ECE 546 – Jose Schutt‐Aine 1
ECE 546Lecture ‐13
Scattering ParametersSpring 2020
Jose E. Schutt-AineElectrical & Computer Engineering
University of [email protected]
ECE 546 – Jose Schutt‐Aine 2
Transfer Function Representation
Use a two-terminal representation of system for input and output
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Y-parameter Representation
1 11 1 12 2
2 21 1 22 2
I y V y VI y V y V
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Y Parameter Calculations
2 2
1 211 21
1 10 0V V
I Iy yV V
To make V2= 0, place a short at port 2
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Z Parameters
1 11 1 12 2
2 21 1 22 2
V z I z IV z I z I
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Z-parameter Calculations
2 2
1 211 21
1 10 0I I
V Vz zI I
To make I2= 0, place an open at port 2
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H Parameters
1 11 1 12 2
2 21 1 22 2
V h I h VI h I h V
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H Parameter Calculations
To make V2= 0, place a short at port 2
2 2
1 211 21
1 10 0V V
V Ih hI I
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G Parameters
1 11 1 12 2
2 21 1 22 2
I g V g IV g V g I
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G-Parameter Calculations
2 2
1 211 21
1 10 0I I
I Vg gV V
To make I2= 0, place an open at port 2
ECE 546 – Jose Schutt‐Aine 11
TWO‐PORT NETWORK REPRESENTATION
- At microwave frequencies, it is more difficult to measure total voltagesand currents.
- Short and open circuits are difficult to achieve at high frequencies.
- Most active devices are not short- or open-circuit stable.
1 11 1 12 2V Z I Z I
2 21 1 22 2V Z I Z I 1 11 1 12 2I Y V Y V
2 21 1 22 2I Y V Y V
Z Parameters Y Parameters
ECE 546 – Jose Schutt‐Aine 12
1 11I = i r
o
E EZ 2 2
2I = i r
o
E EZ
- Total voltage and current are made up of sums of forward andbackward traveling waves.
- Traveling waves can be determined from standing-wave ratio.
Use a travelling wave approach
Wave Approach
1 1 1i rV E E 2 2 2i rV E E
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11a = i
o
EZ
22a = i
o
EZ
11b = r
o
EZ
22b = r
o
EZ
Zo is the reference impedance of the system
b1 = S11 a1 + S12 a2
b2 = S21 a1 + S22 a2
Wave Approach
ECE 546 – Jose Schutt‐Aine 14
111 a2=0
1
S = | ba
221
1 | a2=0
S = ba
112 a1=0
2
S = | ba
222
2 | a1=0
S = ba
To make ai = 01) Provide no excitation at port i2) Match port i to the characteristic impedance of the reference lines.
CAUTION : ai and bi are the traveling waves in the reference lines.
Wave Approach
ECE 546 – Jose Schutt‐Aine 15
2
2 2
1111 22( X )S = S =
X
2
2 2
1112 21( )XS = S =
X
c ref
c ref
Z ZZ Z
R j L G j C
cR j LZG j C
S‐Parameters of TL
lX e
ECE 546 – Jose Schutt‐Aine 16
2
2 2
1111 22( X )S = S =
X
2
2 2
1112 21( )XS = S =
X
c ref
c ref
Z ZZ Z
LC
cLZC
S‐Parameters of Lossless TL
j lX e
If Zc = Zref
011 22S = S = j l
12 21S = S = e
ECE 546 – Jose Schutt‐Aine 17
N-Port S Parameters
1 11 12 1
2 21 22 2
n nn n
b S S ab S S a
b S a
b = Sa
ii
oi
VaZ
If bi = 0, then no reflected wave on port i port is matched
ii
oi
VbZ
iV
iV
oiZ
: incident voltage wave in port i
: reflected voltage wave in port i
: impedance in port i
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N-Port S Parameters
1o
o
ZZ
a + b Z a - b
v = Zi 1
oZi a - b
Substitute (1) and (2) into (3)
Defining S such that b = Sa and substituting for b
o oZ ZU + S a U - S a
1oZ Z U + S U - S o oZ Z -1S Z + U Z U
SZ ZS
(3)(2)(1) oZv a + b
U : unit matrix
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N-Port S Parameters
-1i = k a - b
If the port reference impedances are different, we define k as
v = k(a + b)
1
2
o
o
on
Z
Z
Z
k
-1k(a + b) Zk a - b
-1Z = k U + S U - S k -1 -1S = Zk + k Zk - k
SZZS
and and
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NormalizationAssume original S parameters as S1 with system k1. Then the representation S2 on system k2 is given by
-1-1 -1
2 1 1 1 1 2 2 1 1 1 1 2 2S = k (U + S )(U - S ) k k + k k (U + S )(U - S ) k k - k
Transformation Equation
If Z is symmetric, S is also symmetric
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Dissipated Power 1
2dP T T * *a U - S S a
The dissipation matrix D is given by:T *D = U - S S
Passivity insures that the system will always be stable provided that it is connected to another passive network
For passivity‐ (1) the determinant of D must be‐ (2) the determinant of the principal minors must be 0
0
ECE 546 – Jose Schutt‐Aine 22
Dissipated Power
T *S S U
When the dissipation matrix is 0, we have a lossless network
The S matrix is unitary.
2 211 21 1S S
2 222 12 1S S
For a lossless two‐port:
If in addition the network is reciprocal, then2
12 21 11 22 12and 1S S S S S
ECE 546 – Jose Schutt‐Aine 23
Lossy and Dispersive Line
2
11 22 2 2
11
S S
2
21 12 2 2
11
S S
le
c o
c o
Z ZZ Z
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Frequency-Domain Formulation*
* J. E. Schutt-Aine and R. Mittra, "Scattering Parameter Transient analysis of transmission lines loaded with nonlinear terminations," IEEE Trans. Microwave Theory Tech., vol. MTT-36, pp. 529-536, March 1988.
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1 11 1 12 2( ) ( ) ( )B S A S A
2 21 1 22 2( ) ( ) ( )B S A S A
Frequency-Domain
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Time-Domain Formulation
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Time-Domain Formulation
1 11 1 12 2( ) ( )* ( ) ( )* ( )b t s t a t s t a t
2 21 1 22 2( ) ( )* ( ) ( )* ( )b t s t a t s t a t
1 1 1 1 1( ) ( ) ( ) ( ) ( )a t t b t T t g t
2 2 2 2 2( ) ( ) ( ) ( ) ( )a t t b t T t g t
( )( )
oi
i o
ZT tZ t Z
( )( )( )
i oi
i o
Z t ZtZ t Z
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Time-Domain Solutions
'2 22 1 1 1 1
1
1 ( ) (0) ( ) ( ) ( ) ( )( )
( )t s T t g t t M t
a tt
'1 12 2 2 2 2( ) (0) ( ) ( ) ( ) ( )
( )t s T t g t t M t
t
'1 11 2 2 2 2
2
1 ( ) (0) ( ) ( ) ( ) ( )( )
( )t s T t g t t M t
a tt
'2 21 1 1 1 1( ) (0) ( ) ( ) ( ) ( )
( )t s T t g t t M t
t
ECE 546 – Jose Schutt‐Aine 29
Time-Domain Solutions
' ' ' '1 11 2 22 1 12 2 21( ) 1 ( ) (0) 1 ( ) (0) ( ) (0) ( ) (0)t t s t s t s t s
' (0) (0)ij ijs s
1 11 12( ) ( ) ( )M t H t H t
2 21 22( ) ( ) ( )M t H t H t
1
1
( ) ( ) ( )t
ij ij jH t s t a
' '1 11 1 12 2 1( ) (0) ( ) (0) ( ) ( )b t s a t s a t M t
' '2 21 1 22 2 2( ) (0) ( ) (0) ( ) ( )b t s a t s a t M t
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Special Case – Lossless Line
11 22( ) ( ) 0s t s t 12 21( ) ( ) ls t s t tv
1 2( ) lM t a tv
2 1( ) lM t a tv
1 1 1 1 2( ) ( ) ( ) la t T t g t t a tv
2 2 2 2 1( ) ( ) ( ) la t T t g t t a tv
1 2( ) lb t a tv
2 1( ) lb t a tv
Wave Shifting Solution
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Time-Domain Solutions
1 1 1( ) ( ) ( )v t a t b t
2 2 2( ) ( ) ( )v t a t b t
1 11
( ) ( )( )o o
a t b ti tZ Z
2 22
( ) ( )( )o o
a t b ti tZ Z
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SimulationsLine length = 1.27mZo = 73
v = 0.142 m/ns
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0 100 2000
1
2
3
4
5
6
Near End
Time (ns)
Volts
0 100 200-2
0
2
4
6
Far End
Time (ns)
Volts
Simulations
ECE 546 – Jose Schutt‐Aine 34
0 10 20 30 40 50-1
0
1
2
3
4
5
lossylossless
Near End
Time (ns)
Volts
0 10 20 30 40 50-2
0
2
4
6
lossylossless
Far End
Time (ns)
Volts
Simulations
Line length = 25 in
L = 539 nH/m
C = 39 pF/m
Ro = 1 k (GHz)1/2
Pulse magnitude = 4V
Pulse width = 20 ns
Rise and fall times = 1ns
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N-Line S-Parameters*
B1 = S11 A1 + S12 A2 B2 = S21 A1 + S22 A2
* J. E. Schutt-Aine and R. Mittra, "Transient analysis of coupled lossy transmission lines with nonlinear terminations," IEEE Trans. Circuit Syst., vol. CAS-36, pp. 959-967, July 1989.
ECE 546 – Jose Schutt‐Aine 36
Scattering Parameters for N-Line
-1-111 22S = S = T Γ -ΨΓΨ 1-ΓΨΓΨ T
2 -1-121 12 oS = S = E E 1-Γ Ψ 1-ΓΨΓΨ T
-1-1 -1 -1 -1 -1 -1
o o o m o o o mΓ = 1+ EE Z H H Z 1- EE Z H H Z
-1-1 -1 -1 -1
o o o m oT = 1+ EE Z H H Z EE -lΨ = W
ECE 546 – Jose Schutt‐Aine 37
Scattering Parameter MatricesEo : Reference system voltage eigenvector matrix
E : Test system voltage eigenvector matrix
Ho : Reference system current eigenvector matrix
H : Test system current eigenvector matrix
Zo : Reference system modal impedance matrix
Zm : Test system modal impedance matrix
ECE 546 – Jose Schutt‐Aine 38
Eigen Analysis
* Diagonalize ZY and YZ and find eigenvalues.* Eigenvalues are complex: i = i + ji
W(u)
e1u j1u
e2u j2u
en u jnu
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Solution
mV = EV
mI = HI
( ) ( ) ( )x x x mV = W A W
( ) ( ) ( )x x x -1m mI = Z W A W
-1 -1m mZ = Λ EZH
-1 -1 -1c m mZ = E Z H = E Λ EZ
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Solutions
111 1 1 11 1 1 1 1( ) 1 ( ) ' (0) ( ) ( ) ( ) ( )a t t s T t g t t M t
1 111 1 11 2 221 ( ) ' (0) 1 ( ) ' (0)t s t s
1 21 2 2 2 2 ( ) ' (0) ( ) ( ) ( ) ( )t s T t g t t M t
112 2 2 22 2 2 2 2( ) 1 ( ) ' (0) ( ) ( ) ( ) ( )a t t s T t g t t M t
1 112 2 22 1 111 ( ) ' (0) 1 ( ) ' (0)t s t s
1 12 1 1 1 1( ) ' (0) ( ) ( ) ( ) ( )t s T t g t t M t
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Solutions
-1 -1' ' '1 1 11 2 22 1 21 2 12( ) 1- 1- ( ) (0) 1- ( ) ' (0) ( ) (0) ( ) (0)t t s t s t s t s
-1-1 ' ' '2 2 22 1 11 2 12 1 21( ) 1- 1- ( ) ' (0) 1- ( ) (0) ( ) (0) ( ) (0)t t s t s t s t s
' '2 21 1 22 2 2( ) (0) ( ) (0) ( ) ( )b t s a t s a t M t
' '1 11 1 12 2 1( ) (0) ( ) (0) ( ) ( )b t s a t s a t M t
ECE 546 – Jose Schutt‐Aine 42
11 1 1 1 1 1( ) ( ) ( ) ( ) ( ) ( )m ov t a t b t v t E a t b t
12 2 2 2 2 2( ) ( ) ( ) ( ) ( ) ( )m ov t a t b t v t E a t b t
Solutions
V3(z)
V2(z)
V1(z)
...
L, C
z=0 z=l
ECE 546 – Jose Schutt‐Aine 43
Lossless Case – Wave Shifting
21 12( ) ( ) ( - )ms t s t t
1 2( ) ( - )mM t a t
2 1( ) ( - )mM t a t
1 1 1 1 2( ) ( ) ( ) ( ) ( - )ma t T t g t t a t
2 2 3 3 1( ) ( ) ( ) ( ) ( - )ma t T t g t t a t
1 2( ) ( - )mb t a t
2 2( ) ( - )mb t a t
ECE 546 – Jose Schutt‐Aine 44
(t m )
(t m1)(t m2 )
(t mn )
ai (t m )
a1(t m1)a2 (t m2 )
an (t mn )
Solution for Lossless Lines
ECE 546 – Jose Schutt‐Aine 45
2111 2(1 )
leY lZ ec
Zc : microstrip characteristic impedance : complex propagation constantl : length of microstrip
Y11 can be unstable
2111 221
l( e )S le
Z Zc oZ Zc o
S11 is always stable
Y-Parameter S-Parameter
Test line: Zc,
shortZoZo
ReferenceLine
ReferenceLine
Test line: Zc,
TestLine
Why Use S Parameters?
ECE 546 – Jose Schutt‐Aine 46
c ref
c ref
Z ZZ Z
cR j LZG j C
Zref is arbitraryWhat is the best choice for Zref ?
refLZC
cLZC
11 0S 12j LCd
oS e X
At high frequencies
Thus, if we choose
Choice of Reference
ECE 546 – Jose Schutt‐Aine 47
50.0 0.0 0.0 0.00.0 50.0 0.0 0.00.0 0.0 50.0 0.00.0 0.0 0.0 50.0
S-Parameter measurements (or simulations) aremade using a 50-ohm system. For a 4-port, the reference impedance is given by:
Zo =
11 1o oS ZZ I ZZ I
1oZ I S I S Z
Z: Impedance matrix (of blackbox)S: S-parameter matrixZo: Reference impedanceI: Unit matrix
Choice of Reference
ECE 546 – Jose Schutt‐Aine 48
328.0 69.6 328.9 69.669.6 328.8 69.6 328.9328.9 69.6 328.8 69.669.6 328.9 69.6 328.8
50.0 0.0 0.0 0.00.0 50.0 0.0 0.00.0 0.0 50.0 0.00.0 0.0 0.0 50.0
Method: Change reference impedance from uncoupled to coupled system to get new S-parameter representation
Zo =
Zo =
Uncoupled system
Coupled system
as an example…
Reference Transformation
ECE 546 – Jose Schutt‐Aine 49
0
0.5
1
1.5
0 2 4 6 8 10
S11 - Linear Magnitude
S11 - 50 Ohm
S11 - Zref
S11
Frequency (GHz)
50.0 0.0 0.0 0.00.0 50.0 0.0 0.00.0 0.0 50.0 0.00.0 0.0 0.0 50.0
using
Zo =
as reference…
328.0 69.6 328.9 69.669.6 328.8 69.6 328.9328.9 69.6 328.8 69.669.6 328.9 69.6 328.8
using
as reference…
Zo =
Harder toapproximate
Easier to approximate (up to 6 GHz)
Choice of Reference
ECE 546 – Jose Schutt‐Aine 50
0
0.5
1
1.5
0 2 4 6 8 10
S11 - Linear Magnitude
S11 - 50 Ohm
S11 - Zref
S11
Frequency (GHz)
50.0 0.0 0.0 0.00.0 50.0 0.0 0.00.0 0.0 50.0 0.00.0 0.0 0.0 50.0
using
Zo =
as reference…
328.0 69.6 328.9 69.669.6 328.8 69.6 328.9328.9 69.6 328.8 69.669.6 328.9 69.6 328.8
using
as reference…
Zo =
Harder toapproximate
Easier to approximate (up to 6 GHz)
Choice of Reference
ECE 546 – Jose Schutt‐Aine 51
0
0.05
0.1
0.15
0.2
0.25
0.3
0 2 4 6 8 10
S12 - Linear Magnitude
S12 - 50 Ohm
S12 - Zref
S12
Frequency (GHz)
50.0 0.0 0.0 0.00.0 50.0 0.0 0.00.0 0.0 50.0 0.00.0 0.0 0.0 50.0
using
Zo =
as reference…
328.0 69.6 328.9 69.669.6 328.8 69.6 328.9328.9 69.6 328.8 69.669.6 328.9 69.6 328.8
using
as reference…
Zo =
Easier to approximate (up to 6 GHz)
Harder toapproximate
Choice of Reference
ECE 546 – Jose Schutt‐Aine 52
0
0.2
0.4
0.6
0.8
1
1.2
0 2 4 6 8 10
S31 - Linear Magnitude
S31 - 50 Ohm
S31 - ZrefS3
1
Frequency (GHz)
50.0 0.0 0.0 0.00.0 50.0 0.0 0.00.0 0.0 50.0 0.00.0 0.0 0.0 50.0
using
Zo =
as reference…
328.0 69.6 328.9 69.669.6 328.8 69.6 328.9328.9 69.6 328.8 69.669.6 328.9 69.6 328.8
using
as reference…
Zo =
Harder toapproximate
Easier to approximate
Choice of Reference
ECE 546 – Jose Schutt‐Aine 53
.
0
0.05
0.1
0.15
0.2
0.25
0.3
0 0.5 1 1.5 2 2.5 3
S11 Magnitude
Zref=ZoZref=80 ohmsZref=100 ohms
Frequency (GHz)
S11
Choice of Reference
ECE 546 – Jose Schutt‐Aine 54
.
0.7
0.75
0.8
0.85
0 0.5 1 1.5 2 2.5 3
S21 Magnitude
Zref=ZoZref=80 ohmsZref=100 ohms
Frequency (GHz)
S21
Choice of Reference
ECE 546 – Jose Schutt‐Aine 55
Modeling of Discontinuities
1. Tapered Lines
2. Capacitive Discontinuities
ECE 546 – Jose Schutt‐Aine 56
General topology of tapered microstrip with dw :width at wide end,dn: width at narrow end, lw: length of wide section, ln : length ofnarrow section, lt: length of tapered section.
Tapered Microstrip
ECE 546 – Jose Schutt‐Aine 57
( ) ( )21 1 22( ) ( )* ( ) ( )* ( )j j
j j ju t s t u t s t w t
( 1) ( 1)11 12 1( ) ( )* ( ) ( )* ( )j j
j j jw t s t u t s t w t
* J. E. Schutt-Aine, IEEE Trans. Circuit Syst., vol. CAS-39, pp. 378-385, May 1992.
Tapered Line Analysis Using S Parameters*
ECE 546 – Jose Schutt‐Aine 58
-1
0
1
2
3
4
5
0 1 2 3 4 5 6
Vol
ts
Time (ns)
Small EndExcitation at small end
-1
0
1
2
3
4
5
0 1 2 3 4 5 6
Vol
ts
Time (ns)
Wide EndExcitation at small end
-1
0
1
2
3
4
5
0 1 2 3 4 5 6
Vol
ts
Time (ns)
Small EndExcitation at wide end
-1
0
1
2
3
4
5
0 1 2 3 4 5 6
Vol
ts
Time (ns)
Wide EndExcitation at wide end
Tapered Transmission Line
ECE 546 – Jose Schutt‐Aine 59
-1
0
1
2
3
4
5
0 1 2 3 4 5Time (ns)
Near End
.0564 mils/in
.1128 mils/in
.2257 mils/in
6 7 8
volts
-1
0
1
2
3
4
5
0 1 2 3 4 5Time (ns)
Far End
.0564 mils/in
.1128 mils/in
.2257 mils/in
6 7 8
volts
Varying tapering rate
Tapered Transmission Line
ECE 546 – Jose Schutt‐Aine 60
Zo
Zo
ZoC
-1
-0.5
0
0.5
1
1.5
2
2.5
0 10 20 30 40 50
Near End -- C=4 pF
Vol
ts
Time (ns)
-1
-0.5
0
0.5
1
1.5
2
2.5
0 10 20 30 40 50
Far end -- C=4 pF
Vol
ts
Time (ns)
Capacitive Load
ECE 546 – Jose Schutt‐Aine 61
Zo
Zo
ZoC
-1
-0.5
0
0.5
1
1.5
2
2.5
0 10 20 30 40 50
Near end -- C=40 pF
Vol
ts
Time (ns)
-0.5
0
0.5
1
1.5
2
2.5
0 10 20 30 40 50
Far end -- C=40 pF
Vol
ts
Time (ns)
Capacitive Load
ECE 546 – Jose Schutt‐Aine 62
Zo
Zstub
ZsL
- Stubs of TL with nonlinear loads- Reduce speed and bandwidth - Limit driving capabilities
Multidrop Buses
ECE 546 – Jose Schutt‐Aine 63
Transmission Lines with Capacitive Discontinuities
ECE 546 – Jose Schutt‐Aine 64
i r tV V V i r i r t
o o
V V V V VEZ R R Z
r c c iV T E V
Capacitive Discontinuity
ECE 546 – Jose Schutt‐Aine 65
( ) ' ( )21 1 22( ) ( )* ( ) ( )* ( )j j
j j ju t s t u t s t w t
' "( ) ( ) ( )j j ju t u t u t
' "( ) ( ) ( )j j jw t w t u t
Scattering Parameter Analysis
ECE 546 – Jose Schutt‐Aine 66
- 1
0
1
2
3
4
0 11.25 22.5 33.75 45
V o l ts
Ti me ( ns)
Near End
- 1
0
1
2
3
4
0 11.25 22.5 33.75 45
V o l ts
Ti me ( ns)
Far End
Capacitive Loading
ECE 546 – Jose Schutt‐Aine 67
-1
0
1
2
3
4
5
0 1 2 3 4 5Time (ns)
loading period 600 mils300 mils150 mils
6 7 8
volts
-1
0
1
2
3
4
0 1 2 3 4 5Time (ns)
capacitive loading1 pF2 pF4 pF
6 7 8
volts
Computer-simulated near end responses for capacitively loaded transmission line with l = 3.6 in, w = 8 mils, h = 5 mils. Pulse parameters are Vmax = 4 V, tr = tf = 0.5 ns, tw = 4 ns. Left: Varying P with C = 2 pF. Right: Varying C with P = 300 mils.
Capacitive Loading