Microwave Network Analysis
|
|
|
- Lindsay McBride
- 6 years ago
- Views:
Transcription
1 Microwave Network Analysis S R Zinka zinka@hyderabadbits-pilaniacin Department of Electrical & Electronics Engineering BITS Pilani, Hyderbad Campus May 7, 2015 RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
2 Outline 1 N -Port Parameters 2 S Paramters 3 Even & Odd Mode Analysis 4 Interconnected Networks 5 V, I, and Z *** 6 Discontinuities and Bends *** RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
3 Outline 1 N -Port Parameters 2 S Paramters 3 Even & Odd Mode Analysis 4 Interconnected Networks 5 V, I, and Z *** 6 Discontinuities and Bends *** RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
4 Z, Y, h, g Paramters z 11 z 22 h 11 z 12 I 2 z 21 I 1 h 12 V 2 h 21I 1 h 22 g 22 y 11 y 12V 2 y 21V 1 y 22 g 11 g 12I 2 g 21V 1 * h 12, h 21, g 12, g 21 are dimensionless quantities * h 22, y 11, y 22, g 11 are admittance values So, actual resistance values are 1/h 22, etc RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
5 Series-Series Connection [Z] 1 Combined Port 1 Combined Port 2 [Z] 2 [Z] = [Z] 1 [Z] 2 (1) RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
6 Shunt-Shunt Connection [Y] 1 Combined Port 1 Combined Port 2 [Y] 2 [Y] = [Y] 1 [Y] 2 (2) RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
7 Series-Shunt Connection [h] 1 Combined Port 1 Combined Port 2 [h] 2 [h] = [h] 1 [h] 2 (3) RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
8 Shunt-Series Connection [g] 1 Combined Port 1 Combined Port 2 [g] 2 [g] = [g] 1 [g] 2 (4) RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
9 Z Paramters of N-port Networks V 2, I 2 V 3, I 3 V 2, I 2 t 2 t 3 V 3, I 3 V 1, I 1 t 1 S V 4, I 4 V 1, I 1 t 4 V 4, I 4 t N V N, I N V N, I N RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
10 Z Paramters of N-port Networks V 2, I 2 V 3, I 3 V 2, I 2 t 2 t 3 V 3, I 3 V n = V n V n V 1, I 1 V 1, I 1 t 1 t N V N, I N S V N, I N t 4 V 4, I 4 V 4, I 4 V 1 V 2 V N = I n = I n I n Z 11 Z 12 Z 1N Z 21 Z N1 Z NN I 1 I 2 I N RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
11 Y Paramters of N-port Networks V 2, I 2 V 3, I 3 V 2, I 2 t 2 t 3 V 3, I 3 V 1, I 1 t 1 S V 4, I 4 V 1, I 1 t 4 V 4, I 4 t N V N, I N V N, I N RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
12 Y Paramters of N-port Networks V 2, I 2 V 3, I 3 V 2, I 2 t 2 t 3 V 3, I 3 V n = V n V n V 1, I 1 V 1, I 1 t 1 t N V N, I N S V N, I N t 4 V 4, I 4 V 4, I 4 I 1 I 2 = I N I n = I n I n Y 11 Y 12 Y 1N Y 21 Y N1 Y NN V 1 V 2 V N RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
13 Reciprocal Networks z 11 z 22 z 12 I 2 z 21 I 1 A network is said to be reciprocal if the voltage appearing at port 2 due to a current applied at port 1 is the same as the voltage appearing at port 1 when the same current is applied to port 2 RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
14 Reciprocal Networks z 11 z 22 z 12 I 2 z 21 I 1 A network is said to be reciprocal if the voltage appearing at port 2 due to a current applied at port 1 is the same as the voltage appearing at port 1 when the same current is applied to port 2 Z 12 = Z 21 RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
15 Reciprocal Networks z 11 z 22 z 12 I 2 z 21 I 1 A network is said to be reciprocal if the voltage appearing at port 2 due to a current applied at port 1 is the same as the voltage appearing at port 1 when the same current is applied to port 2 Z 12 = Z 21 More generally, Z ij = Z ji RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
16 Reciprocal Networks Cont d y 11 y 12V 2 y 21V 1 y 22 Exchanging voltage and current results in an equivalent definition of reciprocity Y 12 = Y 21 More generally, Y ij = Y ji RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
17 Reciprocal and Loss-less Networks If the network is lossless, then the net real power delivered to the network must be zero Thus, Re ( ) 1 ( ) P avg = 2 Re [I] H [V] = 1 ) ([I] 2 Re H [Z] [I] ( N = 1 2 Re j=1 i=1 ) N Ii Z iji j = 0 (5) RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
18 Reciprocal and Loss-less Networks Cont d Let s re-write (5) as shown below: 1 N ( N 2 Re j=1 i=1 1 2 Re ( I 1 2 Z 11 I 1 Z 12 I 2 I 2 Z 21 I 1 I i Z iji j ) = 0 ) = 0 1 ) ( I 2 Re 1 2 Z 11 Z 12 (I1 I 2 I2 I 1) = 0, Z 12 = Z Re I 1 2 Z 11 Z 12 I1 I 2 I2 I 1 = 0 }{{} real I 1 2 Re (Z 11) I1 I 2 I2 I 1 Re (Z 12) = 0 }{{} real So, for reciprocal and loss-less networks, Re ( Z ij ) = 0, for any i, j RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
19 Outline 1 N -Port Parameters 2 S Paramters 3 Even & Odd Mode Analysis 4 Interconnected Networks 5 V, I, and Z *** 6 Discontinuities and Bends *** RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
20 S Parameters V 2, I 2 V 3, I 3 V 2, I 2 t 2 t 3 V 3, I 3 V 1, I 1 t 1 S V 4, I 4 V 1, I 1 t 4 V 4, I 4 t N V N, I N V N, I N RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
21 S Parameters V 2, I 2 V 3, I 3 V 1, I 1 V 1, I 1 V 2, I 2 t 1 t 2 S t 3 V 3, I 3 t 4 V 4, I 4 V 4, I 4 V 1 V 2 V N = S 11 S 12 S 1N S 21 S N1 S NN V 1 V 2 V N t N V N, I N V N, I N RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
22 S to Z Paramters V 1 V 2 V N ([U] [S]) V 1 V 2 V N V 1 V 2 V N [V] = [Z] [I] Z 11 Z 12 Z 1N V 1 = Z 21 1 V 2 Z 0 V Z N1 Z N NN Z 11 Z 12 Z 1N = 1 Z 21 Z 0 ([U] [S]) Z N1 Z NN [U] [S] = 1 Z 0 [Z] ([U] [S]) [Z] ([U] [S]) = Z 0 ([U] [S]) V 1 V 2 V N V 1 V 2 V N [Z] = Z 0 ([U] [S]) ([U] [S]) 1 (6) RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
23 Z to S Paramters From the previous slide, [U] [S] = 1 [Z] ([U] [S]) Z 0 Z 0 [S] [Z] [S] = [Z] Z 0 [U] (Z 0 [U] [Z]) [S] = [Z] Z 0 [U] [S] = ([Z] Z 0 [U]) 1 ([Z] Z 0 [U]) (7) RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
24 S to Y Parameters Since [Z] = Z 0 ([U] [S]) ([U] [S]) 1, [Y] = [Z] 1 = [Z 0 ([U] [S]) ([U] [S]) 1] 1 = Y 0 ([U] [S]) ([U] [S]) 1 (8) Similarly, one can convert one type of N port network parameters into another For further details, please refer to the table provided in Pozar s book RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
25 S Parameters of Reciprocal Networks [ V ] = 1 2 ([V] Z 0 [I]) = 1 2 ([Z] Z 0 [U]) [I] [ V ] = 1 2 ([V] [I]) = 1 2 ([Z] Z 0 [U]) [I] Combining the above set of equations gives [ V ] = 1 2 ([Z] Z 0 [U]) [I] [ V ] = ([Z] Z 0 [U]) ([Z] Z 0 [U]) 1 [ V ] [S] = ([Z] Z 0 [U]) ([Z] Z 0 [U]) 1 ( [S] t = ([Z] Z 0 [U]) t) 1 ([Z] Z0 [U]) t [S] t = (([Z] Z 0 [U])) 1 ([Z] Z 0 [U]) [Z] = [Z] t for reciprocal networks (9) Comparing (7) and the above equation gives [S] t = [S] (10) RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
26 S Parameters of Loss-less Reciprocal Networks [S] [S] H = [Z] Z 0 [U] [Z] Z 0 [U] [ ] [Z] Z0 [U] H [Z] Z 0 [U] = [Z] Z 0 [U] [Z] H Z 0 [U] [Z] Z 0 [U] [Z] H Z 0 [U] = [Z] Z 0 [U] [Z] Z 0 [U] [Z] Z 0 [U] [Z] Z 0 [U], [Z]H = [Z] for lossless reciprocal networks = [Z] Z 0 [U] [Z] Z 0 [U] [Z] Z 0 [U] [Z] Z 0 [U] = [U] So, [S] of any loss-less reciprocal network is always a unitary matrix, ie, [S] [S] H = [U] RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
27 Shift in Reference Planes θ 1 V' 1 V' 1 V 1 V 1 Port 1 z 1 = l 1 z 1 = 0 N-port network [S], [S'] θ n V' n V' n V n V n Port n z n = l n z n = 0 V n V n = V n e jθn = V n e jθn RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
28 Shift in Reference Planes Cont d V 1 V 2 V N = S 11 S 12 S 1N S 21 S N1 S NN V 1 V 2 V N V 1 ejθ 1 V 2 ejθ 2 V N ejθ N = [S] V 1 e jθ 1 V 2 e jθ 2 V N e jθ N e jθ e jθ 2 0 e jθ N V 1 V 2 V N = [S] e jθ e jθ 2 0 e jθ N V 1 V 2 V N RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
29 Shift in Reference Planes Cont d V 1 V 2 V N = e jθ e jθ 2 0 e jθ N 1 [S] e jθ e jθ 2 0 e jθ N V 1 V 2 V N V 1 V 2 V N = e jθ e jθ 2 0 e jθ N [S] e jθ e jθ 2 0 e jθ N }{{} [S ] V 1 V 2 V N RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
30 Outline 1 N -Port Parameters 2 S Paramters 3 Even & Odd Mode Analysis 4 Interconnected Networks 5 V, I, and Z *** 6 Discontinuities and Bends *** RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
31 Symmetric T and Pi Networks R 2 R 1 R 1 R 1 R 1 R 2 R 2 /2 R 2 /2 R 1 R 3 R 1 R 3 2R 2 2R 2 RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
32 Symmetric T and Pi Networks R 2 R 1 R 1 R 1 R 1 R 2 R 2 /2 R 2 /2 R 1 R 1 R 1 R 1 2R 2 2R 2 RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
33 Even Mode Analysis v - R 1 R 2 /2 R 2 /2 R 1 R 1 R 1 v v 2R 2 2R v - Open Open RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
34 Odd Mode Analysis R 2 /2 R 2 /2 R 1 R 1 v R 1 R 1 -v v 2R 2 2R 2 -v Short Short RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
35 Even & Odd Mode Analysis - A Simple Example R 2 v R 1 R R 2 /2 R 2 /2 R 2 /2 R 2 /2 v/2 R 1 R 1 R 1 R 1 v/2 v/2 -v/ Open Short RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
36 Even & Odd Mode Analysis - A Simple Example R 2 /2 R 2 /2 R 2 /2 R 2 /2 v/2 R 1 R 1 R 1 R 1 v/2 v/2 -v/ Open Short S 11 = 1 2 (Se 11 So 11 ) S 21 = 1 2 (Se 11 So 11 ) RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
37 Even & Odd Mode Analysis - A Simple Example R 2 /2 R 2 /2 R 2 /2 R 2 /2 v/2 R 1 R 1 R 1 R 1 v/2 v/2 -v/ Open Short S 11 = 1 2 (Se 11 So 11 ) S 21 = 1 2 (Se 11 So 11 ) S 22 = S 11 S 12 = S 21 RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
38 Outline 1 N -Port Parameters 2 S Paramters 3 Even & Odd Mode Analysis 4 Interconnected Networks 5 V, I, and Z *** 6 Discontinuities and Bends *** RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
39 Method I RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
40 Cascaded Networks RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
41 ABCD Parameters I 1 I 2 Port 1 V A B 1 V C D 2 Port 2 [ V1 I 1 ] = [ A B C D ] [ V2 I 2 ] RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
42 Cascaded Networks I 1 I 2 I 3 A 1 B 1 A 2 B 2 V 1 V C 2 1 D 1 C 2 D 2 V 3 [ ] [ ] [ ] [ ] V1 A1 B = 1 A2 B 2 V3 I 1 C 1 D 1 C 2 D 2 I 3 RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
43 ABCD Parameters of Some Useful Two-Port Circuits Z A = 1 C = 0 B = Z D = 1 Y A = 1 C = Y B = 0 D = 1 Z 0, l N : 1 A = cos β C = jy 0 sin β A = N C = 0 B = j Z 0 sin β D = cos β B = 0 D = 1 N Y Y 3 A = 1 2 Y 3 Y 1 Y 2 C = Y 1 Y 2 Y 1Y 2 Y 3 B = 1 Y 3 D = 1 Y 1 Y 3 Z 1 Z 2 Z 3 A = 1 Z 1 Z 3 C = 1 Z 3 B = Z 1 Z 2 Z 1 Z 2 Z 3 D = 1 Z 2 Z 3 RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
44 Method II RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
45 More Complicated Networks RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
46 Signal Flow Graph Method Port 1 b 1 a 1 a 2 [S] b 2 Port 2 (a) a 1 S 21 b 2 S 11 S 22 b 1 S 12 a 2 (b) RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
47 Signal Flow Graph Method Cont d a b a Γ l Γ l (a) b Z s V i b V s Γ s a b Γ s (b) a RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
48 Signal Flow Graph Method Rules V 1 S 21 S 32 V 2 V 3 V 1 S 21S 32 V 3 S a S a S b V 1 V 2 V 1 V 2 S b S 22 S 21 S 32 V 1 V 2 V 3 S 21 1 S 22 S 32 V 1 V 2 V 3 S 42 V 4 S 21 V' 2 S 42 V 4 S 21 S 32 S 21 S 32 V 1 V 2 V 3 V 1 V 2 V3 RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
49 Mason s Gain Formula Mason s gain formula is as follows: G = N k=1 G k k = 1 L i L i L j L i L j L k ( 1) m where: N is the total number of forward paths between input and output G k is the path gain of the kth forward path between input and output L i is the loop gain of each closed loop in the system L i L j is the product of the loop gains of any two non-touching loops L i L j L k is the product of the loop gains of any three pairwise non-touching loops k is the cofactor value of for the kth forward path, with the loops touching the kth forward path removed RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
50 Example Usage of Mason s Gain Formula RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
51 Method III RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
52 Interconnected Networks - A Matrix Approach RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
53 Interconnected Networks - A Matrix Approach [ ] [ bo [Soo] [S = oi ] b i [S io ] [S ii ] ] [ ao a i ] (11) RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
54 Interconnected Networks - A Matrix Approach [ ] [ bo [Soo] [S = oi ] b i [S io ] [S ii ] ] [ ao a i ] (11) b i = [C] a i (12) RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
55 Interconnected Networks - A Matrix Approach Cont d We should find [S] which is defined by the equation Substituting (12) and (13) in (11) gives, [ ] [ [S] ao [Soo] [S = oi ] [C] a i [S io ] [S ii ] b o = [S] a o (13) ] [ ao a i ] Solving the above matrix equation gives [S] = [S oo] [S oi ] ([C] [S ii ]) 1 [S io ] RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
56 Outline 1 N -Port Parameters 2 S Paramters 3 Even & Odd Mode Analysis 4 Interconnected Networks 5 V, I, and Z *** 6 Discontinuities and Bends *** RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
57 Equivalent Voltages and Currents TEM Lines RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
58 Equivalent Voltages and Currents non-tem Lines RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
59 Physical Interpretation of Impedance RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
60 Even and Odd Properties of Z, Γ RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
61 Outline 1 N -Port Parameters 2 S Paramters 3 Even & Odd Mode Analysis 4 Interconnected Networks 5 V, I, and Z *** 6 Discontinuities and Bends *** RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
62 Diaphragms & Irises Symmetrical inductive diaphragm Asymmetrical inductive diaphragm Equivalent circuit Symmetrical capacitive diaphragm Asymmetrical capacitive diaphragm Equivalent circuit Rectangular resonant iris Circular resonant iris Equivalent circuit E Change in height Z 01 Z 02 Equivalent circuit E Change in width Z 01 Z 02 Equivalent circuit RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
63 Microstrip-line Discontinuities Z 0 Z 0 C C g Z 0 Z 0 Z 0 C p C p Z 0 Z 01 Z 02 L L Z 01 C Z 02 L 2 Z 02 Z 01 Z 02 Z 01 L 1 C L 3 Z 03 Z 03 Z 0m L Z 0c C 1 C 2 Z 0m Z 0c RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
64 Microstrip Bends W W W a W a r 3 W Right-angle bend Swept bend Mitered bends Mitered step Mitered T-junction RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
Microwave Network Analysis
Prof. Dr. Mohammad Tariqul Islam titareq@gmail.my tariqul@ukm.edu.my Microwave Network Analysis 1 Text Book D.M. Pozar, Microwave engineering, 3 rd edition, 2005 by John-Wiley & Sons. Fawwaz T. ILABY,
Microwave Network Analysis Lecture 1: The Scattering Parameters
Microwave Network Analysis Lecture : The Scattering Parameters ELC 305a Fall 0 Department of Electronics and Communications Engineering Faculty of Engineering Cairo University Outline Review on Network
Scattering Parameters
Berkeley Scattering Parameters Prof. Ali M. Niknejad U.C. Berkeley Copyright c 2016 by Ali M. Niknejad September 7, 2017 1 / 57 Scattering Parameters 2 / 57 Scattering Matrix Voltages and currents are
Networks and Systems Prof. V. G. K. Murti Department of Electrical Engineering Indian Institution of Technology, Madras
Networks and Systems Prof. V. G. K. Murti Department of Electrical Engineering Indian Institution of Technology, Madras Lecture - 32 Network Function (3) 2-port networks: Symmetry Equivalent networks Examples
TASK A. TRANSMISSION LINE AND DISCONTINUITIES
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
ECE Microwave Engineering
ECE 537-635 Microwave Engineering Fall 8 Prof. David R. Jackson Dept. of ECE Adapted from notes by Prof. Jeffery T. Williams Notes Power Dividers and Circulators Power Dividers and Couplers A power divider
Sinusoidal Steady State Analysis (AC Analysis) Part II
Sinusoidal Steady State Analysis (AC Analysis) Part II Amin Electronics and Electrical Communications Engineering Department (EECE) Cairo University elc.n102.eng@gmail.com http://scholar.cu.edu.eg/refky/
Solutions to Problems in Chapter 6
Appendix F Solutions to Problems in Chapter 6 F.1 Problem 6.1 Short-circuited transmission lines Section 6.2.1 (book page 193) describes the method to determine the overall length of the transmission line
ECE 5260 Microwave Engineering University of Virginia. Some Background: Circuit and Field Quantities and their Relations
ECE 5260 Microwave Engineering University of Virginia Lecture 2 Review of Fundamental Circuit Concepts and Introduction to Transmission Lines Although electromagnetic field theory and Maxwell s equations
Design of all-pole microwave filters. Giuseppe Macchiarella Polytechnic of Milan, Italy Electronic and Information Department
Design of all-pole microwave filters Giuseppe Macchiarella Polytechnic of Milan, Italy Electronic and Information Department In-line filters with all-equal resonators R L eq, f L eq, f L eq, f L eq, f
A two-port network is an electrical network with two separate ports
5.1 Introduction A two-port network is an electrical network with two separate ports for input and output. Fig(a) Single Port Network Fig(b) Two Port Network There are several reasons why we should study
Single- and Multiport Networks. RF Electronics Spring, 2018 Robert R. Krchnavek Rowan University
Single- and Multiport Networks RF Electronics Spring, 208 Robert R. Krchnavek Rowan University Objectives Generate an understanding of the common network representations of Z, Y, h, and ABCD. To be able
0-2 Operations with Complex Numbers
Simplify. 1. i 10 2. i 2 + i 8 3. i 3 + i 20 4. i 100 5. i 77 esolutions Manual - Powered by Cognero Page 1 6. i 4 + i 12 7. i 5 + i 9 8. i 18 Simplify. 9. (3 + 2i) + ( 4 + 6i) 10. (7 4i) + (2 3i) 11.
0-2 Operations with Complex Numbers
Simplify. 1. i 10 1 2. i 2 + i 8 0 3. i 3 + i 20 1 i esolutions Manual - Powered by Cognero Page 1 4. i 100 1 5. i 77 i 6. i 4 + i 12 2 7. i 5 + i 9 2i esolutions Manual - Powered by Cognero Page 2 8.
ECE Spring Prof. David R. Jackson ECE Dept. Notes 26
ECE 6345 Spring 05 Prof. David R. Jackson ECE Dept. Notes 6 Overview In this set of notes we use the spectral-domain method to find the mutual impedance between two rectangular patch ennas. z Geometry
Analog Circuits and Systems
Analog Circuits and Systems Prof. K Radhakrishna Rao Lecture 5 Analog Signal Processing using One Port Networks, Passive Two Ports and Ideal Amplifiers 1 One Port Devices Passive devices like R, L, C and
QUESTION BANK SUBJECT: NETWORK ANALYSIS (10ES34)
QUESTION BANK SUBJECT: NETWORK ANALYSIS (10ES34) NOTE: FOR NUMERICAL PROBLEMS FOR ALL UNITS EXCEPT UNIT 5 REFER THE E-BOOK ENGINEERING CIRCUIT ANALYSIS, 7 th EDITION HAYT AND KIMMERLY. PAGE NUMBERS OF
Sinusoidal Steady State Analysis (AC Analysis) Part I
Sinusoidal Steady State Analysis (AC Analysis) Part I Amin Electronics and Electrical Communications Engineering Department (EECE) Cairo University elc.n102.eng@gmail.com http://scholar.cu.edu.eg/refky/
Two Port Networks. Definition of 2-Port Network A two-port network is an electrical network with two separate ports for input and output
Two Port Networks Definition of 2-Port Network A two-port network is an electrical network with two separate ports for input and output What is a Port? It is a pair of terminals through which a current
ECE 202 Fall 2013 Final Exam
ECE 202 Fall 2013 Final Exam December 12, 2013 Circle your division: Division 0101: Furgason (8:30 am) Division 0201: Bermel (9:30 am) Name (Last, First) Purdue ID # There are 18 multiple choice problems
Lecture 23 Date: Multi-port networks Impedance and Admittance Matrix Lossless and Reciprocal Networks
Lecture 23 Date: 30.0.207 Multi-port networks mpedance and Admittance Matrix Lossless and Reciprocal Networks ntroduction A pair of terminals through which a current may enter or leave a network is known
TRANSMISSION LINES AND MATCHING
TRANSMISSION LINES AND MATCHING for High-Frequency Circuit Design Elective by Michael Tse September 2003 Contents Basic models The Telegrapher s equations and solutions Transmission line equations The
Lecture 17 Date:
Lecture 17 Date: 09.03.017 The Quadrature Hybrid We began our discussion of dividers and couplers by considering important general properties of three- and four-port networks. This was followed by an analysis
Consider a simple RC circuit. We might like to know how much power is being supplied by the source. We probably need to find the current.
AC power Consider a simple RC circuit We might like to know how much power is being supplied by the source We probably need to find the current R 10! R 10! is VS Vmcosωt Vm 10 V f 60 Hz V m 10 V C 150
Solutions to Problems in Chapter 5
Appendix E Solutions to Problems in Chapter 5 E. Problem 5. Properties of the two-port network The network is mismatched at both ports (s 0 and s 0). The network is reciprocal (s s ). The network is symmetrical
RF AND MICROWAVE ENGINEERING
RF AND MICROWAVE ENGINEERING EC6701 RF AND MICROWAVE ENGINEERING UNIT I TWO PORT RF NETWORKS-CIRCUIT REPRESENTATION 9 Low frequency parameters-impedance,admittance, hybrid and ABCD. High frequency parameters-formulation
Electric Circuit Theory
Electric Circuit Theory Nam Ki Min nkmin@korea.ac.kr 010-9419-2320 Chapter 18 Two-Port Circuits Nam Ki Min nkmin@korea.ac.kr 010-9419-2320 Contents and Objectives 3 Chapter Contents 18.1 The Terminal Equations
Lecture 11 Date:
Lecture 11 Date: 11.09.014 Scattering Parameters and Circuit Symmetry Even-mode and Odd-mode Analysis Generalized S-Parameters Example T-Parameters Q: OK, but how can we determine the scattering matrix
Lecture 19 Date:
Lecture 19 Date: 8.10.014 The Quadrature Hybrid We began our discussion of dividers and couplers by considering important general properties of three- and fourport networks. This was followed by an analysis
192 Chapter 4: Microwave Network Analysis
92 hapter 4: Microwave Network nalysis TLE 4.2 onversions etween Two-Port Network Parameters S Z Y S S (Z Z 0 )(2 + Z 0 ) (Y 0 Y )(Y 0 + Y 22 ) + Y 2 Y 2 + /Z 0 Z 0 + /Z 0 + Z 0 + S S 2Z 2 Z 0 2 2 2Y 2
Introduction. A microwave circuit is an interconnection of components whose size is comparable with the wavelength at the operation frequency
Introduction A microwave circuit is an interconnection of components whose size is comparable with the wavelength at the operation frequency Type of Components: Interconnection: it is not an ideal connection
EE292: Fundamentals of ECE
EE292: Fundamentals of ECE Fall 2012 TTh 10:00-11:15 SEB 1242 Lecture 18 121025 http://www.ee.unlv.edu/~b1morris/ee292/ 2 Outline Review RMS Values Complex Numbers Phasors Complex Impedance Circuit Analysis
Case Study: Parallel Coupled- Line Combline Filter
MICROWAVE AND RF DESIGN MICROWAVE AND RF DESIGN Case Study: Parallel Coupled- Line Combline Filter Presented by Michael Steer Reading: 6. 6.4 Index: CS_PCL_Filter Based on material in Microwave and RF
Power System Engineering Prof. Debapriya Das Department of Electrical Engineering Indian Institute of Technology, Kharagpur
Power System Engineering Prof. Debapriya Das Department of Electrical Engineering Indian Institute of Technology, Kharagpur Lecture 41 Application of capacitors in distribution system (Contd.) (Refer Slide
TWO-PORT. C.T. Pan 1. C.T. Pan
TWO-PORT CRCUTS C.T. Pan 5. Definition of Two-Port Circuits 5. Classification of Two-Port Parameters 5.3 Finding Two-Port Parameters 5.4 Analysis of the Terminated Two-Port Circuit 5.5 nterconnected Two-Port
CHAPTER 5 : REDUCTION OF MULTIPLE SUBSYSTEMS
CHAPTER 5 : REDUCTION OF MULTIPLE SUBSYSTEMS Objectives Students should be able to: Reduce a block diagram of multiple subsystems to a single block representing the transfer function from input to output
Lecture 12. Microwave Networks and Scattering Parameters
Lecture Microwave Networs and cattering Parameters Optional Reading: teer ection 6.3 to 6.6 Pozar ection 4.3 ElecEng4FJ4 LECTURE : MICROWAE NETWORK AND -PARAMETER Microwave Networs: oltages and Currents
Rana Pratap Yadav *, Sunil Kumar, and S. V. Kulkarni Institute for Plasma Research, Bhat, Gandhinagar , India
Progress In Electromagnetics Research B, Vol. 56, 5 49, 013 AN ANALYSIS OF JUNCTION DISCONTINUITY EF- FECTS IN THE MULTI-ELEMENT COUPLED LINES AND ITS DIMINUTION AT DESIGNING STAGE Rana Pratap Yadav *,
Lecture 7: Two-Ports (2)
Lecture 7: Two-Ports (2) Amin Arbabian Jan M. Rabaey EE142 Fall 2010 Sept. 16 th, 2010 Announcements HW3 Posted, due Thursday September 23 rd, 330pm EE142 drop box. Office hours Two-Port notes by Professor
ARTIFICIAL TRANSMISSION LINE WITH LEFT/RIGHT-HANDED BEHAVIOR BASED ON WIRE BONDED INTERDIGITAL CAPACITORS
Progress In Electromagnetics Research B, Vol. 11, 245 264, 29 ARTIFICIAL TRANSMISSION LINE WITH LEFT/RIGHT-HANDED BEHAVIOR BASED ON WIRE BONDED INTERDIGITAL CAPACITORS J. J. Sánchez-Martínez, E. Márquez-Segura,
Transmission and Distribution of Electrical Power
KINGDOM OF SAUDI ARABIA Ministry Of High Education Umm Al-Qura University College of Engineering & Islamic Architecture Department Of Electrical Engineering Transmission and Distribution of Electrical
EE 205 Dr. A. Zidouri. Electric Circuits II. Two-Port Circuits Two-Port Parameters. Lecture #42
EE 05 Dr. A. Zidouri Electric Circuits Two-Port Circuits Two-Port Parameters Lecture #4-1 - EE 05 Dr. A. Zidouri The material to be covered in this lecture is as follows: o ntroduction to two-port circuits
Power lines. Why do birds sitting on a high-voltage power line survive?
Power lines At large distances, the resistance of power lines becomes significant. To transmit maximum power, is it better to transmit high V, low I or high I, low V? (a) high V, low I (b) low V, high
Alternating Current Circuits
Alternating Current Circuits AC Circuit An AC circuit consists of a combination of circuit elements and an AC generator or source. The output of an AC generator is sinusoidal and varies with time according
ECE 546 Lecture 13 Scattering Parameters
ECE 546 Lecture 3 Scattering Parameters Spring 08 Jose E. Schutt-Aine Electrical & Computer Engineering University of Illinois jesa@illinois.edu ECE 546 Jose Schutt Aine Transfer Function Representation
Lecture 14 Date:
Lecture 14 Date: 18.09.2014 L Type Matching Network Examples Nodal Quality Factor T- and Pi- Matching Networks Microstrip Matching Networks Series- and Shunt-stub Matching L Type Matching Network (contd.)
320-amp-models.tex Page 1 ECE 320. Amplifier Models. ECE Linear Active Circuit Design
320ampmodels.tex Page 1 ECE 320 Amplifier Models ECE 320 Linear Active Circuit Design 320ampmodels.tex Page 2 2Port Networks A 2port network is any circiut with two pairs of wires connecting to the outside
ELG4125: Power Transmission Lines Steady State Operation
ELG4125: Power Transmission Lines Steady State Operation Two-Port Networks and ABCD Models A transmission line can be represented by a two-port network, that is a network that can be isolated from the
EXP. NO. 3 Power on (resistive inductive & capacitive) load Series connection
OBJECT: To examine the power distribution on (R, L, C) series circuit. APPARATUS 1-signal function generator 2- Oscilloscope, A.V.O meter 3- Resisters & inductor &capacitor THEORY the following form for
INVESTIGATION OF MICROWAVE TRI-RESONATOR STRUCTURES
SCHOOL OF ELECTRONIC, ELECTRICAL AND COMPUER ENGINEERING THE UNIVERSITY OF BIRMINGHAM INVESTIGATION OF MICROWAVE TRI-RESONATOR STRUCTURES Negassa Sori Gerba A thesis submitted to the University of Birmingham
NONLINEAR AND ADAPTIVE (INTELLIGENT) SYSTEMS MODELING, DESIGN, & CONTROL A Building Block Approach
NONLINEAR AND ADAPTIVE (INTELLIGENT) SYSTEMS MODELING, DESIGN, & CONTROL A Building Block Approach P.A. (Rama) Ramamoorthy Electrical & Computer Engineering and Comp. Science Dept., M.L. 30, University
Introduction to Network Analysis of Microwave Circuits
1 Introduction to Network Analysis of Microwave Circuits ABSTRACT Network presentation has been used as a technique in the analysis of lowfrequency electrical electronic circuits. The same technique is
Basics of Wave Propagation
Basics of Wave Propagation S. R. Zinka zinka@hyderabad.bits-pilani.ac.in Department of Electrical & Electronics Engineering BITS Pilani, Hyderbad Campus May 7, 2015 Outline 1 Time Harmonic Fields 2 Helmholtz
PHYS 241 EXAM #2 November 9, 2006
1. ( 5 points) A resistance R and a 3.9 H inductance are in series across a 60 Hz AC voltage. The voltage across the resistor is 23 V and the voltage across the inductor is 35 V. Assume that all voltages
EE Power Gain and Amplifier Design 10/31/2017
EE 40458 Power Gain and Amplifier Design 10/31/017 Motivation Brief recap: We ve studied matching networks (several types, how to design them, bandwidth, how they work, etc ) Studied network analysis techniques
Refinements to Incremental Transistor Model
Refinements to Incremental Transistor Model This section presents modifications to the incremental models that account for non-ideal transistor behavior Incremental output port resistance Incremental changes
Lecture 23: Basic Properties of Dividers and Couplers.
Whites, EE 48/58 Lecture 23 age of 9 Lecture 23: Basic roperties of Dividers and Couplers. For the remainder of this course we re going to investigate a plethora of microwave devices and circuits both
ANTENNAS and MICROWAVES ENGINEERING (650427)
Philadelphia University Faculty of Engineering Communication and Electronics Engineering ANTENNAS and MICROWAVES ENGINEERING (65427) Part 2 Dr. Omar R Daoud 1 General Considerations It is a two-port network
TWO PORT NETWORKS Introduction: A port is normally referred to a pair of terminals of a network through which we can have access to network either for a source for measuring an output We have already seen
IMPEDANCE and NETWORKS. Transformers. Networks. A method of analysing complex networks. Y-parameters and S-parameters
IMPEDANCE and NETWORKS Transformers Networks A method of analysing complex networks Y-parameters and S-parameters 1 ENGN4545/ENGN6545: Radiofrequency Engineering L#7 Transformers Combining the effects
Two Port Network. Ram Prasad Sarkar
Two Port Ram Prasad Sarkar 0 Two Post : Post nput port Two Post Fig. Port Output port A network which has two terminals (one port) on the one side and another two terminals on the opposite side forms a
mywbut.com Lesson 16 Solution of Current in AC Parallel and Seriesparallel
esson 6 Solution of urrent in Parallel and Seriesparallel ircuits n the last lesson, the following points were described:. How to compute the total impedance/admittance in series/parallel circuits?. How
Electromagnetic Theorems
Electromagnetic Theorems Daniel S. Weile Department of Electrical and Computer Engineering University of Delaware ELEG 648 Electromagnetic Theorems Outline Outline Duality The Main Idea Electric Sources
Lecture #3. Review: Power
Lecture #3 OUTLINE Power calculations Circuit elements Voltage and current sources Electrical resistance (Ohm s law) Kirchhoff s laws Reading Chapter 2 Lecture 3, Slide 1 Review: Power If an element is
Networks and Systems Prof. V. G. K. Murti Department of Electrical Engineering Indian Institute of Technology, Madras
Networks and Systems Prof. V. G. K. Murti Department of Electrical Engineering Indian Institute of Technology, Madras Lecture - 34 Network Theorems (1) Superposition Theorem Substitution Theorem The next
Smith Chart Figure 1 Figure 1.
Smith Chart The Smith chart appeared in 1939 as a graph-based method of simplifying the complex math (that is, calculations involving variables of the form x + jy) needed to describe the characteristics
Equivalent Circuit Model Extraction for Interconnects in 3D ICs
Equivalent Circuit Model Extraction for Interconnects in 3D ICs A. Ege Engin Assistant Professor, Department of ECE, San Diego State University Email: aengin@mail.sdsu.edu ASP-DAC, Jan. 23, 213 Outline
Waves on Lines. Contents. ! Transmission Lines! The Smith Chart! Vector Network Analyser (VNA) ! Measurements
Waves on Lines If the wavelength to be considered is significantly greater compared to the size of the circuit the voltage will be independent of the location. amplitude d! distance but this is not true
UNIVERSITY OF BOLTON. SCHOOL OF ENGINEERING, SPORTS and SCIENCES BENG (HONS) ELECTRICAL & ELECTRONICS ENGINEERING EXAMINATION SEMESTER /2018
ENG018 SCHOOL OF ENGINEERING, SPORTS and SCIENCES BENG (HONS) ELECTRICAL & ELECTRONICS ENGINEERING MODULE NO: EEE6002 Date: 17 January 2018 Time: 2.00 4.00 INSTRUCTIONS TO CANDIDATES: There are six questions.
Solution of Final Exam Problems 1
of Final Exam Problems Prob. Two different electrical devices have the same power consumption, but one is meant to be operated on 20 V AC and the other on 220 V AC. (a) What is the ratio of their resistances?
TECHNICAL REPORT: CVEL Maximum Radiated Emission Calculator: I/O Coupling Algorithm. Chentian Zhu and Dr. Todd Hubing. Clemson University
TECHNICAL REPORT: CVEL-13-045 Maximum Radiated Emission Calculator: I/O Coupling Algorithm Chentian Zhu and Dr. Todd Hubing Clemson University August 4, 013 Table of Contents Abstract... 3 1. Introduction...
Surface Mount Chip Capacitors
Features High '' Factor at high frequencies High RF power capabilities Low High self resonant frequencies Excellent stability across temperature range Small size High Frequency Measurement and Performance
EECS 117 Lecture 3: Transmission Line Junctions / Time Harmonic Excitation
EECS 117 Lecture 3: Transmission Line Junctions / Time Harmonic Excitation Prof. Niknejad University of California, Berkeley University of California, Berkeley EECS 117 Lecture 3 p. 1/23 Transmission Line
2.004 Dynamics and Control II Spring 2008
MIT OpenCourseWare http://ocwmitedu 00 Dynamics and Control II Spring 00 For information about citing these materials or our Terms of Use, visit: http://ocwmitedu/terms Massachusetts Institute of Technology
Outline. Thermal noise, noise power and noise temperature. Noise in RLC single-ports. Noise in diodes and photodiodes
3. Noise 1 Outline Thermal noise, noise power and noise temperature Noise in RLC single-ports Noise in diodes and photodiodes -port and multi-port noise parameters Noise temperature and noise parameter
Synthesis of passband filters with asymmetric transmission zeros
Synthesis of passband filters with asymmetric transmission zeros Giuseppe Macchiarella Polytechnic of Milan, Italy Electronic and Information Department Passband filters with asymmetric zeros Placing asymmetric
Conventional Paper-I Part A. 1. (a) Define intrinsic wave impedance for a medium and derive the equation for intrinsic vy
EE-Conventional Paper-I IES-01 www.gateforum.com Conventional Paper-I-01 Part A 1. (a) Define intrinsic wave impedance for a medium and derive the equation for intrinsic vy impedance for a lossy dielectric
Lab #8 Two Port Networks
Cir cuit s 212 Lab Lab #8 Two Port Networks Network parameters are used to characterize a device. Given the network parameters of a device, the voltage and current characteristics of the device can be
Annexure-I. network acts as a buffer in matching the impedance of the plasma reactor to that of the RF
Annexure-I Impedance matching and Smith chart The output impedance of the RF generator is 50 ohms. The impedance matching network acts as a buffer in matching the impedance of the plasma reactor to that
SECTION 5: POWER FLOW. ESE 470 Energy Distribution Systems
SECTION 5: POWER FLOW ESE 470 Energy Distribution Systems 2 Introduction Nodal Analysis 3 Consider the following circuit Three voltage sources VV sss, VV sss, VV sss Generic branch impedances Could be
ECE414/514 Electronics Packaging Spring 2012 Lecture 6 Electrical D: Transmission lines (Crosstalk) Lecture topics
ECE414/514 Electronics Packaging Spring 2012 Lecture 6 Electrical D: Transmission lines (Crosstalk) James E. Morris Dept of Electrical & Computer Engineering Portland State University 1 Lecture topics
NETWORK ANALYSIS ( ) 2012 pattern
PRACTICAL WORK BOOK For Academic Session 0 NETWORK ANALYSIS ( 0347 ) 0 pattern For S.E. (Electrical Engineering) Department of Electrical Engineering (University of Pune) SHREE RAMCHANDRA COLLEGE OF ENGG.
Transmission lines. Shouri Chatterjee. October 22, 2014
Transmission lines Shouri Chatterjee October 22, 2014 The transmission line is a very commonly used distributed circuit: a pair of wires. Unfortunately, a pair of wires used to apply a time-varying voltage,
Impedance Matching and Tuning
C h a p t e r F i v e Impedance Matching and Tuning This chapter marks a turning point, in that we now begin to apply the theory and techniques of previous chapters to practical problems in microwave engineering.
Power Transformer Transient Modeling Using Frequency Response Analysis
Power Transformer Transient Modeling Using Frequency Response Analysis by Hosam Salem Alharbi A thesis submitted to the Faculty of Graduate Studies of The University of Manitoba in partial fulfilment of
and Ee = E ; 0 they are separated by a dielectric material having u = io-s S/m, µ, = µ, 0
602 CHAPTER 11 TRANSMISSION LINES 11.10 Two identical pulses each of magnitude 12 V and width 2 µs are incident at t = 0 on a lossless transmission line of length 400 m terminated with a load. If the two
EELE 3332 Electromagnetic II Chapter 11. Transmission Lines. Islamic University of Gaza Electrical Engineering Department Dr.
EEE 333 Electromagnetic II Chapter 11 Transmission ines Islamic University of Gaza Electrical Engineering Department Dr. Talal Skaik 1 1 11.1 Introduction Wave propagation in unbounded media is used in
Two-Port Networks Admittance Parameters CHAPTER16 THE LEARNING GOALS FOR THIS CHAPTER ARE THAT STUDENTS SHOULD BE ABLE TO:
CHAPTER16 Two-Port Networks THE LEARNING GOALS FOR THIS CHAPTER ARE THAT STUDENTS SHOULD BE ABLE TO: Calculate the admittance, impedance, hybrid, and transmission parameter for two-port networks. Convert
TC 412 Microwave Communications. Lecture 6 Transmission lines problems and microstrip lines
TC 412 Microwave Communications Lecture 6 Transmission lines problems and microstrip lines RS 1 Review Input impedance for finite length line Quarter wavelength line Half wavelength line Smith chart A
) Rotate L by 120 clockwise to obtain in!! anywhere between load and generator: rotation by 2d in clockwise direction. d=distance from the load to the
3.1 Smith Chart Construction: Start with polar representation of. L ; in on lossless lines related by simple phase change ) Idea: polar plot going from L to in involves simple rotation. in jj 1 ) circle
A New, Accurate Quasi-Static Model for Conductor Loss in Coplanar Waveguide
A New, Accurate Quasi-Static Model for Conductor Loss in Coplanar Waveguide M. S. Islam, E. Tuncer, D. P. Neikirk Dept. of Electrical & Computer Engineering Austin, Texas Conductor loss calculation techniques
General Physics (PHY 2140)
General Physics (PHY 2140) Lecture 10 6/12/2007 Electricity and Magnetism Induced voltages and induction Self-Inductance RL Circuits Energy in magnetic fields AC circuits and EM waves Resistors, capacitors
RF engineering basic concepts: S-parameters
RF engineering basic concepts: -parameters F. Caspers CERN, Geneva, witzerland Abstract The concept of describing RF circuits in terms of waves is discussed and the -matrix and related matrices are defined.
Outline. Week 5: Circuits. Course Notes: 3.5. Goals: Use linear algebra to determine voltage drops and branch currents.
Outline Week 5: Circuits Course Notes: 3.5 Goals: Use linear algebra to determine voltage drops and branch currents. Components in Resistor Networks voltage source current source resistor Components in
Lecture 11 - AC Power
- AC Power 11/17/2015 Reading: Chapter 11 1 Outline Instantaneous power Complex power Average (real) power Reactive power Apparent power Maximum power transfer Power factor correction 2 Power in AC Circuits
The Impedance Matrix
0/0/09 The mpedance Matrix.doc /7 The mpedance Matrix Consider the -port microwave device shown below: z ( z ) z z port z z port 0 -port 0 microwave 0 device P z z P z port z P z ( z ) z port 0 ( z ) z
Two-Port Networks Introduction
Two-Port Networks ntroduction n general, a network ma have n ports. The current entering one terminal leaves through the other terminal so that the net current entering all ports equals zero. Thus, a two-port
Analytic Solutions for Periodically Loaded Transmission Line Modeling
Analytic Solutions for Periodically Loaded Transmission Line Modeling Paul G. Huray, huray@sc.edu Priya Pathmanathan, Intel priyap@qti.qualcomm.com Steve Pytel, Intel steve.pytel@ansys.com April 4, 2014
Some of the different forms of a signal, obtained by transformations, are shown in the figure. jwt e z. jwt z e
Transform methods Some of the different forms of a signal, obtained by transformations, are shown in the figure. X(s) X(t) L - L F - F jw s s jw X(jw) X*(t) F - F X*(jw) jwt e z jwt z e X(nT) Z - Z X(z)
PH2200 Practice Exam II Summer 2003
PH00 Practice Exam II Summer 00 INSTRUCTIONS. Write your name and student identification number on the answer sheet and mark your recitation section.. Please cover your answer sheet at all times.. This