CALIFORNIA STATE POLYTECHNIC UNIVERSITY, POMONA

ELECTRICAL COMPUTER ENGINEERING DEPARTMENT

ECE 448

RF/MICROWAVE CIRCUIT DESIGN CLASS NOTES

RF/MICROWAVE CIRCUIT DESIGN-1

Narayan Mysoor, Ph.D, P.E

SCATTERING (5) PARAMETERS

To characterize a microwave structure or network having an input port and an output port, any of the scatteringS,

impedance Z, and admittance Y can be employed. A two-port network can be completely represented by four

scattering S (or Z, or Y) parameters, two independent and two dependent variables. Denoting the characteristic

impedance by Z0 and defining

And

(Voltage wave incident on port 1) al = ;z;

(Voltage wave reflected on port 1) bl = ----'------==------'---;z;

(Voltage wave incident on port 2) az =

(Voltage wave reflected on port 2) hz = -------------

jl;

The two port network can be described by the relations

And

The scatteringS parameters are then given as

(E.1)

(E.2)

(E.3)

(E.4)

(E. 5)

(E.6)

(E. 7)

(E. 8)

(E. 9)

(E.10)

For the input excited by a signal with matched source and the output terminated in a matched load, S11 and S21 represent the input reflection coefficient and the forward transmission coefficient (insertion loss), respectively.

Similarly for the output excited with a matched source and the input terminated in a matched load, S22 and S12 denote the output reflection coefficient and the reverse transmission coefficient, respectively.

REFERENCES

Hewlett Packard, S-parameter Design, Application Note 154. Hewlett Packard, Palo Alto, CA 1972

Input return loss at port-1 = 20 log IS11 1 dB

Input VSWR = [ 1 + IS11Il I [ 1 - ISull

Forward insertion Loss or gain from port-1 to port-2 = 20 log IS21 I dB

Output return loss at port-2 = 20 log IS22 I dB

Output VSWR = [ 1 + 1Sz2 ll I [ 1-ISzzl]

Tf* 821 Tb*

a1 - - b2 811 822 b1 - - a2 Tf

812 Tb

FIGURE 1: S-MATRIX EQUIVALENT CIRCUIT REPRESENTATION OF A 2-PORT NETWORK

Dual Directional Coupler

Device Under Test

Dual Directional Coupler

500 Load

FIGURE 2: MICROWAVE TEST SET FOR EVALUATING Sll AND 521 OR 522 AND 512

IMPEDANCE (Z) PARAMETERS

Similar representation of a two port network with impedance Z and admittance Y parameter is possible. Thus in z representation,

(E.ll)

And

(E. 12)

Where "V" and "I" represent the voltage and current variables at respective port 1 or 2. The parameters Z12, Z21 and Z22 are measured by applying an voltage source to one port, while the other port is open circuited.

V1j Zu =I 1 l z=O

V2 1 z21 =I 1 lz =O

Vzl z22 =I 2 /1 =0

v1 j zl2 =I 2 /1 =0

ADMITTANCE (Y) PARAMETERS

(E.13)

(E. 14)

(E. 15)

(E. 16)

(E.17)

(E. 18)

In this case, Y parameters are measured by exciting one port with a voltage source, while the other port is short

circuited.

Hence

(E.19)

/21 y 21 =v-

1 Vz= O

(E. 20)

(E. 21)

(E. 22)

EQUIVALENT CIRCUIT REPRESENTATION OF A GENERAL TWO-PORT RECIPROCAL MICROWAVE CIRCUIT

TF TB

a1 - - b2 b1 - - a2 TF TB

FIGURE 3: S PARAMETERS

I Z11-Z12 I I I

I I Z22-Z12 I I I

Te*

Zo I Z12 I

Zo

l Te

FIGURE 4: T EQUIVALENT CIRCUIT

Yo Yo

TF -----+----~--------------~----+----- Te

FIGURE 5: PI-EQUIVALENT CIRCUIT REPRESENTATIONS OF A TWO-PORT NETWORK

GENERALIZED 2-PORT SCATTERING PARAMETERS (S1]

Zo=50 ohms

Network [S] Zo=50 ohms

FIGURE 6: MEASURED [S) IN SO OHMS TERMINATIONS

Find [51 ) in terms of given load and source impedances as shown in Figure 7.

Zs

ZL

FIGURE 7: NETWORK WITH GIVEN LOAD AND SOURCE IMPEDANCES

Solution:

Where

Ai = (1 - ri?.JC1 ~ lrd2) =The iithelement of the diagonal matrix corresponding to the ithport 1- ri

z: -zi r = ---

t z; +Z =The iithelement of the diagonal reflection coefficient matrix corresponding to the ithport

Z'i =The arbitrary terminating impedance of the ithport

Zi =The original terminating impedance of the ithport which the measurements were made.

(Reference: High Frequency Amplifiers, R. S. Carson, Wiley, 1975)

GENERALIZED TWO-PORT SCATTERING PARAMETERS [S1]

For a 2-Port network the new generalized scattering parameters [51 ] in terms of measured parameters [S] and

arbitrary load and generator impedance are given as

(G.l)

(G. 2)

(G.3)

(G.4)

SIMPLE GENERALIZED TWO-PORT SCATTERING PARAMETERS [S1 ]

For a two-port network using f 5 = Zs- Zo and rL = zL- Zo in equations (G-1) to (G-4), we obtain Z5 +Z0 ZL+Z0 simple equations for

, S12Sz1fs Szz = Szz + 1 f S - s 11

(G. 5)

(G.6)

(G.7)

(G. B)

SIGNAL FLOW GRAPHS OF TWO-PORT NETWORKS

Variables are designated as nodes.

S-parameters are designated as branches.

Branches enter dependent variable nodes.

Branches exit independent variable nodes.

Each node is equal to the sum of the branches entering that node.

Note: Modern computers have diminished that importance of signal flow graphs, but the concepts are

still valuable.

S21 b2 a1----- - ...

S12

b1 = S11a1+S12a2

Independent variable a1 and a2 Dependent variable b1 and b2

a1 S21 c2

S11 S22

b1 S12 a2

b2 = S21a1+S22a2

S22

a2

Two-port Network

FIGURE 8: COMPLETE FLOW GRAPH FOR A TWO-PORT NETWORK

FLOW GRAPH OF Two-PORT NETWORK

~ a1 ~ b2 Zs ~r-b_1 _____ ....,~ a2

Zo 2-port network Zo ZL

rs

FIGURE 9: TWO-PORT NETWORK

bs a1'=a1 a1 S21 b2 b2'=b2

S11 S22

rs

b1'=b1 b1 S12 a2 a2'=a2

Source Flow Graph Network Load Flow Graph

bs a1 S21 b2

rs rL

FIGURE 10: COMBINED SIGNAl FlOW GRAPH

NON-TOUCHING LOOP RULE (MASON'S RULE)

A First Order Loop is defined as the product of the branches encountered in a journey starting from a

node and moving in the direction of arrows back to that original node.

A Second Order Loop is the product of any two-non-touching first order loops.

A Third Order Loop is the product of any three non-touching first order loops.

The non-touching loop rule determines the ratio of two variables, one being dependent, the other being

independent. (Example: b2/al)

P1 [1- H(l)1 + LL(2)1 - H(3)1 + ] + P2 [1- H(2)1 + ]

T=----------------------~------~--------------1-H(l) + LL(2)- LL(3) + ...

Where:

H(l): is the sum of all first order loops

H(n): is the sum of all nth order loops

P1 P2 Pn: are the paths connecting the variables in question

H(l)1 : is the sum of those first order loops that do not touch path P1

LL(n)n: is the sum of those nth order loops that do not touch path Pn

T: is t he ratio of dependent to independent variables

Example 1:

Output reflection coefficient of a two-port network w ith an arbitrary source impedance and ZL = Z0

rs 511

Find T:

521

522 \

512

, hz T = s22 =-

Gz

Path connecting the variables in question are: P1 - S22 and P2 - S12 f 5S21

First order loops L(l) = f 5S11

5 '22 =?

Example 2:

Use the loop rule to determine the transducer gain, ratio bz (see Fig. x) in term of network scattering parameters bs

and f 5 and fL

bs a1 821 b2

rs

FIGURE 11: FLOWGRAPH OF A TWO-PORT NETWORK EMBEDDED BETWEEN A SOURCE AND A LOAD

Need to find bz : bs

Path connecting the variables in question

First order loop:

Second order loop:

MULTI-PORT NETWORK

Example:

Directional Couplers

( Power Input to 1 )

Coupling Factor (dB) = lOlog1o p 0

f 4 ower utput rom

( Power Input to 4 )

Directivity (dB) = 10log10 p 0 f 3 ower utput rom

( Power Input to 1 )

Isolation (dB) = 10log10 p 0 3 ower utput at

( Power Input to 1 )

Insertion Loss (dB) = lOlog1o P 0 2 ower utput at

Directivity = Isolation- Coupling Factor

$-MATRIX OF DIRECTIONAL COUPLER

(2n+ 1 }~g/4

PORT1 PORT2

-~,~,:,~--- --r-~ --- ----------- --~- - - ~- ----- ~0

Which leads to the following $-Matrix

Symmetry of the Junction may now used to simplify the matrix: 512 = 534

The final matrix involves only two variables

To establish the realizability of the junction, apply the unitary condition: S 5* =I

This gives

Satisfies Conservation of Energy

Suggests One Possible solution.

Possible solution at a suitable pair of terminal is

And

Where p and q are real numbers.

The scattering matrix for the Symmetrica l Directional Coupler is therefore,

5 = (~ - 0 jq

~ ~q 16) jq 0 p 0 p 0

Microwave Power Source

Power Meter

Amplifier Design

Double Stub Tuner

Device Test Jig

Power Meter

Device Bias - T

FIGURE 13: PRACTICAL REALIZATION OF A MICROWAVE FET AMPLIFIER

STABILITY CRITERIA

S'11

Zs

rs Zin

Circuit is Unstable (Oscillators) if:

Loop impedance is zero.

The system determinant is zero

The characteristic equation is zero

Two-port Network (MESFET)

r Vs ~ FIGURE 14: TWO-PORT NETWORK

{Note that these are just three forms of the same statement.)

Circuit is Stable if:

Loop impedance is greater than zero, e.g., the circuit is lossy.

For the circuit, Z5 + Zin > 0 for stability,

Where

Solving yields:

Similarly for the other port:

Z = l+fs 5 1- f s

1Sl1 fs l < 1 provides stability at input port 1.

IS~ 2 rLI < 1 will insure stability at the output port 2.

(Note that if either port is unstable, both ports are unstable)

ZL

TYPES OF STABILITY

1. Absolutely Unstable if:

2. Conditionally Stable if:

I 1 JSuJ > lfsl

1 JS~2J > jfJ

JS~2J > 1

For some positive rea l terminations (typica l "real world")

3. Unconditionally Stable if:

JS~2J < 1

For All positive real terminations:

Be sure to check stability at all ports and frequencies

STABiliTY CIRCLES

For stability, the boundary conditions are

IS' I = s + 12 21 L < 1 I s s r I 11 11 1 - rLs22

I s s r I IS' I = s + 12 21 s < 1 22 22 1 _ r s s 11 To find the limiting values, set

IS~2I = 1

And solve for rL and fs

The solution for rL will lie on a circle of center

And radius

Similarly on the f 5 plan

The solution for f 5 will lie on a circle of center

And radius

Where C1 = S11 - flS;2

This value of n. yeilds IS'11I=1

FIGURE 15: A TYPICAL STABILITY CIRCLE ON THE rL PLANE

FOUR DIFFERENT REGIONS OF STABILITY

Potentially Unstable

Stable

Case 1: IS111 < 1 The Smith Chart Center is STABLE

Stable

Case 2: !Sui > 1 The Smith Chart Center is UNSTABLE

UNCONDITIONAL STABILITY

For unconditional stablility, the stability circles must be completely outside as shown in Fig16 (a) or totally enclose

the Smith chart as sown in Fig16 (b).

Fig (a)

Potentially Unstable

Fig {b)

FIGURE {16) rL PLANE: FIG. (A): THE STABILilY CIRCLE OUTSIDE THE SMITH CHART, ITS INTERIOR BEING UNSTABLE

FIG. (B): THE STABILilY CIRCLE ENCLOSING THE SMITH CHART, ITS INTERIOR BEING STABLE

Thus for unconditional stability: llrs1 1- R51 1 > 1 for IS11 1 < 1

llrsz l- Rs2 1 > 1 for IS22I < 1

UNCONDITIONAL STABILITY AND STABILITY FACTOR (K)

In summary, the conditions, in terms of the transistor parameters, for unconditional (or inherent) stability are:

IS12S21I < 1 -1Sul2

IStzSztl < 1-ISd2

OR

l-11 < 1,

Alternately, defining the stability factor, K:

Thus for Unconditional Stability:

K > 1,

ISul < 1} ISd < 1

I Sui < 1, and ISzz I < 1

Note: These equations describe the necessary, and the sufficient conditions.

TRANSDUCER POWER GAIN ( Gr) For the device terminated with arbitrary load and source impedances, the transducer power gain is defined as

Power Delivered to the Load G = ------...,-------,-------

r Power Available from the source

Where S~ 1 is given by equation G.S as

rtand rz have been replaced by rLand rs in equations G.l through G.4, where

Note: Gr (dB)= lOlogJS~1 l 2dB = 20 log S~1 dB

SIMULTANEOUS CONJUGATE MATCH

Input Match

Gs

Device Networ1<

Go

rsM = (S'11 )*

ZL

rLM = (8'22)*

If the transistor is found to be unconditionally stable, it can be simultaneously complex conjugate matched. This

leads to maximum power gain in the circuit.

The load and generator reflection coefficients to conjugate match the input and output ports are

s s ~ ~ - S' - s + 12 21 SM LM - 22 - 22 1 _ S r usM

Solving these two equations simultaneously for rLM and rsM yields

Ci (s1 ,/Bt- 4IC11 2) f sM = ZIC1I 2

C~ (s2 ,/B~- 4IC212) rLM = ZIC2I2

Where the subscript M represents the matched or optimum condition.

If the computed values for B1 and B2 are negative, the plus sign should be used. Conversely, if B1 and B2 are

positive, the minus sign should be used, where

Bt = 1 + 1Sul2 -IS22I2 -lt.l2

Bz = 1 + 1Szzl 2 - 1Sul2 -lt.l2

For complex conjugate matching, the maximum transducer gain becomes, from Gr = IS~ 1 1 2

Where the minus sign is used if B is positive and the plus sign is used if B is negative.

UNILATERAL TRANSDUCER GAIN

When 512 ~ 0, the device is termed unilateral because there is no internal feedback. Most well-designed devices

are unilateral. Setting 512 = 0 in Gr = 15~ 1 1 2 defines the unilateral transducer gain as

Power Delivered I G =-----,--

ru Power Available s12

=o

_ (1-lfsl2) 2 (1 -lfd2)

- 11- 5ufsl2 1521 l 11- 522fd2

= GsGoGL = G5 (dB) + G0 (dB) + GL(dB)

rs 8'11

Go

rLM 8'22

GL Zo

FIGURE 17: SUBDIVISION OF UNILATERAL AMPLIFIER INTO THREE SEPARATE GAIN STAGES

MAXIMUM UNILATERAL TRANSDUCER GAIN

When f 5 = 5;11 Gs is maximum, i.e., G =-1-s,max 1-ISalz

Similarly, when rL = 52z, GL is maximum, given by GL,max = 1-l;z2J2

Thus for complex conjugate image matching, the maximum unilateral transducer gain becomes,

- 1 2 1 Gru,max- 11 _ 5111

zl5z1 1 11 - 5zz12

= Gs,maxGoGL,max = Gs,max(dB) + G0 (dB) + GL,max(dB)

8*11 822

Input 811 Impedance Matching

8*22 Output Impedance Matching

Zo

EXAMPLE OF NARROW BAND AMPLIFIE R DESIGN

Design an amplifier to operate between a matched source and load impedance of 500 and to provide maximum

gain at 750 MHz. The scattering parameter of the transistor are:

S11 = 0.277 L- 59

S22 = 0.848L- 31

CASE 1: BILATERAL OR NON-UNILATERAL DESIGN

The scattering coefficient are now used to calculate

C1 = S11 - LlSi2 = 0.12L- 135.4

C2 = S22- LlSi1 = 0.768L- 33.8

1 IS 12 IS 12 + 181 2 K = - 11

21- 22! = 1.033 > 1,and llll < 1 s12s21

In addition,

1- IS11 12 = 0.924

1 - 1Szzl2 = 0.281

are satisfied.

Hence, the transistor is unconditionally stable. And complex conjugate matching can be used to realize the

maximum transducer gain.

81 and 8 2 are both positive, hence negative signs must be used to calculate r5M, rLM and Crmax

The transducer power gain is

Input Match ing Circu it :

Output Ma tching Circuit :

Gr.max = ~~:;I (K J K2 - 1) = 19.087 = 12.807 dB

Ls

8.44nH

Cs 8.913pF

FIGURE 18: BILATERAL DESIGN AT 750 M HZ

CL

684.5fF

LL 22.6nH

XLs = WL5 = (0.4 + 0.395)Z0 ~ L5 = 8.44nH

1 XcL = - C = 6.2Z0 ~ CL = 0.6845pF

W L

ZL son

r~~-~ -;::. o.73 /t3S.+" NORMALIZED IMPEDANCE AND ADMITTANCE COORDINATES

r:_H =- ().%.) I 33>, 8 0 . 0/-1. o!' . 0~ u, ~

o.3-l

0.16

o. ~~s-

CASE 2: U NILATERAL DESIGN

The transistor is assumed unilateral, i.e., S12 = 0

The stability conditions in this case are:

ISul < 1

ISzzl < 1

The above are satisfied for the given transistor.

Complex conjugate matching can be applied to realize the maximum unilateral gain.

This yields

lrd = s 22 = o.B48L32

The unilateral amplifier gain is

- 1 2 1 GTu.max -

11_ Sui2 1Sztl 11

_ Szzl2 = 14.21 = 11.53dB

Where as GTmax = 12.807 dB, a difference of about 1.3 dB.

Matching network can be designed by network synthesis.

Example 2: MESFET Amplifier Design at 6 GHz (Transmission line matching circuit)

MESFET $-Parameters at 6 GHz in a 50 ohm system:

Z0 =50 il

S11 = 0.614L- 167.4

S12 = o.o46L65

S21 = 2.187 L32.4

S22 = o.716L- 83

K = 1.1296 > 1 and IS11 1 < 1 & ISzz l < 1

The FET is UNCONDITIONALLY STABLE

C1 = S11 - t.Si 2 = 0.371L- 169.75

Cz = S22 - t.S~1 = 0.507 L - 84.4 7

Gr,max = ~~:~I ( K .j K2 - 1) = 14.58 dB fsM = 0.868 L 169.75

fLM = 0.9 L 84.97

RF coupling capacitor Zo=SOO

Zo=SOO

L2=0.056Ag

Zo=SOO

L1 =0.204Ag rsM

Open-Stub

DC Bypass capacitor

(/) w 1-

Example 3: MESFET Amplifier Design at 6 GHz (Lumped element matching circuits)

MESFET S-Parameters at 6 GHz in a 50 ohm system:

Z0 =50 .fl

S11 = 0.614L-167.4

S21 = 2.187 L32.4

S22 = 0.716L-83

K = 1.1296 > 1, IS111 < 1 & ISd < 1

The FET is unconditionally stable.

CB

!!.. = 0.34195L113.16

81 = 0.747

8 2 = 1.019

cl = o.37L-169.75

C2 = 0.507 L- 84.47

Gr,max = ~~:: 1 ( K J K2 - 1) = 14.58 dB

Ls

0.464nH

Cs 1.883pF

fsM = 0.868L169.75

fLM = 0.9L84.47

Lload CB2

Lload2 0.45nH

ZL 500

Input Matching Circuit:

B c,sh = wCs,sh = 3.55Yo = 3~~5

3.55 Cs,sh = 50 X 2rr X 6 X 109 = 1883 pF

X s,se = wLs,se = (0.26 + 0.09)Z0 0.35 X 50

Ls.se = 2rr X 6 X 109 = 0.464 nH

Output Matching Circuit:

1 B Lsh = -- = 2.95Y0 ' WLJ...sh

50 LL,sh = 2.95 X 2rr X 6 X 109 = 0.4SnH

XL,se = wLL,se = (1.09 - 0.30S)Zo 0.785 X 50

LL se = 2 6 09

= 1.041 nH rrx x1

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