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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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 21 1 2 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
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
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
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
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
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
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
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
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
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
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 1 0 0 0 e jθ 2 0 e jθ N V 1 V 2 V N = [S] e jθ 1 0 0 0 e jθ 2 0 e jθ N V 1 V 2 V N RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
Shift in Reference Planes Cont d V 1 V 2 V N = e jθ 1 0 0 0 e jθ 2 0 e jθ N 1 [S] e jθ 1 0 0 0 e jθ 2 0 e jθ N V 1 V 2 V N V 1 V 2 V N = e jθ 1 0 0 0 e jθ 2 0 e jθ N [S] e jθ 1 0 0 0 e jθ 2 0 e jθ N }{{} [S ] V 1 V 2 V N RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
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
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
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
Even Mode Analysis v - R 1 R 2 /2 R 2 /2 R 1 R 1 R 1 v v 2R 2 2R 2 - - v - Open Open RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
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
Even & Odd Mode Analysis - A Simple Example R 2 v R 1 R 1 0 - - 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/2 - - - - Open Short RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
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/2 - - - - 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
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/2 - - - - 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
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
Method I RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
Cascaded Networks RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
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
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
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
Method II RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
More Complicated Networks RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
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
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
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
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
Example Usage of Mason s Gain Formula RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
Method III RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
Interconnected Networks - A Matrix Approach RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
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
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
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
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
Equivalent Voltages and Currents TEM Lines RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
Equivalent Voltages and Currents non-tem Lines RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
Physical Interpretation of Impedance RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
Even and Odd Properties of Z, Γ RF & Microwave Engineering, Dept of EEE, BITS Hyderabad
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
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
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
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