TASK A. TRANSMISSION LINE AND DISCONTINUITIES

Transcription

1 TASK A. TRANSMISSION LINE AND DISCONTINUITIES Task A. Transmission Line and Discontinuities... 1 A.I. TEM Transmission Line... A.I.1. Circuit Representation of a Uniform Transmission Line... A.I.. Time Harmonic Voltage and Current Waves in a Transmission Line... 4 A.I.3 Transmission Line Termination... 6 A.II. Physical Transmission Lines... 8 A.II.1. TEM Transmission Lines... 8 Properties of TEM Lines:... 8 Commonly Used TEM Lines: a) Coaial Line: b) Planar Transmission Lines: Stripline: Microstrip: A.II.. Non-TEM Transmission Lines... 0 Rectangular Waveguide:... 0 Circular Waveguide:... 4 A.II.3 Characteristic Impedance in Waveguide... 6 A.II.3.1 Rectangular Waveguide... 6 A.III Transmission Line Discontinuities A.III.1 Two-Port Discontinuities

2 A.I. TEM Transmission Line A.I.1. Circuit Representation of a Uniform Transmission Line Circuit Representation of a Uniform Transmission Line is shown in Figure A.I.1.1 [1,Rizzi], where R = Series Resistance per unit length of line (ohm/m) G = Shunt conductance per unit length of line (mho/m) L = Series inductance per unit length of line (ohm/m) C = Shunt capacitance per unit length of line (mho/m) v + R z L z v+ v + i i+ i G z C z +z - - Figure A.I.1.1 Circuit Representation of a Uniform Transmission Line. Applying the Kirchhoffs's voltage law and the Kirchhoffs's voltage law, result in the following system of equations for the voltage and current ( z 0) v z v = R i + L t ( A.I.1.1)

3 I = GV + z d V C d t ( A.I.1.) By solving the above system of equations a second order partial differential equation for voltage (Equation A.I.1.3) and a second order differential equation for voltage (Equation A.I.1.4) are obtained v = LC z v + t v t ( RC + GL) + RGv ( A.I.1.3) i z i = LC t + i t ( RC + GL) + RGi ( A.I.1.4) For a lossless line (G=0 and R=0) Equations A.I.1.1 through A.I.1.4 reduces to Equations A.I.1.5 through A.I.1.8. v i = L (A.I.1.5) z t Z f f c TM = η 1 (A.I.1.6) v v = LC z t (A.I.1.7) i z i = LC t (A.I.1.8) Equations A.I.1.7 and A.I.1.8 are the wave equations for homogeneous, lossless, and linear medium.the general solution for such wave equations is y(t,z)= f + (z - vt) + f - (z + vt) (A.I.1.9) where y() is the physical quantity of interest, e.g., voltage (v) or current (i), and f ± () are arbitrary functions representing the progression of the quantity in time and distance, and u p is the wave velocity. Here 3

4 1 u p = (A.I.1.10) LC A.I.. Time Harmonic Voltage and Current Waves in a Transmission Line When the change of a physical quantity with the time is as a sinusoidal wave with the angular frequency ω, it is possible to apply the phasor forms of that quantity. In this case, partial differential Equations A.I.1.1 through A become the ordinary differential Equations A.I..1 through A.I..4 respectively: d V = (R + jωl) I = ZI (A.I..1) d z I = (G + jωc) V = Y V (A.I...) z d V = (R + jωl)(g + jωc) V = ZYV z (A.I..3) d I = (R + jωl) (G + jωc) V = ZYV (A.I...4) d z with Z = R+ jωl and Y = G + jωc. The solution for the voltage phasor of the Equation A.I..3 is + + γ z + γ z + V= V0 e + V 0 e = V + V (A.I..5) γ= ZY ( R + jωl)( G + jωc) ) = (A.I..6) where γ= α+jβ is the propagation constant of the line, α is the attenuation constant (Np/m), and β is the phase constant (rad/m) of the line. V ± = V ± e µ γ z 0 are the voltage 4

5 waves in ± z (forward and reverse) directions. From A.I..1 and A.I.. it is easy to show that + γ z + γ z + I= I0 e _ I0 e = I I (A.I..7) where I + 0 = V + 0/Z 0 and I - 0 = V 0 /Z 0. The quantity Z 0 = V I + + = I V (A.I..8) is a line parameter called the line characteristic impedance and is related to the various lumped element components of Figure A.I.1.1 through the following relation: Z 0 = Z Y R + jωl = (A.I..9) G + JωC The reciprocal of Z 0, Y 0 =1/Z 0, is called the line characteristic admittance. 5

6 A.I.3 Transmission Line Termination Figure A.I.3.1 shows a uniform transmission line terminated to a load Zl located at z=0. Z l E G Z L z = -l z = 0 Figure A.I.3.1 A terminated uniform transmission line Recalling Equations A.I..5, A.I..7 and V λ = knowing that Z L =V (0)/I (0), it can f be easily shown that Γ L = I - /I + = V - /V + = (Z l -Z 0 )/(Z l +Z 0 ) (A.I.3.1) Parameter Γ L is called the load reflection coefficient. It is also possible to define the reflection coefficient Γ and the input impedance Z in in an arbitrary point z. Equations A.I..5, A.I..7 and A.I.3.1 yield Γ(z)= I - (z)/i + (z) =V - (z)/v + (z) = (Z in (z)-z 0 )/(Z in (z)+z 0 ) = Γ L ep(jz) (A.I.3.) and 6

7 1+ΓLe Z in = 1 Γ e L γ z γ z (A.I.3.3) or Z in = Z 0 Z Z L 0 Z 0 Z L tanh tanh ( γ z) ( γ z) (A.I.3.4) Eample A.I..1. The voltage and current distribution inside a uniform lossless transmission line terminated to a load 7

8 A.II. Physical Transmission Lines For analytical basis see Propagation of E&M Waves in Uniform Cylindrical Waveguides. A.II.1. TEM Transmission Lines Properties of TEM Lines: 1) The number of TEM modes is N-1 where N is number of conductors. Eample: Microstrip line N = 1 TEM mode Coaial Cable N = 1 TEM mode Parallel Plate Waveguide N = 1 TEM mode Coupled Mircostrip N = 3 TEM modes Even mode of eatation Odd mode of eatation ) Electric field and magnetic field are normal to the direction of propagation; i.e. E t H t k z E r ; H φ ; k z E ; H y ; k z no aial field for coaial. for parallel plate waveguide 3) Ratio electric field over magnetic field is always a constant everywhere in space. η is characteristic impedance of wave. 8

9 ρ E t ρ ρ η = ρ ; E t = η H t â z H t and ρ H t ρ ρ a z Ε = η t ρ ρ 4) E = 0= H where t t t = i.e. for the Cartesian coordinates z t = + and for cylindrical coordinates it is y t 1 = r r 1 r + r r φ These relationships implies that E ρ and H ϖ are calculated using the Laplace s Equation, hence it behaves quasi -static. 5) A unique value for voltage and current is found for TEM lines. [Note: A path dependent voltage and current is found for metallic waveguides.] ρ ρ V = E dl = Unique ρ ρ I = H dl = Unique 6) Knowing V and I, then the chrematistic impendence of a line is defined as: Z 0 = I V = L / length C / length L = Inductance per unit length of line length C = Capacitance per unit of line length 7) Equal values of phase and group velocities that are related to L and C. ω dω [Note: V p = ; Vg = k dk ω o ]. V p = V g = 1 L / length C / length 8) Equation partition of electric and magnetic energy: 9

10 W E z = 1 ρ ε E dσ ; W H z = 1 ρ µ H dσ W = W W Ε = WΗ = WΕ + Η P f = P z = 1 Re ρ E t η dσ = ρ Re η H 1 t dσ. 9) Power transmitted in z is the total energy per unit length multiplied by the group velocity P z = W V z g If TEM line is lossy these equations are modified; however in practice, these results are very good approimation for the low loss TEM Lines. Commonly Used TEM Lines: a) Coaial Line: For analytical basis see Propagation of E&M Waves in Uniform Cylindrical Waveguides. A homogeneously filled, low loss coaial cable (Figure A.II.1.1) is an eample of a physical TEM transmission line The dielectric with relative permittivity ε r fills the region between inner and outer conductors (diameters a and b). 10

11 b a z ε r Front Side Figure A.II.1.1. A homogeneously filled, low loss coaial cable. As the term TEM (i.e., Transverse Electric and Magnetic) represents, for a dominant mode, there are no electric and magnetic components in z direction inside the line. This mode is able to propagate from zero frequency. If the line is low loss, the following equations are always valid for the phase velocity, wavelength, and propagation constant with enough accuracy V = C ε r C = m s (A.II.1.1) λ = f λ = εr εr C 0 ( A.II.1.) ω β= C π ε r = (A.II.1.3) λ For a coaial cable the characteristic impedance is given by Z 0 = 60 ε r b ln a (A.II.1.4) 11

12 The electric and magnetic fields of a coaial cable in cylindrical coordinate system are given by: ρ V a r E = e b r ln a 0 jβz ρ εr V0 a H = b r 10πln a φ e jβz (A.II.1.5) (A.II.1.6) where, V 0 is a constant and a φ and a r are unit vectors in cylindrical coordinates. A coaial line can operate single mode up to a certain point that the first higher order non-tem mode starts propagating. This mode is known as TE 11 and an approimate frequency that it starts propagating is f c C (A.II.1.7) π ( a + b) εr For instance, a 50 Ω teflon filled line (ε r =.05) and a = 0.91 mm and b =.98, will support TE 11 mode from about 35.0 GHz. Eample A.II.1.1 Field distribution inside a uniform coaial line for the dominant (TEM) and the second order (TE11) modes The attenuation (NP/m) of a coaial line is obtained from the following equation R m ε r 1 1 π α = α c + α d = + + ε r tan( δ) (A.II.1.8) b a b λ 0 40πln a where α c and α d are the conductor loss and dielectric loss respectively, and 1

13 ωµ R m = is the surface resistance of the conductor. 6 Eample A.II.1. The loss of a coaial cable Eample A.II.1.3 ω β diagram of the TEM and the TE11 modes of a coaial cable 13

14 b) Planar Transmission Lines: Several different forms of waveguiding structures made from parallel metal strips on a dielectric substrate have found use in microwave and millimeter-wave circuits as well as in high speed digital circuits. In this section, we will eamine in some detail three types, called stripline, microstrip, and coplanar waveguide. Emphasis will be on the lowest-order mode, which is a TEM wave in a homogeneusly filled stripline and a quasi- TEM wave in microstrip and coplanar waveguide. Stripline: The stripline consists of a conducting strip lying between, and parallel to, two wide conducting planes, as shown in Figure A.II.1..a The region between the strip and the planes is filled with a uniform dielectric. Such a structure, with a uniform dielectric and more than one conductor, can support a TEM wave. If the strip width w is much greater than the spacing d and the two-plane lines connected in parallel. More precise results are found from the capacitance per unit length. For a TEM wave, the phase 1/ velocity is ν = ( ) ρ µε, and from the transmission-line formalism, it is given by ν ( LC) 1/ ρ =. Then the characteristic impedance is L LC µε Z o = = = (A.II.1.9) C C C Thus, for systems with uniform ε and µ the characteristic impedance can be found from the capacitance, which can be determined in a number of ways, as we saw in Chapters 1 and 7. An approimate epression for the charateristic impedance of the stripline, assuming a zero-thichness strip, has been found by conformal transformation to be 14

15 Z o ( k) K η 4 K k ( 1 ) (A.II.1.10) (a) (b) Figure A.II.1. (a) Stripline, (b) Microstrip where η = µ ε, k is given by 1 πw k = cosh 4d (A.II.1.11) and K(k) is the complete elliptic integral of the first kind (see E. 4.7a). A convenient approimate epression for w Z o, accurate over the range > 0. 56, is d ηπ Z o (A.II.1.1) π w 8ln ep 4d Approimate techniques used to obtain corrections to Z o for t > 0 are discussed 1/ by Hoffmann. The velocity of propagation does not depend on thickness and is ( µ ε) as for all TEM waves. Both velocity of propagation and characteristic impedance, neglecting loss, are independent of frequency and may be used up to the cutoff frequency of modes between the ground plates, Eq 8.3(14). 15

16 An approimate epression for attenuation resulting from conductor surface resistivity R s is πw 8d ln R + s d t π α c = nepers/ m (A.II.1.13) ηd πw ln + 4d which is valid if w > 4d and t < 5 d. Approimations for other dimensions are given in footnote 4. Attenuation from lossy dielectrics is eactly as in equation 8.5(). Microstrip: Probably the most widely used thin-strip line is one formed with the strip lying on top of insulator with a conductive backing. A different dielectric (usually air) is above the insulator and strip (Figure A.II.1..b). In such an arrangement, there cannot be a true TEM wave. The reason is that such a wave, as shown in connection with equation 8.3(1), k = requires that γ + 0. The propagation constant γ is a single quantity for the wave, so 1 γ + k and γ + k cannot both be zero if k1 k Eact solution of this problem is complicated by the finite width of the strip and the two different dielectrics. As mentioned above for the stripline, the lowest-order approimation for the micostrip is a section of parallel-plane guide neglecting fringing. A much better, though still approimate, approach is to consider that the lowest-order wave is approimately a TEM wave so the distribution of fields in the transverse plane is nearly the same as that for static fields. This so-called quasi static approach employs calculations made with static fields to determine the transmission-line parameters for propagation of the lowest-order 16

17 mode, even though it is not a pure TEM wave. The approimation is very useful in practical applications but has some limitations that we will mention below. A common approach to obtaining simple, accurate epression is to first find the characteristic impedance Z oo if an electrode structure identical to the one of interest but with the strip electrode having zero thickness and the dielectric being free space everywhere. This problem can be solved eactly by the method of conformal mapping, but the epressions are very comple and more useful results are the various approimations to the eact epressions. A particularly useful epression is w w Ζoo = (A.II.1.14) d d which is accurate to <0.3% for all w >0.06. Then to get the characteristic impedance d of the actual line, it is corrected by the so-called effective dielectric constant ε eff, which if filling the entire space, would give the same capacitance as that of the actual structure. Since the inductance is unaffected by the presence of a dielectric, correction for the capacitance gives the correction for the characteristic impedance. The Static approimation to the effective relative dielectric constant is also found by conformal mapping methods; one of several useful approimations is ( ε ) r 1 1 ε eff = (A.II.1.15) 10d 1+ w which is always between unity and ε r. Applying Equations (A.II.1.14) and (A.II.1.15) in Ζoo Ζ o = (A.II.1.16) ε eff 17

18 we obtain the static approimation for the actual characteristic impedance. The results for a variety of insulator dielectric constants are shown in figure 8.6c. Corrections for nonzero strip thickness have been published. 1/ Since the phase velocity in a TEM wave is = ( µε) c ν, the ratio = εeff in the quasi static case As frequency increases, the longitudinal field components become increasingly important; this can be represented by a frequency-dependent effective dielectric constant ε eff (f) to epress the variation of phase velocity or phase constant. Approimate epressions for ε eff (f) and for attenuation from conductor and dielectric losses have been obtained from numerical calculations or empirically. Some results follow: p v p Eample A.II.1.4 Attenuation In a Microstrip Transmission Line Also see A.S1 more on microstrip equations and applications A.S3 Microstrip Discontinuity Compensation A.S4 Microstrip Applications 18

19 19

20 A.II.. Non-TEM Transmission Lines Rectangular Waveguide: For analytical basis see Propagation of E&M Waves in Uniform Cylindrical Waveguides. Figure A.II..1 illustrates a rectangular waveguide. The modes are either TΕ (H z = 0) or TM (Ε z = 0). The fields are given in Table A.II..1 y b z a Figure A.II..1 A rectangular waveguide. 0

21 Table A.II..1 parameter and field inside a rectangular wave guide. Ε H H Ε Ε z = jk = y = y k = Z c k TM = Z H z = 0 Z TM = η TM mn f c η sin(k f y cos )sin )cos( k yy) ( k ) sin( k y) sin(k H TM y H f c 1 f y ( k y) y jk c Hz = η Ε = Ε = y H H y k k Ε = Z Ε =0 z TΕ mn fc cos(k )cos f y sin TΕ Ε = Z Z TΕ = ( k ) sin( k yy) ( k ) cos( k y) cos y TE η f 1 f c y ( k y) y µ η = γ = ( k c k ) k =ω µ ε ε c ( K K ) k = + f c y ma k = a Vk c k c = = = π π µε k y ( k + k ) π µε ηπ = b y k c is called the cutoff wave number f c is the cutoff frequency and γ is the propagation constant. The other important parameters are the phase velocity V p ω V = = (A.II..1) β f c 1 f and group velocity 1

22 V dω dβ f f c g = = V 1 (A.II..) V is the plane wave speed in a medium with dielectric constant ε. V = 1 (A.II..3) µε The wavelength inside the waveguide for a mode with cutoff frequency f c is λ λ g = (A.II..4) fc 1 f and fc β = k 1 (A.II..5) f where V λ = is the wavelength and k = ω/v is the phase constant of the plane wave. f Furthermore, π β = (A.II..6) λ g Eample A.II..1 Electric field distribution inside a rectangular waveguide for various modes (y plane). Eample A.II.. Magnetic field distribution inside a rectangular waveguide for various modes (y plane). Eample A.II..3 ω β diagram if the first few modes in X-band rectangular wave guide.

23 Eample A.II..4 phase and group velocity of TE10 mode in rectangular waveguide. Eample A.II..5 TE mode attenuation of an -band rectangular wave guide. For waveguide ecitation see: ecitation of waveguides (from Pozar) 3

24 Circular Waveguide: For analytical basis see Propagation of E&M Waves in Uniform Cylindrical Waveguides. Figure A.II.. illustrates a hollow circular waveguide. The modes are either TΕ (H z = 0) or TM (Ε z = 0). The fields are given in Table A.II... The relations A.II..1 through A.II..4 are also valid for the circular waveguide. y a r φ Figure A.II.. A hollow circular waveguide. Table A.II.. parameter and field inside a circular waveguide. TM TE Ε η f c = j Jn f u = k c r ( u) cos mφ = η H z H r = ( u) m J n sin mφ = Ε u r H φ = J ( u) ' n cos mφ = E φ 4

25 Ε Z r TM Ε Ε r φ Ε φ = H φ = ( = Ε ryte ) = H r = ( = Ε φyte ) Z Z Z TE TM TE TM: k c = p nm ; J m (p nm ) = 0 TE: k c = a p nm a ; J m ( p nm ) = 0 Z TM = η f c 1 f Z TΕ = η f 1 f c p nm n m p nm m n

26 Eample A.II..6 Electric field distribution inside a circular waveguide for various modes (y plane). Eample A.II..7 Magnetic field distribution inside a circular waveguide for various modes (y plane). Eample A.II..8 ω-β diagram of the first few modes in an -band circular waveguide. Eample A.II..9 phase and group velocity of TE11 mode in circular waveguide. Eample A.II..10 TE11 mode attenuation of an -band circular waveguide. A.II.3 Characteristic Impedance in Waveguide A.II.3.1 Rectangular Waveguide In order to apply transmission-line theory to waveguides, one must be able determine the phase constant (β) or guide wavelength (λ g ) as well as the characteristic impedance (Ζ O ) of the particular guide configuration. Equations (A.II..4) (A.II..6) provide the required relationships for β and λ g. Characteristic impedance is defined in Equation (A.I..8) as V I + +. Where V + is the voltage between the conductors and I + is the conduction current in the propagation direction for the traveling wave. For TEM lines Ζ o is uniquely defined since the value of V + ρ ρ = Ε dl is independent of the integration 6

27 path. In the waveguide case, V + is function of the integration path and therefore many definitions of Ζ o are possible. For rectangular waveguide in the TE 10 mode, from Table A.II..1 E E y λc π = E 0 sin sin( ωt βz) (A.II.3.1) λ a g λ c π = H 0 sin sin( ωt βz) (A.II.3.) λ a g π = H cos sin( ωt βz) (A.II.3.3) a E y 0 where λ c = a and E 0 = H 0 Z TE. The electric field is maimum at a = and its value is Ε v =Ε o λ λ c g. Integrating directly from y = 0 to y = b yields the voltage across the centerline of the waveguide. Namely, V cl λ c λ c = Εob =Η oζteb (A.II.3.4) λ λ g g However, this represents only one possible voltage value between the points y = 0, a = and y = b, a =. For instance, one could integrate Ε ρ dl by choosing a path along the conducting walls. In this case V CL = 0. Other paths leading to different values of voltage are indicated in Figure A.II.3.1. Clearly, voltage and hence Ζ o does not have a unique value in waveguide. One possible definition is the ratio of V g as defined in Equation (A.II.3.4) to the longitudinal current I z, where 7

28 a a I = K d = Η d (A.II.3.5) z o z o a V cl =0 λ V cl =Εob λ c g V cl Εo λ = b λ c g a/4 Figure A.II.3.1 Possible integration paths for determining V CL in rectangular waveguide. Integration along the bottom wall (y = 0) yields I z Therefore, g a λ c π a λ c = Η o sin d = Η o λ (A.II.3.6) a π λ o g b a µ R 59 VCL π b εr Ζ o = = ΖTE = ohms (A.II.3.7) I z a f c 1 f where Ζ TE is given by Table A.II..1. Since V CL and I z are in phase, Ζ o is real for f > f c. The above epression is known as the voltage-current definition of Ζ o. Other commonly used definions are based upon power flow along the waveguide. These are 8

29 1. Power-Current Definition:. Power-Voltage Definition: Ζ o Ζ = P = I P z V t 3. Modified Power-Voltage Definition: Ζ o = Where V CL is given by Equation (A.II.3.4) and I z by Equation (A.II.3.6). An epression for power flow in the direction can be developed can be developed with the aid of Poynting s theorem. With Ε and Η y in phase. V t P P S ρ ρ ρ a b ( Ε Η) ds= Ε Η dyd = o o y (A.II.3.8) and therefore or a λc π P = Η 0 ΖTE b sin d (A.II.3.9) λg a o 1 c p 0 TE ab λ = Η Ζ (A.II.3.10) λ g Applying this equation to the three definitions results in the following epressions for Ζ o in rectangular guide. Power-Current Definition: Power-Voltage Definition: Ζ Ζ o o b µ λ R g = 465 ohms a ε λ R b µ λ R g = 754 ohms (A.II.3.11) a ε λ R 9

30 Modified Power-Voltage Definition: Ζ o b µ λ R g = 377 ohms a ε λ R Eample A.II

31 A.III Transmission Line Discontinuities See Discontinuities and Modal Analysis (from Pozar) A.III.1 Two-Port Discontinuities When a transmission line changes its shape or terminates to a line with different dimension a discontinuity will be produced. In this case, part of the energy from the dominant mode will be coupled to the higher order modes. These modes are usually evanescent in the frequency of operation and can introduce capacitive and / or inductive effects. By either necessity or design, microwave networks often consist of transmission lines with various types of transmission line discontinuities. In some cases discontinuities are an unavoidable result of mechanical or electrical transitions from one medium to another (e.g., a junction between two waveguides, or a coa-to-microstrip transition), and the discontinuity effect is unwanted but may be significant enough to warrant characterization. In other cases discontinuities may be deliberately introduced into the circuit to perform a certain electrical function (e.g., reactive diaphragms in waveguide or stubs in microstrip line for matching or filter circuits). In any event, a transmission line discontinuity can be represented as an equivalent circuit at some point on the transmission line. Depending on the type of discontinuity, the equivalent circuit may be a simple shunt or series element across the line or, in the more general case, a T- or Π- equivalent circuit may be required. The component values of an equivalent circuit depend on the parameters of the line and the discontinuity, as well as the frequency of operation. In some cases the equivalent circuit involves a shift in the phase reference planes on the 31

32 transmission lines. Once the equivalent circuit of a given discontinuity is known, its effect can be incorporated into the analysis or design of the network using the theory developed previously in this chapter. The basic procedure for obtaining equivalent circuits for transmission line discontinuities; starts with a field theory solution to a canonical discontinuity problem and develop a circuit model, with component values. This is thus another eample of our objective of replacing complicated field analyses with circuit concepts. Figures A.III.1.1 and A.III.1. show some common transmission line discontinuities and their equivalent circuits. As shown in Figure A.III.1.1 a-c, thin metallic diaphragms (or "irises") can be placed in the cross-section of a waveguide to yield equivalent shunt inductance, capacitance, or a resonant combination. Similar effects occur with step changes in the height or width of the waveguide, as shown in Figures A.III.1.1 d,e. Similar discontinuities can also be made in circular waveguide. The best reference for waveguide discontinuities and their equivalent circuits is The Waveguide Handbook [7]. Some typical microstrip discontinuities and transitions are shown in Figure A.III.1.; similar geometries eist for stripline and other printed transmission lines such as slotline, covered microstrip, coplanar waveguide, etc. Since printed transmission lines are newer, relative to waveguide, and much more difficult to analyze, more research work-is needed to accurately characterize printed transmission line discontinuities; some approimate results are given in reference [8]. For a two-port discontinuity, these effects can be well represented by a T or Π network (cf Section C.I.3). The study of these effects can be done either by approimate 3

33 analytical techniques (e.g. Variational technique, see Harington) or numerical techniques (Finite Element, Finite difference, mode matching, see Gupta or Itoh), and in general are very complicated. Following are some eamples illustrating some two-port discontinuities. Figure A.III.1.1 Rectangular waveguide discontinuities 33

34 Figure A.III.1. Some common microstrip discontinuities. (a) Open ended microstrip.(b) Gap in microstrip. (c) Change in width (d) T-junction (e) Coa-tomicrostrip junction 34

35 A.S Modal Analysis of an H-Plane Step in Rectangular Waveguide Eample A.III.1.1 Capacitive iris inside a rectangular wave guide. Eample A.III.1. Inductive iris inside a rectangular wave guide. Also see A.S3 Microstrip Discontinuity Compensation Eample D.III..1 Aperture in transverse wall of a rectangular waveguide. Eample A.III.1.3 Microstrip step discontinuity Eample A.III.1.4 Microstrip capacitive gap Eample A.III.1.5 Microstrip Mitered Bend Eample A.III.1.6 Microstrip Curved Bend Eample A.III.1.7 Microstrip line loaded with an open stub or a radial stub. 35

36 A.III. Three-Port Discontinuities Eample A.III..1 Coaial T Eample A.III. Microstrip T A.III. Four-Port Discontinuities Eample A.III.3.1 Microstrip Cross Junction Eample A.III.3. Waveguide Magic T Eample D.III.. Aperture in broad wall of a rectangular waveguide. 36