Basics on High Frequency Circuit Analysis -...

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

April 25, 2018

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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 wires easily for efficient power transmission measured voltage and current not dependent on position along 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+ -


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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


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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)


Common Transmission Media (cont)

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


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From Lumped Circuits to Distributed Circuits

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


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)


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The interconnect is modeled using an infinite number of RLCG sections




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


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Transmission Line Equations

Time-Domain (Telegrapher Equations)

Telegrapher Equations in Frequency-Domain

ttzi

LtzRiz

tzv

),(),(

),(

ttzv

CtzGvz

tzi

),(),(

),(

)()()(

zILjRdz

zdV

)()()(

zVCjGdz

zdI


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


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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)


Transmission Line Symbol

(Lossy) Transmission line

Length along the line

Z0,

l

Z0,

0l

ZL

z


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Characteristic Impedance

Characteristic impedance of the TL

CjGLjR

Z

0

0

0

0

00 I

VIV

Z


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


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Input Impedance

Input impedance along the line

)tanh()tanh(

)()(

)(L0

0L0in lZZ

lZZZ

lIlV

lZ

Z0,

0l

ZLZin


Modeling Uniform Interconnects



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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


Interconnect Shunt Capacitance


F/m) 10854.8( 120


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Fringing Capacitance Effect



Interconnect Series Inductance


H/m) 104( 70


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Interconnect Series DC Resistance



Skin Effect



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EM-Simulation of Conductor Current Distribution



Proximity Effect



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Proximity and Skin Effects

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


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


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Edge and Indy Effects – Example (cont)



rH

W

L

L

L

W





rH

W

L

L

L

W


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EM simulation using SonnetAt 1 GHz:

rH

W

L

L

L

W


Interconnect Shunt Conductance



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Loss Tangent of Typical Materials



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

)



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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



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)



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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


Lightwave Analogy to RF Energy

RF

Incident

Reflected

Transmitted

Lightwave



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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



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



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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



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



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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


Standing Wave Ratio – EM Simulation (cont)

Using Zport1 = Zport2 = 50

f = 0.1 GHz f = 30 GHz


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Standing Wave Ratio – EM Simulation (cont)

Using Zport1 = 50 , Zport2 = 1 K

f = 0.1 GHz f = 30 GHz


Transmission Parameters

V TransmittedV

Incident

Transmission Coefficient = =VTransmitted

VIncident=

DUT

Gain (dB) = 20 Log V Trans

V Inc

= 20 log

Insertion Loss (dB) = - 20 Log V Trans

V Inc

= - 20 log

Insertion Phase (deg) = V Trans

V Inc

=


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N-Ports Networks (Linear Circuits)



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


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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


Z-Parameters for 2-Port Networks

2

1

V

VV

2

1

I

II

ZIV

2221

1211

ZZ

ZZZ

Equivalent circuit:


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Y-Parameters for 2-Port Networks

2

1

V

VV

2

1

I

II

YVI

2221

1211

YY

YYY

Equivalent circuit:


H-Parameters (Hybrid) for 2-Port Networks

2

1

2

1

V

I

I

VH

2221

1211

HH

HHH

Equivalent circuit:


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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 = Zok)

V1

I1I1

+V1+

I1V1

Zo1

V 2

I 2

I 2+

V 2+

I 2

V 2

Z o2

Vk

IkIk

+ Vk+

Ik Vk

Zok

VN

I N

IN+

VN+

IN VN

ZoN

N-port Linear Network


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


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Measuring S-Parameters

S 11 = Reflected

Incident=

b1

a 1 a2 = 0

S 21 =Transmitted

Incident=

b2

a 1 a2 = 0

S 22 = Reflected

Incident=

b2

a 2 a1 = 0

S 12 =Transmitted

Incident=

b1

a 2 a1 = 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


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


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An RF Prototype


Hybrid Microwave Integrated Circuits (cont)

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


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Monolithic Microwave Integrated Circuits (MMIC)

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


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


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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

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