111111111111111111111111 000000000000000000000000000000000000000000 11 00 111111111111111111111111111111111111111111 000000000000000000000000000000000000000000 5 111111111111111111111111111111111111111111 000000000000000000000000000000000000000000 0.2 111111111111111111111111111111111111111111 000000000000000000000000000000000000000000 111111111111111111111111111111111111111111 000000000000000000000000000000000000000000 111111111111111111111111111111111111111111 000000000000000000000000000000000000000000
1.0 2.0 Short Circuit l 2 = 0.089 λ & 1.6 y L= 0.25 + j 0.25 11 00 ystub = −j 1.6 OHIO S ATE T
UNIVERSITY T.H.E Patrick Roblin 77 The Ohio State University % 1.0 ' Design Example with Plane B $
Z* L z* = 2 − j 2
L Consider the load: ZL = 100 + j 100. Using Z0 = 50 we have zL = 2:0 ; j 2:0
y stub = − j 1.6 RG Z0 l2 β β RG l1 Z0 ytotal =1 − j 1.6 yG = 1 Plane B & Design Solution: Line length: `1 = 0:2193 Stub length: `1 = 0:089 Load Measurement Moving toward the load OHIO S ATE T
UNIVERSITY T.H.E Patrick Roblin 78 The Ohio State University % '
y* = 0.25 + j 0.25
L Plane B Design: as easy as 1, 2, 3
1.0 $
Short Circuit 0.2 5 111 000
0.25 2
1.0 2.0 Open Circuit z* =2 − j 2
L load 1.6
193 0.2 3
ytotal =1 − j 1.6 11 00 1 11 00
l 2 = 0.089 λ the λ Tow = l1 111 000 111 000 111 000
2.0
1.6 & ard ystub =− j 1.6 OHIO S ATE T
UNIVERSITY T.H.E Patrick Roblin 79 The Ohio State University % '
ADS Tutorial $ & OHIO S ATE T
UNIVERSITY T.H.E Patrick Roblin 80 The Ohio State University % ...
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This note was uploaded on 04/30/2011 for the course ECE 710 taught by Professor Roblin during the Spring '11 term at Ohio State.
 Spring '11
 Roblin

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