Basics on High Frequency Circuit Analysis

Basics on High Frequency Circuit Analysis Dr. José Ernesto Rayas-Sánchez

April 22, 2020

1

1

Basics on High Frequency Circuit Analysis

Dr. José Ernesto Rayas Sánchez

Most of the figures of this presentation were taken from Agilent Technologies Educator’s Corner: 1999 RF Design and Measurement Seminar, David Ballo, Joe Civello, Ed Henicle, Sara Meszaros, Andy Potter, BoydShaw, My Le Truong

2Dr. J. E. Rayas Sánchez

Electrical Size

Low frequencies wavelengths >> wire length current (I) travels down the wires easily for efficient power transmission measured voltage and current not dependent on position along the wire

High frequencies wavelength or < length of transmission medium need transmission lines for efficient power transmission matching to characteristic impedance (Z0) is very important for low

reflection and maximum power transfer measured envelope voltage dependent on position along line

I


April 22, 2020

2


The Need of Transmission Line Theory

For analog circuits:

If the physical length of the transmission media is larger 10% of the wavelength of the highest frequency of interest

For digital circuits:

If the propagation time in the longest transmission path is larger than 10% of the fastest transition time


Common Transmission Media

Uniform Interconnects

microstrip

h

w

coplanar

w1

w2

r

waveguide

twisted-paircoaxial

b

a

h

w

(Hewlett-Packard's RF Design and Measurement Seminar, 2000)

r


April 22, 2020

3


Common Transmission Media (cont.)

Practical interconnects can be decomposed in segments of uniform interconnects (usually necessary)

Components(Chip + Pkg)

PCB(Motherboard)

PCB(add-in card)

Connector

(H. Heck, 2005)



Practical interconnects have discontinuities and imperfections

(M. Resso, 2016)


April 22, 2020

4



High-speed BGA (Ball Grid Array) packagesChip

Board

Flip chip interconnect (bump)

Signal layer

Signal layer

Power/ground plane

Power/ground plane

Through via

Burried via

Complete signal path (chip to board)

Chip carrier (substrate)

BGA ball

Blind via


From Lumped Circuits to Distributed Circuits

(A. Weisshaar, Tutorial on High-Speed Interconnects, IMS June 2004, Fort Worth, TX)


April 22, 2020

5


Transmission Line Model

The interconnect is modeled using an infinite number of RLCG sections

(R. Ludwig and P. Bretchko, RF Circuit Design, Prentice Hall, 2000)



The interconnect is modeled using an infinite number of RLCG sections



April 22, 2020

6



Generic equivalent circuit for each section (R, L, C and G are per unit length)

The interconnect is modeled using an infinite number of these sections, making


0z


Transmission Line Equations

Time-Domain (Telegrapher Equations)

Telegrapher Equations in Frequency-Domain

ttzi

LtzRiz

tzv

),(),(

),(

ttzv

CtzGvz

tzi

),(),(

),(

)()()(

zILjRdz

zdV

)()()(

zVCjGdz

zdI


April 22, 2020

7


Transmission Line Equations (cont.)

Wave equation (frequency-domain)

whereis the complex propagation constant

jCjGLjR ))((

Solutions to the wave equation are

zz eVeVzV 00)( zz eIeIzI 00)(

0)()( 2

2

2

zVdz

zVd 0)()( 2

2

2

zIdz

zId

incident wavesreflected waves


Transmission Line Equations (cont.)

Solutions in the frequency domain

Solutions in the time domain

zz eVeVzV 00)(zz eIeIzI 00)(

zz eztVeztVtzv )cos(||)cos(||),( 00

Phase velocity, wave velocity or propagation speed

2p

f

v

jCjGLjR ))((

Wavelength

dtdz

vp

(speed at which a constant phase point travels down the line)


April 22, 2020

8


Transmission Line Symbol

(Lossy) Transmission line

Length along the line

Z0,

l

Z0,

0l

ZL

z


Characteristic Impedance

Characteristic impedance of the TL

CjGLjR

Z

0

0

0

0

00 I

VIV

Z


April 22, 2020

9


Reflection Coefficient

Reflection coefficient along the line, l

Reflection coefficient at the load,

l

l

l

l eVV

eVeV

l

2

0

0

0

0)(

0L

0L

0

0

0

0)0(ZZZZ

II

VV

ll

l

l

l

l eII

eIeI

l

2

0

0

0

0)(

Z0,

0l

ZL


Input Impedance

Input impedance along the line

)tanh()tanh(

)()(

)(L0

0L0in lZZ

lZZZ

lIlV

lZ

Z0,

0l

ZLZin


April 22, 2020

10


Modeling Uniform Interconnects



Modeling Uniform Interconnects (cont.)

Parasitic effects associated to each transmission media

– Capacitance between conductors, C

– Resistance of conductors (conductor losses), R

– Inductance of conductor loops, L

– Dielectric conductivity (dielectric losses), G

R, C, L, and G must be determined per unit length


April 22, 2020

11


Interconnect Shunt Capacitance


F/m) 10854.8( 120


Fringing Capacitance Effect



April 22, 2020

12


Interconnect Series Inductance


H/m) 104( 70


Interconnect Series DC Resistance



April 22, 2020

13


Skin Effect



EM-Simulation of Conductor Current Distribution



April 22, 2020

14


Proximity Effect



Proximity and Skin Effects

E. Bogatin, “Essential principles of signal integrity,” IEEE Microwave Magazine, vol. 12, pp. 34-41, Aug. 2011.


April 22, 2020

15


Edge and Indy Effects – Example

W = 10 mil L = 40 milH = 5 mil r = 4.2dielectric loss tan = 0.017PCL-FR-226 Polyclad Laminates

EM simulation using SonnetAt 1 MHz:

rH

W

L

L

L

W


Edge and Indy Effects – Example (cont.)



rH

W

L

L

L

W


April 22, 2020

16





rH

W

L

L

L

W




EM simulation using SonnetAt 1 GHz:

rH

W

L

L

L

W


April 22, 2020

17


Interconnect Shunt Conductance



Loss Tangent of Typical Materials



April 22, 2020

18


Power Transfer Efficiency

RS

RL

Maximum power is transferred when RL = RS

RL / RS

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5 6 7 8 9 10

Lo

ad

Po

wer

(no

rmaliz

ed

)



Power Transfer Efficiency (cont.)

For complex impedances, maximum power transfer occurs when ZL = ZS* (conjugate match)

Zs = R + jX

ZL = Zs* = R - jX

Zo

Zo

Rs

RL

+jX

-jX

At high frequencies, maximum power transfer occurs when RS = RL = Zo



April 22, 2020

19


The Smith Chart

-90 o

0o180

o+-

.2

.4

.6

.8

1.0

90o

0 +R

+jX

-jX

Smith Chart maps rectilinear impedanceplane onto polar plane

Rectilinear impedance plane

Polar plane

Z = ZoL

= 0

Constant X

Constant R

Z = L

= 0 O

1

Smith Chart

(open)

LZ = 0

= ±180 O1

(short)



0.2

0.5

1.0

2.0

5.0

+0.2

-0.2

+0.5

-0.5

+1.0

-1.0

+2.0

-2.0

+5.0

-5.0

0.0

The Smith Chart – Interpretation

0L Z LZ

0L ZZ

0L jZZ

0L jZZ

(open circuit)(short circuit)

(pure inductive)

(pure capacitive)

(matched)Circles of constant XL

Circles of constant RL

0

180101

901

901


April 22, 2020

20


Lightwave Analogy to RF Energy

RF

Incident

Reflected

Transmitted

Lightwave



Transmission Line Terminated with Zo

For reflection, a transmission line terminated in Zo behaves like an infinitely long transmission line

Zs = Zo

Zo

Vrefl = 0 (all the incident power is absorbed in the load)

Vinc

Zo = characteristic impedance of transmission line



April 22, 2020

21


Transmission Line Terminated with Short, Open

Zs = Zo

Vrefl

V inc

For reflection, a transmission line terminated in a short or open reflects all power back to source

In phase (0°) for openOut of phase (180°) for short



Transmission Line Terminated with 25

Zs = Zo

ZL = 25

Vrefl

V inc

Standing wave pattern does not go to zero as with short or open



April 22, 2020

22


Reflection Parameters

dB

No reflection(ZL = Zo)

RL

SWR

0 1

Full reflection(ZL = open, short)

0 dB

1

=Z L - Z O

ZL + OZ

ReflectionCoefficient

=Vreflected

Vincident=

= Return loss = -20 log(),

SWR = Emax

Emin

=1 + 1

Standing Wave RatioEmax

Emin



Standing Wave Ratio – EM Simulation

Microstrip lineLaminate: Rogers RO3003

H = 5 mil

r = 3

tan() = 0.0013 at 10 GHz

Lossy metals: Cu = 5.8×107 S/m

t = 0.6 mil (half-once copper)

W = 12.5 mil

L = 187.5 mil


April 22, 2020

23


Standing Wave Ratio – EM Simulation (cont.)

Using Zport1 = Zport2 = 50

f = 0.1 GHz f = 30 GHz


Standing Wave Ratio – EM Simulation (cont.)

Using Zport1 = 50 , Zport2 = 1 K

f = 0.1 GHz f = 30 GHz


April 22, 2020

24


Transmission Parameters

VTransmitted

VIncident

Transmission Coefficient = =VTransmitted

V Incident=

DUT

Gain (dB) = 20 Log VTrans

V Inc

= 20 log

Insertion Loss (dB) = ‐ 20 Log VTrans

V Inc

= - 20 log

Insertion Phase (deg) =

VTrans

V Inc

=


N-Ports Networks (Linear Circuits)



April 22, 2020

25


Impedance Matrix Representation (Z)

Each element of matrix Z is given by

NV

V

V

2

1

V

NI

I

I

2

1

I

ZIV

NNNN

N

N

ZZZ

ZZZ

ZZZ

21

22221

11211

Z

jkIj

iij

k

IV

Z

for 0


Admittance Matrix Representation (Y)

Each element of matrix Y is given by

NV

V

V

2

1

V

NI

I

I

2

1

I

YVI

NNNN

N

N

YYY

YYY

YYY

21

22221

11211

Y

jkVj

iij

k

VI

Y

for 0


April 22, 2020

26


Z-Parameters for 2-Port Networks

2

1

V

VV

2

1

I

II

ZIV

2221

1211

ZZ

ZZZ

Equivalent circuit:


Y-Parameters for 2-Port Networks

2

1

V

VV

2

1

I

II

YVI

2221

1211

YY

YYY

Equivalent circuit:


April 22, 2020

27


H-Parameters (Hybrid) for 2-Port Networks

2

1

2

1

V

I

I

VH

2221

1211

HH

HHH

Equivalent circuit:


The Scattering Matrix (S)

NV

V

V

2

1

V

NV

V

V

2

1

V

SVV

NNNN

N

N

SSS

SSS

SSS

21

22221

11211

S

jkVj

iij

k

VV

S

for 0

Vk+ = 0 if we terminate port k with a matched load (ZLk = Z0k)

V1

I1I1

+V1+

I1V1

Z01

Vk

IkIk

+ Vk+

Ik Vk

Z0k


April 22, 2020

28


The Scattering Matrix Representation (S)

They can be more easily obtained at high frequencies:

– Incident and reflected waves can be measured using a Vector Network Analyzer (VNA)

– They do not require “shorts” or “opens” (active devices might oscillate or self-destroy)

They are more directly related to high-frequency effects (, T, IL, RL, SWR, etc.)

We can convert back and forth between S, Y and Z parameters


Measuring S-Parameters

S 11 =Reflected

Incident=

b 1

a 1 a 2 = 0

S 21 =Transmitted

Incident=

b2

a 1 a 2 = 0

S 22 =Reflected

Incident=

b 2

a 2 a 1 = 0

S 12 =Transmitted

Incident=

b1

a 2 a 1 = 0

Incident TransmittedS 21

S 11Reflected

b 1

a 1

b 2

Z 0

Loada2 = 0

DUTForward

1IncidentTransmitted S 12

S 22

Reflected

b 2

a2

b

a1 = 0

DUTZ 0

Load Reverse


April 22, 2020

29


Meaning of the S-parameters (cont.)

S11 : forward reflection coefficient (input match)

S22 : reverse reflection coefficient (output match)

S21 : forward transmission coefficient (gain or loss)

S12 : reverse transmission coefficient (isolation)

ikVi

ikVi

iii

k

kVV

S

for 0

for 0

iiiS

jiij TS jkVji

jkVj

iij

k

k

TVV

S

for 0

for 0


An RF Prototype


April 22, 2020

30


Hybrid Microwave Integrated Circuits (cont.)

M. Pozar (1998), Microwave Engineering. Amherst, MA: John Wiley and Sons.


Monolithic Microwave Integrated Circuits (MMIC)

M. Pozar (1998), Microwave Engineering. Amherst, MA: John Wiley and Sons.


April 22, 2020

31


High Speed Interconnects

J.C. Rautio, Rigorous Evaluation of Worst Case Total Crosstalk in the Time Domain Using Frequency Domain Scattering Parameters, 2001 High-Performance System Design Conference


High Speed Interconnects (cont.)

J.C. Rautio, Rigorous Evaluation of Worst Case Total Crosstalk in the Time Domain Using Frequency Domain Scattering Parameters, 2001 High-Performance System Design Conference

Basics on High Frequency Circuit Analysis

Documents

Transcript of Basics on High Frequency Circuit Analysis