Microwave Network Analysis Lecture 1: The Scattering Parameters

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1 Microwave Network Analysis Lecture : The Scattering Parameters ELC 305a Fall 0 Department of Electronics and Communications Engineering Faculty of Engineering Cairo University

2 Outline Review on Network Parameters Impedance and Admittance Matrices ABCD (Transmission) Matrix Example: Transmission Line Section Introduction The Scattering Parameters Examples Relation to Other Network Parameters Network Properties Shift in Reference Planes Circuit Analysis

3 3 Outline Review on Network Parameters Impedance and Admittance Matrices ABCD (Transmission) Matrix Example: Transmission Line Section Introduction The Scattering Parameters Examples Relation to Other Network Parameters Network Properties Shift in Reference Planes Circuit Analysis

4 Impedance and Admittance Matrices Definitions 4 V V V n n n I I I Terminal (Reference) n n n Plane Linear Network V I I Y V What is the difference between these matrices and the conventional ones?

5 Impedance and Admittance Matrices How to Compute the Matrix Elements 5 ij V I i j I 0 k j k Terminal (Reference) Plane Linear Network Y Y ij Ii V j V 0 k j k The impedance and admittance matrices are the inverses of each other.

6 Impedance and Admittance Matrices Network Properties 6 Symmetric Y ii ii Y jj jj Linear Network Lossless Re 0 Re Y 0 ij ij Reciprocal Y ij ij Y ji ji How to prove the losslessness condition?

7 Transmission (ABCD) Matrix Two-Port Transmission Parameters 7 V A BV I C DI How to compute the ABCD parameters? What is the advantage of using the ABCD matrix? How is the ABCD matrix related to the (or Y) matrix? What are the conditions for reciprocity, symmetry, and losslessness?

8 Transmission (ABCD) Matrix Relation to Impedance Matrix 8 A C AD C C D C BC A B C D Similar relations can be derived for the admittance parameters.

9 Transmission (ABCD) Matrix Example 9, 0 β A B cos β j 0 sin β C D jy0 sin β cos β Determine the impedance, admittance, and transmission parameters for the lossless transmission line section shown.

10 0 Outline Review on Network Parameters Impedance and Admittance Matrices ABCD (Transmission) Matrix Example: Transmission Line Section Introduction The Scattering Parameters Examples Relation to Other Network Parameters Network Properties Shift in Reference Planes Circuit Analysis

11 Introduction What are the Scattering Parameters? A set of reflection and transmission coefficients defined at the reference planes of the network ports. They relate the incident and outgoing waves at the network ports under arbitrary loading/excitation conditions.

Introduction The Wave Amplitudes and Power Incident/Reflected Power on/from Port j Vj Pj I j 0 j 0 j Vj j j 0 j a I Pj a j 0 j Incident Wave Amplitude Vj bj I j P 0 j j b j

12 Introduction The Wave Amplitudes and Power Incident/Reflected Power on/from Port j Vj Pj I j 0 j 0 j Vj j j 0 j a I Pj a j 0 j Incident Wave Amplitude Vj bj I j P 0 j j b j 0 j Reflected Wave Amplitude V V V a b j j j 0 j j j I I I a b j j j j j 0 j a V I j j j 0 j 0 j b V I j j j 0 j 0 j How to compute the total power absorbed by the network?

i T ij S a i S a i S a S a b a i j ij j in n Transmission Coefficient (from Port j to Port i) R sn V sn

13 The Scattering Parameters The Incident and Reflected Wave Amplitudes Reflection Coefficient of Source i Wave Generated by Source i Rs 3 a Γ b a i si i si Vs Incident Wave on Port i Outgoing Wave from Port i b i T ij S a i S a i S a S a b a i j ij j in n Transmission Coefficient (from Port j to Port i) R sn V sn Γ What is the difference between S ij and T ij, S ii and Γ i? i b a i i Reflection Coefficient (at Port i)

14 The Scattering Parameters Evaluation of the Scattering Parameters 4 S ij b a i j a 0 k j k S ij V V i j 0 j 0i a 0k j k b S a S ij V I V I 0 j i i 0i 0i j j 0 j a 0k j k To evaluate the elements of any column of the scattering matrix, a source should be connected to the corresponding port and all the other ports should be terminated in matched loads (why?).

15 Examples Scattering Matrix for a TL Section 5, β S 0 e jβ e jβ 0

16 Examples Scattering Matrix for a T- and Π-Network 6 3 3

17 Examples Different Port Characteristic Impedances 7, β 0

18 Relation to Other Network Parameters Transformation from S to Matrix 8 v V i 0 0 I v V n n 0n i I n n 0n diag 0 0, 0,, 0N a v i b v i n n n n n n V I v i z i a b z a b 0 0 b z U z U a z U S U S S z U z U

19 Network Properties Reciprocity, Losslessness, and Matching 9 Reciprocal S S T Lossless S * T S Internal Matching S ii 0 How to prove the losslessness condition?

20 Shift in Reference Plane Transformation Matrix 0 S PSP P diag e jθ, e jθ,, e jθ N, θn βn n

21 Circuit Analysis Analysis of a Two-Port Network Using the Parameters V g g I I V Two-Port Network V L in =? V I I V I I V I L in V I L

22 Circuit Analysis Analysis of a Two-Port Network Using the S Parameters g V g a b Two-Port Network S a b L Γ in =? b S a S a b S a S a a Γ L b Γ in b SS Γ S a S Γ L L

23 3 Conclusion The scattering parameters and their use in the analysis of microwave circuits. The relation between the S matrix and the other network parameters. Determining the network properties (reciprocity, symmetry, losslessness, and matching) from the S matrix. The effect of shifting the reference plane on the S matrix. Circuit analysis using the scattering parameters.

Module 13: Network Analysis and Directional Couplers

Module 13: Network Analysis and Directional Couplers 13.2 Network theory two port networks, S-parameters, Z-parameters, Y-parameters The study of two port networks is important in the field of electrical