NH-67, TRICHY MAIN ROAD, PULIYUR, C.F , KARUR DT. DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING COURSE MATERIAL

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1 NH-67, TRICHY MAIN ROAD, PULIYUR, C.F , KARUR DT. DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING COURSE MATERIAL Subject Name: Microwave Engineering Class / Sem: BE (ECE) / VII Subject Code: Staff Name: V. Navaneetha krishnan UNIT I MICROWAVE NETWORK CHARACTERIZATION AND PASSIVE COMPONENTS Syllabus Circuit and S parameter representation of N ports - Reciprocity Theorem- Lossless networks and unitary conditions - ABCD parameters-cascaded networks - Relations between S - Y and ABCD parameters - Effect of changing the reference planes in the S matrix - S Matrix of a Directional Coupler - waveguide tees and rat race coupler - Qualitative discussion on: Waveguide Corners- Bends Twists - Matched loads and movable shorts. Overview Microwave network characterization is important in analyzing various microwave networks. Several theorems in electromagnetics are useful like reciprocity theorem to obtain general properties of network matrices representing the microwave circuits. The study of microwave passive components like directional coupler, power divider, circulator becomes important in analyzing various microwave networks.

2 Objectives To study and analyze the microwave circuits using reciprocal theorem To study the characteristics of reciprocal and lossless networks To study the S parameter representation of N ports and unitary condition Effect of changing the reference planes in the S matrix ABCD parameters Relation between S, Y, Z and ABCD parameters To study the different types of microwave passive components like waveguide tees and rat race coupler, waveguide corners, bends, twists, matched loads and movable shorts Circuit and S parameter representation of N ports A representation in accord with direct measurements and with the ideas of incident and reflected, and transmitted waves, is given by the scattering matrix. S matrix provides a complete description of the network as seen at its N ports. It relates the voltage waves incident on the ports to those reflected from the ports It can be calculated using network analysis techniques or can be measured direct with a vector network analyzer Consider the N-port network shown in the figure, where Vn + is the amplitude of the voltage wave incident on port n and Vn - is the amplitude of the voltage wave reflected from port n

3 A specific element of the [S] matrix can be determined as Sii is the reflection coefficient seen looking into port i when all other ports are terminated in matched loads Sij is the transmission coefficient from port j to port i when all other ports are Terminated in matched loads Reciprocity Theorem Lorentz reciprocity theorem for EM fields To obtain the general properties of the network matrices representing the microwave circuits To evaluate the coupling of waveguides from current probes and loops and coupling of waveguides through apertures

4 Geometry for Lorentz reciprocity theorem Consider two separate set of sources J1, M1 and J2, M2 which generate the fields E1, H1 and E2, H2 in the volume V enclosed by closed surface S J and M are electric and magnetic current sources Maxwell s equations Consider the quantity Integrating over the volume V

5 S encloses no sources J1 = J2 = M1 = M2 = 0 and called source free fields S bounds a perfect conductor Let S be the inner surface of a closed perfectly conducting cavity. Then S is a sphere at infinity The fields evaluated on S are very far from the sources and considered locally as plane waves Impedance relation applies This result can also be obtained for the case of closed surface S where the surface impedance boundary condition applies Reciprocal Networks Consider the arbitrary network shown in the below figure, with short circuits placed at all terminal planes except those of ports 1 and 2

6 Let Ea, Ha and Eb, Hb be the fields anywhere in the network due to two independent sources a and b, located somewhere in the network According to the reciprocity theorem S is the closed surface along the boundaries of the network and through the terminal planes of the ports The field due to sources a and b can be evaluated at the terminal planes t1 and t2 Where s1, s2 are the cross sectional areas at the terminal planes of ports 1 and 2

7 The equivalent voltages and currents have been defined so that the power through a given port can be expressed as VI*/2 2 X 2 admittance matrix is used to eliminate the current In order to satisfy the above equation, Y12 = Y21 General result: Yij = Yji Lossless Networks Consider a lossless N-port junction The elements of the impedance and admittance matrices must be pure imaginary for a lossless network. If the network is lossless, then the net real power delivered to the network must be zero. Re{Pav} = 0

8 ABCD parameters Many microwave networks consist of a cascade connection of two or more two port networks It is convenient to define a 2 x 2 matrix or ABCD matrix, for each two-port network. ABC matrix of the cascade connection of two or more two-port networks can be easily found by multiplying the ABCD matrices of the individual two-ports ABCD matrix is defined for a two-port network in terms of the total voltages and currents Cascaded networks In the cascade connection of two-ports networks as shown in the figure below,

9 Relation between S - Y and ABCD parameters Effect of changing the reference planes in the S matrix S parameters relate amplitudes (magnitude and phase) of traveling waves incident on and reflected from a microwave network, phase reference planes must be specified for each port of the network S parameters are transformed when the reference planes are moved from their original locations

10 Consider a N-port microwave network, where the original terminal planes are assumed to be located at z n = 0 for the nth port and where z n is an arbitrary coordinated measured along the transmission line feeding the n th port The scattering matrix for the network with this set of terminal planes is denoted by [S] Consider a new set of reference planes defined at zn = ln, for the nth port and let the new scattering matrix be denoted as [S`] From the theory of traveling waves on lossless transmission lines, riting the above equation in matrix form,

11 The phase of Snn is shifted twice the electrical length of the shift in the terminal plane n, because the wave travels twice over this length upon incidence and reflection S matrix of a Directional Coupler A directional coupler is a four-port waveguide junction as shown in the figure It consists of a primary waveguide 1-2 and a secondary waveguide 3-4 When all ports are terminated in their characteristics impedances, there is free transmission of power, without reflection, between port a and port2, and there is no transmission of power between port 1 and port 3 The characteristics of a directional coupler can be expressed in terms of its coupling factor and its directivity

12 In a directional coupler all four ports are completely matched. The diagonal elements of the S matrix are zeros. There is no coupling between port 1 and port 3 and between port 2 and port 4. Thus the S matrix of a directional coupler becomes By the zero property of the S matrix, From the unity property of the S matrix

13 Let where p is positive and real. Let where q is positive and real. The S matrix of a directional coupler is reduced to

14 Waveguide tees A waveguide junction with three independent ports is referred to as a tee junction S parameter Characterized by matrix of third order containing nine elements Six elements should be independent Explained by three theorems of tee junction A short circuit may always be placed in one of the arms of the threeport junction in such a way that no power can be transferred through the other arms If the junction is symmetric about one of its arms, a short circuit can always be placed in that so that no reflections occur in power transmission between the other two arms It is impossible for a general three-port junction of arbitrary symmetry to be present matched impedances at all three arms E-plane tee (series tee) Waveguide tee in which the axis of its side arm is parallel to the E field of the main guide If the collinear arms are symmetric about the side arm, there are two different transmission characteristics

15 E-plane tee can be perfectly matched with Screw tuners, Inductive or capacitive windows at the junction The diagonal components of the S matrix is zero, because there will be no reflection When the waves are fed into the side arm, the waves appearing at port 1 and 2 of the collinear arm will be in opposite phase Symmetry property of the S matrix Zero property of the S matrix Unity property of the S matrix

16 H-plane tee (shunt tee) Axis of its side arm is shunting the E field or parallel to the H field of the main guide. If two input waves are fed into port 1 and port 2 of the collinear arm, the output wave at port 3 will be in phase and additive If the input wave is fed into port 3, the wave will split equally into port1 and port 2 Rat race coupler Annular line of proper electrical length to sustain standing waves, to which four arms are connected at proper intervals by means of series or parallel junctions. When a wave is fed into port 1, it will not appear at port 3 The difference of phase shift for the waves traveling in the clockwise and anti-clockwise directions is 180 degree Waves are cancelled at port 3

17 Waves fed into port 2 will not emerge at port 4 and so on. The S matrix is given by Phase cancellation occurs only at a designated frequency for an ideal hybrid ring Actual hybrid rings Small leakage couplings Zero elements in the matrix are not quite equal to zero Waveguide corners, Bents, Twists Normally used to change the direction of the guide through an arbitrary angle To minimize from the discontinuities, it is desirable to have the mean length L between continuities equal to an odd number of quarter wavelengths If the mean length L is an odd number of quarter wavelengths, the reflected waves from both ends of the waveguide section are completely cancelled Waveguide corners E Plane corner H Plane corner

18 Waveguide bends The minimum radius of curvature for a small reflection is given by R = 1.5 b for an E bend R = 1.5 a for a H bend Twists Summary The microwave circuits using reciprocal theorem were studied and analyzed The characteristics of reciprocal and lossless networks were studied N port microwave network was represented in terms S parameter and unitary condition was obtained Effect of changing the reference planes in the S matrix was analyzed ABCD parameters were derived for cascaded networks Relation between S, Y, Z and ABCD parameters were derived The different types of microwave passive components like waveguide tees and rat race coupler, waveguide corners, bends, twists were studied

19 Key Terms Reciprocal and lossless networks Unitary matrix S, Y, Z and ABCD parameters Waveguide tees E-plane Tee H-plane Tee Magic Tee Rat-race coupler Waveguide corners Waveguide bends Waveguide twists Key Term Quiz 1. The reciprocity theorem for electromagnetic fields is called as 2. The condition for the reciprocal network is 3. For a lossless network, the real power delivered to the network must be 4. The scattering matrix relates and 5. A specific element of the [S] matrix is determined as 6. S parameters relate both and of the traveling waves. 7. ABCD matrix of the cascade connection of two or more two-port networks is found be the ABCD matrices of the individual two-ports. 8. In E-plane tee, the axis of its side arm is to the E field of the main guide. 9. In H-plane tee, the axis of its side arm is to the E field of the main guide. 10. The waveguide components used to change the direction of the guide through an arbitrary angle are, and.

20 Multiple Choice 1. The condition for the reciprocity theorem is (a) Yij = Yjj (b) Yij = Yji (c) Yii = Yjj (d) Yji = Yjj 2. For a lossless network, the real power delivered to the network must be (a) zero (b) unity (c) imaginary (d) none of the above 3. The scattering matrix relates the amplitude of incident and reflected waves in (a) magnitude (b) phase (c) both amplitude and phase (d) none of the above 4. The diagonal elements of the directional coupler are zero because of (a) ports terminated with matched loads (b) symmetry (c) unitary (d) all 5. Commonly used waveguide tees are (a) E-plane tee (b) H-plane tee (c) hybrid ring (d) all the above 6. The waveguide components used to change the direction of the guide through an arbitrary angle are (a) corners (b) bends (c) twists (d) all the above 7. The characteristics of directional coupler can be expressed in terms of (a) coupling factor (b) directivity (c) both (d) none of the above 8. In a rat-race circuit, diagonal elements are not zero because of (a) symmetry (b) unitary condition (c) leakage couplings (d) none of the above 9. For the waveguide E bend, the minimum radius of curvature for a small reflection is given by (a) 1.5b (b) 1.0b (c) 2.0b (d) 0.5b 10. For the waveguide H bend, the minimum radius of curvature for a small reflection is given by (a) 1.5a (b) 1.0a (c) 2.0a (d) 0.5a

21 Review Questions 1. Explain the reciprocal theorem for electromagnetic fields. 2. Derive the S parameter representation for N-port microwave network and prove the condition for reciprocity and lossless networks. 3. Derive the relations between S, Y, Z and ABCD parameters. 4. Derive the unitary condition for microwave networks. 5. Describe the effect of changing the reference planes in the S matrix. 6. Derive the S matrix for a directional coupler. 7. Derive the S matrix for E-plane tee and H-plane tee. 8. Write short notes on a. waveguide corners b. waveguide bends c. waveguide twists