Microwave Network Analysis - Arraytool · PDF file07.05.2015 · Microwave Network...

N -Port Parameters S Paramters Even & Odd Mode Analysis Interconnected Networks V, I, and Z *** Discontinuities and Bends ***

Microwave Network Analysis

S. R. [email protected]

Department of Electrical & Electronics EngineeringBITS Pilani, Hyderbad Campus

May 7, 2015

Course Outline RF & Microwave Engineering, Dept. of EEE, BITS Hyderabad

mailto:[email protected]


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 ***



Z, Y, h, g Paramters

−+ −+ −+

−+

z11

z12I2 z21I1

z22

y11

y12V2 y21V1

y22

h11

h12V2

h21I1

h22

g11

g12I2 g21V1

g22

* h12, h21, g12, g21 are dimensionless quantities * h22, y11, y22, g11 are admittance values. So, actual resistance values are 1/h22, etc.



Series-Series Connection

CombinedPort 1

CombinedPort 2

[Z]1

[Z]2

[Z] = [Z]1 + [Z]2 (1)



Shunt-Shunt Connection

Com

bine

dPo

rt 1

Com

bine

dPo

rt 2

[Y]1

[Y]2

[Y] = [Y]1 + [Y]2 (2)



Series-Shunt Connection

Com

bine

dPo

rt 1

Com

bine

dPo

rt 2

[h]1

[h]2

[h] = [h]1 + [h]2 (3)



Shunt-Series Connection

Com

bine

dPo

rt 1

Com

bine

dPo

rt 2

[g]1

[g]2

[g] = [g]1 + [g]2 (4)



Z Paramters of N-port Networks

tN

t4

t3

S

t2

t1

VN, IN+ + VN, – IN

– –

V4, – I4– –

V4, I4+ +

V3, I3+ +

V3,– I3– –

V2, – I2– –

V2, I2+ +

V1, I1+ +

V1, – I1– –

......

Vn = V+n + V−n

In = I+n − I−n

V1V2

...VN

=

Z11 Z12 · · · Z1N

Z21

......

...ZN1 · · · · · · ZNN

I1I2

...IN



Y Paramters of N-port Networks

tN

t4

t3

S

t2

t1


– –

V4, – I4– –

V4, I4+ +

V3, I3+ +

V3,– I3– –

V2, – I2– –

V2, I2+ +

V1, I1+ +

V1, – I1– –

......

Vn = V+n + V−n

In = I+n − I−n

I1I2

...IN

=

Y11 Y12 · · · Y1N

Y21

......

...YN1 · · · · · · YNN

V1V2

...VN



Reciprocal Networks

−+ −+

z11

z12I2 z21I1

z22

A network is said to be reciprocal if the voltage appearing at port 2 due to a current applied at port1 is the same as the voltage appearing at port 1 when the same current is applied to port 2.

Z12 = Z21

More generally,

Zij = Zji.



Reciprocal Networks ... Cont’d

y11

y12V2 y21V1

y22

Exchanging voltage and current results in an equivalent definition of reciprocity.

Y12 = Y21

More generally,

Yij = Yji.



Reciprocal and Loss-less Networks

If the network is lossless, then the net real power delivered to the network must be zero. Thus,

Re(Pavg

)=

12

Re([I]H [V]

)=

12

Re([I]H [Z] [I]

)=

12

Re

(N

∑j=1

N

∑i=1

I∗i ZijIj

)= 0. (5)



Reciprocal and Loss-less Networks ... Cont’d

Let’s re-write (5) as shown below:

12

Re

(N

∑j=1

N

∑i=1

I∗i ZijIj

)= 0

⇒ 12

Re(|I1|2 Z11 + I∗1 Z12I2 + I∗2 Z21I1 + · · ·

)= 0

⇒ 12

Re(|I1|2 Z11 + Z12 (I∗1 I2 + I∗2 I1) + · · ·

)= 0, ∵ Z12 = Z21

⇒ 12

Re

|I1|2 Z11 + Z12

I∗1 I2 + I∗2 I1︸︷︷︸real

+ · · ·

= 0

⇒ |I1|2 Re (Z11) +

I∗1 I2 + I∗2 I1︸︷︷︸real

Re (Z12) + · · · = 0.

So, for reciprocal and loss-less networks, Re(Zij)= 0, for any i, j.



Outline


2 S Paramters



5 V, I, and Z ***




S Parameters

tN

t4

t3

S

t2

t1


– –

V4, – I4– –

V4, I4+ +

V3, I3+ +

V3,– I3– –

V2, – I2– –

V2, I2+ +

V1, I1+ +

V1, – I1– –

.....

V−1V−2

...V−N

=

S11 S12 · · · S1N

S21

......

...SN1 · · · · · · SNN

V+1

V+2...

V+N



S to Z Paramters

[V] = [Z] [I]

⇒

V+

1V+

2...

V+N

+

V−1V−2

...V−N

=

Z11 Z12 · · · Z1N

Z21

......

...ZN1 · · · · · · ZNN

1

Z0

V+1

V+2...

V+N

−

V−1V−2

...V−N

⇒ ([U] + [S])

V+

1V+

2...

V+N

=1

Z0

Z11 Z12 · · · Z1N

Z21

......

...ZN1 · · · · · · ZNN

([U]− [S])

V+

1V+

2...

V+N

⇒ [U] + [S] =1

Z0[Z] ([U]− [S])

⇒ [Z] ([U]− [S]) = Z0 ([U] + [S])

⇒ [Z] = Z0 ([U] + [S]) ([U]− [S])−1 . (6)



Z to S Paramters

From the previous slide,

⇒ [U] + [S] =1

Z0[Z] ([U]− [S])

⇒ Z0 [S] + [Z] [S] = [Z]− Z0 [U]

⇒ (Z0 [U] + [Z]) [S] = [Z]− Z0 [U]

⇒ [S] = ([Z] + Z0 [U])−1 ([Z]− Z0 [U]) . (7)



S to Y Parameters

Since [Z] = Z0 ([U] + [S]) ([U]− [S])−1,

[Y] = [Z]−1 =[Z0 ([U] + [S]) ([U]− [S])−1

]−1

= Y0 ([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.



S Parameters of Reciprocal Networks

[V+]=

12([V] + Z0 [I]) =

12([Z] + Z0 [U]) [I][

V−]=

12([V]− [I]) =

12([Z]− Z0 [U]) [I]

Combining the above set of equations gives

[V−]=

12([Z]− Z0 [U]) [I]

⇒[V−]= ([Z]− Z0 [U]) ([Z] + Z0 [U])−1 [V+

]⇒ [S] = ([Z]− Z0 [U]) ([Z] + Z0 [U])−1

⇒ [S]t =(([Z] + Z0 [U])t

)−1([Z]− Z0 [U])t

⇒ [S]t = (([Z] + Z0 [U]))−1 ([Z]− Z0 [U]) . ∵ [Z] = [Z]t for reciprocal networks (9)

Comparing (7) and the above equation gives

[S]t = [S] . (10)



S Parameters of Loss-less Reciprocal Networks

[S] [S]H =[Z]− Z0 [U]

[Z] + Z0 [U]

[[Z]− Z0 [U]

[Z] + Z0 [U]

]H

=[Z]− Z0 [U]

[Z] + Z0 [U]

[Z]H − Z0 [U]

[Z]H + Z0 [U]

=[Z]− Z0 [U]

[Z] + Z0 [U]

− [Z]− Z0 [U]

− [Z] + Z0 [U], ∵ [Z]H = − [Z] for lossless reciprocal networks

=[Z]− Z0 [U]

[Z] + Z0 [U]

[Z] + Z0 [U]

[Z]− Z0 [U]= [U] .

So, [S] of any loss-less reciprocal network is always a unitary matrix, i.e., [S] [S]H = [U].



Shift in Reference Planes

zn = ln zn = 0

z1 = l1 z1 = 0

V'n–

V'n+

V'1–

V'1+

V n–

V n+

V 1–

V 1+

Port 1

N-portnetwork

[S ], [S ']

Port n

θ1

θn

V′+n = V+

n ejθn

V′−n = V−n e−jθn



Shift in Reference Planes ... Cont’d

V−1V−2

...V−N

=

S11 S12 · · · S1N

S21

......

...SN1 · · · · · · SNN

V+1

V+2...

V+N

⇒

V′−1 e+jθ1

V′−2 e+jθ2

...V′−N e+jθN

= [S]

V′+1 e−jθ1

V′+2 e−jθ2

...V′+N e−jθN

⇒

e+jθ1 0 · · · 0

0 e+jθ2...

.... . .

...0 · · · · · · e+jθN

V′−1

V′−2...

V′−N

= [S]

e−jθ1 0 · · · 0

0 e−jθ2...

.... . .

...0 · · · · · · e−jθN

V′+1

V′+2...

V′+N



Shift in Reference Planes ... Cont’d

⇒

V′−1

V′−2...

V′−N

=

e+jθ1 0 · · · 0

0 e+jθ2...

.... . .

...0 · · · · · · e+jθN

−1

[S]

e−jθ1 0 · · · 0

0 e−jθ2...

.... . .

...0 · · · · · · e−jθN

V′+1

V′+2...

V′+N

⇒

V′−1

V′−2...

V′−N

=

e−jθ1 0 · · · 0

0 e−jθ2...

.... . .

...0 · · · · · · e−jθN

[S]

e−jθ1 0 · · · 0

0 e−jθ2...

.... . .

...0 · · · · · · e−jθN

︸︷︷︸

[S′ ]

V′+1

V′+2...

V′+N



Outline


2 S Paramters



5 V, I, and Z ***




Symmetric T and Pi Networks

R1R1

R2

R2

R1 R1

R3R1

R2/2 R3R1

2R2 2R2

R2/2



Symmetric T and Pi Networks

R1R1

R2

R2

R1 R1

R1R1

R2/2 R1R1

2R2 2R2

R2/2



Even Mode Analysis

R1R1

R2/2 R1R1

2R2 2R2

R2/2

Open Open

+

-

+

-

+

-

+

-v vvv



Odd Mode Analysis

R1R1

R2/2 R1R1

2R2 2R2

R2/2

Short Short

+

-

+

-

+

-

+

-v -vv-v



Even & Odd Mode Analysis - A Simple Example

R1R1

R2/2 R2/2

Open

+

-

+

-v/2 R1R1

R2/2 R2/2

Short

+

-

+

-v/2 v/2 -v/2

R1R1

R2+

-v

+

-0



Even & Odd Mode Analysis - A Simple Example

R1R1

R2/2 R2/2

Open

+

-

+

-v/2 R1R1

R2/2 R2/2

Short

+

-

+

-v/2 v/2 -v/2

S11 =12(Se

11 + So11)

S21 =12(Se

11 − So11)

S22 = S11

S12 = S21



Outline


2 S Paramters



5 V, I, and Z ***




Method I



Cascaded Networks



ABCD Parameters

+–

I1 I2

V1 V2+–

Port1

Port2

AC

BD

[V1I1

]=

[A BC D

] [V2I2

]



Cascaded Networks

+

–

I1 I2 I3

V1 V2+

–V3

+

–

A1

C1

B1

D1

A2

C2

B2

D2

[V1I1

]=

[A1 B1C1 D1

] [A2 B2C2 D2

] [V3I3

]



ABCD Parameters of Some Useful Two-Port Circuits

Z A = 1C = 0

B = ZD = 1

YA = 1C = Y

B = 0D = 1

Z0,

l

A = cosβC = jY0 sinβ

B = j Z0 sinβD = cosβ

N : 1

A = NC = 0

B = 0

D =1N

Y1 Y2

Y3 A = 1 +Y2Y3

C = Y1 +Y2 +Y1Y2

Y3

B =1

Y3

D = 1 +Y1Y3

Z1 Z2

Z3

A = 1 +Z1Z3

C =1

Z3

B = Z1 + Z2 +Z1 Z2

Z3

D = 1 +Z2Z3



Method II



More Complicated Networks



Signal Flow Graph Method

Port2

Port1 [S]

(a)

(b)

S22

S21

S12

S11

a1 a2

b2b1

a1 b2

b1 a2



Signal Flow Graph Method ... Cont’d

(a)

(b)

Vi

Vs

a

b

b

b

a

a

b

a

Zs

Γl Γl

ΓsΓs



Signal Flow Graph Method ... Rules

V1V3V2V1 V2

S21S21 S32

S42 V4 S21 S42

S32

V3

V4V'

V3V2V1

S21 S32

S22

S21 S32

V3V2V1

S22

S21

S32

2

1 –

V2V1V2V1

Sa + Sb

Sa

Sb

V3V1

S21S32

V3V2V1



Mason’s Gain Formula

Mason’s gain formula is as follows:

G =∑N

k=1 Gk∆k

∆

∆ = 1−∑ Li + ∑ LiLj −∑ LiLjLk + · · ·+ (−1)m ∑ · · ·+ · · ·where:

• N is the total number of forward paths between input and output• Gk is the path gain of the kth forward path between input and output• Li is the loop gain of each closed loop in the system• LiLj is the product of the loop gains of any two non-touching loops• LiLjLk 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



Example Usage of Mason’s Gain Formula



Method III



Interconnected Networks - A Matrix Approach

[bobi

]=

[[Soo] [Soi][Sio] [Sii]

] [aoai

](11)

bi = [C] ai (12)



Interconnected Networks - A Matrix Approach ...Cont’d

We should find [S] which is defined by the equation

bo = [S] ao. (13)

Substituting (12) and (13) in (11) gives,

[[S] ao[C] ai

]=

[[Soo] [Soi][Sio] [Sii]

] [aoai

].

Solving the above matrix equation gives [S] = [Soo] + [Soi] ([C]− [Sii])−1 [Sio].



Outline


2 S Paramters



5 V, I, and Z ***




Equivalent Voltages and Currents ... TEM Lines



Equivalent Voltages and Currents ... non-TEM Lines



Physical Interpretation of Impedance



Even and Odd Properties of Z, Γ



Outline


2 S Paramters



5 V, I, and Z ***




Diaphragms & Irises

Symmetricalinductive

diaphragm

Asymmetricalinductive

diaphragm

Symmetricalcapacitivediaphragm

Asymmetricalcapacitivediaphragm

Rectangularresonant

iris

Circularresonant

iris

Change in height

Change in width

E

EEquivalent circuit

Z01 Z02

Equivalent circuit

Z01 Z02

Equivalent circuit

Equivalent circuit

Equivalent circuit



Microstrip-line Discontinuities

Z0c

Z0m

Z01

Z01 Z02

Z02

Z03

Z0 Z0

Z01

Z01

L1

L

L2

L3

Z02

Z03

C

Z02

Z0

Cp Cp

C

LL

Cg

Z0 Z0

C

Z0

C1 C2Z0c Z0m



Microstrip Bends

W

Right-anglebend

Wa

Mitered bends

Mitered T-junction

Mitered step

Wa

W

Swept bend

r 3 W


Microwave Network Analysis - Arraytool · PDF file07.05.2015 · Microwave Network...

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Transcript of Microwave Network Analysis - Arraytool · PDF file07.05.2015 · Microwave Network...