iii

Contents

1. Introduction .......................................................................................................1

1.1 Background................................................................................................2

1.2 Overview of the Parametric Study ..............................................................4

2. Theory of the Nose-to-Nose Calibration.............................................................4

2.1 Sampler Theory..........................................................................................5

2.2 Nonideal Sampling and the “Impulse Response” ........................................6

2.3 The Kickout Pulse ......................................................................................8

2.4 Nose-to-Nose Calibration ...........................................................................9

3. Large- and Small-Signal Analysis of the Sampling Circuit .................................12

3.1 Diode Models.............................................................................................12

3.1.1 Large-Signal Model ...........................................................................12

3.1.2 Small-Signal Model...........................................................................14

3.2 Diode-Related Errors in the Nose-to-Nose Calibration................................16

3.2.1 Diode Conductance............................................................................18

3.2.2 Nonlinear Diode Junction Capacitance...............................................18

4. SPICE Simulations.............................................................................................19

4.1 SPICE Models............................................................................................20

4.1.1 Riad’s Model .....................................................................................21

4.1.2 Kerley’s Model..................................................................................21

4.2 Development of the Default Sampling-Circuit Model .................................22

4.2.1 Packaging Parasitics ..........................................................................25

4.2.2 Strobe-Excitation-Pulse Rise Time.....................................................26

4.3 SPICE Convergence Studies.......................................................................27

4.3.1 Time Step ..........................................................................................29

4.3.2 Sampling-Cycle Period ......................................................................30

4.3.3 Input Voltage Level ...........................................................................31

4.3.4 Input Pulse Width ..............................................................................31

4.4. More Details on the SPICE Simulations ....................................................32

4.4.1 Impulse Response Calculations..........................................................32

4.4.2 Kickout Pulse Calculations ................................................................33

4.4.3 Nose-to-Nose Calculation and Error ..................................................33

Appendix for Section 4. SPICE Code ...............................................................35

5. Parametric Studies Using SPICE........................................................................39

5.1 Sampling Diode Model Parameters.............................................................39

5.1.1 Description ........................................................................................39

5.1.2 Simulation Results.............................................................................40

5.1.2.1 Diode Model Parameters Except Junction Capacitance..............40

5.1.2.2 Parameters Including Diode Junction Capacitance.....................41

5.1.2.3 Split Diode Model.....................................................................42

5.2 Hold Capacitors..........................................................................................43

5.2.1 Description ........................................................................................43

5.2.2 Simulation Results.............................................................................44

5.3 Strobe-Generator Impedance and Strobe Pulse-Shaping Circuitry...............44

5.3.1 Ideal Source and Transmission Line Pulse Shaping............................44
iv

5.3.1.1 Description................................................................................44

5.3.1.2 Simulation Results ....................................................................46

5.3.2 Strobe-Pulse Rise Time......................................................................46

5.3.2.1 Description................................................................................46

5.3.2.2 Simulation Results ....................................................................46

5.3.3 Strobe-Generator Impedance..............................................................47

5.3.3.1 Description................................................................................47

5.3.3.2 Simulation Results ....................................................................48

5.3.4 Strobe-Excitation-Pulse Amplitude....................................................49

5.3.4.1 Description................................................................................49

5.3.4.2 Simulation Results ....................................................................49

5.4 Diode Imbalance ........................................................................................50

5.4.1 Description ........................................................................................50

5.4.2 Simulation Results.............................................................................51

5.5 Diode Bias Level........................................................................................52

5.5.1 Description ........................................................................................52

5.5.2 Simulation Results.............................................................................53

5.6 Input Network Impedance ..........................................................................54

5.6.1 Description ........................................................................................54

5.6.2 Simulation Results.............................................................................54

Appendix for Section 5. Error Pulses................................................................56

6. Sensitivity Analysis and Standard Uncertainty ...................................................57

6.1 Ranking the Contributions to the Phase Error........................................57

6.2 Estimate of Standard Uncertainty Due to Combined Sampling Circuitry

Variation ....................................................................................................60

6.3 Summary and Conclusions....................................................................62

References .............................................................................................................63
1

 The Impact of Internal Sampling Circuitry on the Phase Error of the

Nose-to-Nose Oscilloscope Calibration

Kate A. Remley

National Institute of Standards and Technology

Radio-Frequency Technology Division

325 Broadway

Boulder, CO 80305

We discuss the dependence of the phase error in the nose-to-nose

oscilloscope calibration on the oscilloscope’s sampling circuitry. We first

present the theory behind the nose-to-nose calibration, and then present

both large- and small-signal analyses of the sampling circuit. Using this

background information, we develop a sampling-circuit model. We then

carry out parametric studies in which we systematically vary the

component values of the internal sampling circuit. From these studies, we

determine which components of the sampling circuitry of the oscilloscope

contribute most significantly to errors in the nose-to-nose calibration. We

then predict an expected range of phase errors from a range of realistic

component values.

Keywords: Electrical Phase; Large-Signal Network Analyzer Calibration;

Nose-to-Nose Calibration; Oscilloscope Calibration; Phase Calibration;

Sampling Circuit; Schottky Diode; SPICE Model.

1. Introduction

The nose-to-nose calibration technique [1-4] was proposed in 1990 to determine

the impulse response of broadband oscilloscopes. Finding and correcting the response of

these high-speed oscilloscopes is important because the typical response time of their

internal sampling circuitry is of the same order as the signal transitions of many highspeed circuits under test. To measure the amplitude response of broadband (defined here

as 20 GHz or greater) sampling oscilloscopes, we currently use the “swept-sine” method

[2, 5], where a series of individual tones of increasing frequency and known power are

fed into the oscilloscope. The power in the tones is measured with a power meter

traceable to NIST. Comparing the known amplitude of the input signal to the signal

measured by the oscilloscope allows us to determine and correct the magnitude response

of the oscilloscope.

However, the phase response of an oscilloscope, that is, the unintentional change

in the phase relationships between frequency components of a signal caused by the

oscilloscope itself, has presented a more difficult challenge. The nose-to-nose calibration

currently represents one practical method for estimating an oscilloscope’s phase

response.
2

In this report, we perform a parametric sensitivity study of the internal sampling

circuitry found in the type of broadband oscilloscope that is used in nose-to-nose

calibrations. This numerical study has three primary objectives: (1) developing a method

to predict the nose-to-nose phase response arising from the internal sampling circuitry of

the oscilloscope that is robust and can be easily extended; (2) ascertaining which

components of the internal sampling circuitry of the oscilloscope contribute most

significantly to errors in the estimated phase response provided by the nose-to-nose

calibration; (3) predicting an expected range of phase errors from a range of realistic

component values. This third point is important because we have no way of knowing the

exact component and parameter values for the sampling circuit due to measurement

limitations, fabrication tolerances, and manufacturer proprietary issues. However, we can

estimate a realistic range for component values based on the literature. Simulating the

nose-to-nose calibration over this range gives us an estimate of the expected range of the

errors in the phase response due to the sampling circuitry. The methodology we develop

can easily be applied to other sampling-circuit representations.

This study is based on numerical simulations of a representative sampling-circuit

model. It yields information we would have difficulty obtaining through other methods.

For example, unambiguous measurements of the response of the internal circuitry of

high-speed samplers is extremely difficult to carry out. As well, simulations enable

parametric variation of circuitry that would be impossible to conduct experimentally due

to limitations in component fabrication and measurement technology. Thus, parametric

variation within numerical simulations offers insight into sampling-circuit-related error

mechanisms in the nose-to-nose calibration that otherwise would not be available.

1.1 Background

Over a decade ago, researchers at a manufacturer of digital oscilloscopes noticed

that when a DC offset was applied to bias lines of their oscilloscope by changing the “DC

offset” setting, a short “kickout” pulse appeared at the input port. This pulse is almost

identical in shape to the time-domain “impulse” response of the oscilloscope itself. When

the input ports of two identical oscilloscopes of this type are connected “nose-to-nose,”

the similarity of the kickout pulse and impulse response can be used to extract the

response of one of the oscilloscopes. In practice, since no two oscilloscopes are truly

identical, a series of three measurements is made on three oscilloscopes to extract the

response. Knowledge of the response can subsequently be used to correct waveforms

measured with the oscilloscope.

One important use for the nose-to-nose calibration is for phase calibration of

certain types of nonlinear vector network analyzers (LSNAs). LSNAs allow measurement

of the phase and amplitude response of nonlinear circuits and systems subjected to

periodic stimuli. The nose-to-nose calibration enables correction of the phase of

measured harmonic components relative to the fundamental. A transfer standard

(typically a “reference” or “comb” generator that is rich in harmonic content) is first

measured using a nose-to-nose-calibrated oscilloscope. The comb generator is then

connected to the LSNA and the phase relationships of the comb generator harmonics are
3

measured on the LSNA. Since the phase relationships of the comb generator are known

from the nose-to-nose-calibrated oscilloscope measurement, correction factors for the

LSNA can be extracted and subsequent LSNA measurements corrected. Correction of the

phase relationship between the fundamental and harmonics of DUTs measured on

LSNAs up to 50 GHz is currently accomplished using this method, although most LSNAs

are currently limited to an upper frequency of 20 GHz.

Other methods for correcting the phase response of broadband sampling

oscilloscopes include measuring a well-characterized waveform with the oscilloscope.

The output of the oscilloscope is a convolution of this known signal and the impulse

response of the oscilloscope. Once the impulse response of the oscilloscope is

deconvolved from the measured signal, correction coefficients are extracted from the

difference between the known and measured signals [6, 7]. Both NIST and NPL

(National Physical Laboratory, in the United Kingdom) have measurement services based

on this approach.

Electro-optic sampling (EOS) systems [8] offer a method of characterizing pulses

with very large bandwidths [9, 10]. Because these systems are based on on-wafer

methods and fast optoelectronic interactions, they have measurement bandwidths in the

terahertz range. These known pulses can then be used to calibrate an oscilloscope to even

higher frequencies than the methods currently in use discussed above.

An experimental method for correcting the phase response of an oscilloscope or

an LSNA involves measuring a reference nonlinear circuit such as a “golden diode” [11]

with the oscilloscope or LSNA we wish to calibrate. If the phase relationship between the

fundamental and the harmonics appearing at the output of the reference circuit is known,

we can measure the reference circuit and correct the measurement. The problem with this

technique is that, to date, no phase standard with sufficient bandwidth, stability, and

known phase response has been developed for broadband oscilloscopes. This is a topic of

current research at NIST and a few other labs.

The nose-to-nose calibration is a variation of the first method described above. In

this case, the “known” pulse is the kickout pulse, which is assumed to have essentially

the same shape as the impulse response of the oscilloscope itself. The method has the

advantage of requiring very little custom equipment and its sources of uncertainty have

been studied in depth. Several researchers have investigated experimental sources of error

in the nose-to-nose calibration [2-5, 12, 13]. Others have provided analytic analyses of

errors introduced by the sampler circuitry of the oscilloscope itself [14, 15]. In this report,

we turn to SPICE-based numerical simulation of a realistic model of the sampling circuit

[16, 17] (SPICE is an acronym for “Simulation Program with Integrated Circuit

Emphasis.” The SPICE3 program that we use was developed by researchers at the

University of California at Berkeley.) Simulation allows us to better identify error

mechanisms involving the internal sampling circuitry and to understand the sensitivity of

the oscilloscope’s phase response to parametric changes in sampling-circuit component

values. We develop procedures to quantify the sensitivity of the phase error to each
4

component and to a combination of components, and then apply these procedures to our

simulation results.

1.2 Overview of the Parametric Study

In Sections 2 and 3, we discuss the theory of the nose-to-nose calibration and

provide large- and small-signal models of the internal sampling circuitry found in the

type of oscilloscope used in nose-to-nose calibrations. In Section 4, we discuss our

implementation of a SPICE model of the sampling circuitry. We calculate a nominal

value for the “phase error” associated with this default sampler configuration, where we

define the phase error as the difference in the phase of the sampling circuit’s impulse

response, derived by the nose-to-nose procedure, from its actual value. For the default

sampling-circuit configuration that we examined, our phase error is approximately 3.3° at

20 GHz. In Section 5, we then vary the parameters of the model one at a time, note the

effect (increase or decrease) on the phase error, and discuss the reasons for the change in

phase error. From this systematic parametric variation, we determine which circuit

components contribute most significantly to the phase error in the nose-to-nose

calibration, and rank these contributions in Section 6.

Finally, even though we don’t know the exact values of sampling-circuit

components we can find the range of predicted phase errors, given a sampling-circuit

model and an expected realistic range of component values. To do this, we systematically

vary all circuit component values over a given range (for example, ±10 % of the value

specified in our default sampling-circuit model) and find the corresponding maximum

and minimum phase error. Assuming that the distribution of phase errors over this range

is uniform, we calculate the standard deviation and the standard uncertainty. For our

sampling circuit, the value of our standard uncertainty is approximately 1.4° (that is, for

our particular set of sampling-circuit parameters, our phase error is approximately 3.3°

and the standard uncertainty of the predicted phase error is ±1.4°). This procedure can

easily be extended to models where component values are known in greater detail,

leading to a better approximation of the phase error in the nose-to-nose calibration due to

internal sampling-circuit effects.

In this work, we have attempted to provide as accurate a model of the

oscilloscope’s sampling circuit as possible, although the model is strictly an

approximation. Thus, while we do provide an estimate of the phase error, we make no

exact statement of the phase-error value arising from the sampling circuitry in a real

oscilloscope. However, as mentioned above, our procedure could easily provide a better

estimate of the phase error if the model and component values were known in greater

detail. We are also able to provide important information on phase-error mechanisms

arising from the internal sampling circuitry that cannot be obtained elsewhere.

2. Theory of the Nose-to-Nose Calibration

In this section, we discuss the fundamental premise of the nose-to-nose

calibration: that the “impulse response” of a sampler is equivalent to the “kickout pulse”
5

that is generated by the sampler when a DC offset is applied. We describe how an

estimate of the impulse response of the sampler is derived from the nose-to-nose

response, and we define a representation of the error in the calibration that tells us how

different the actual impulse response is from the estimate.

To understand the theory of the nose-to-nose calibration, it is useful to understand

the operation of the internal sampling circuits used in the applicable type of broadband

oscilloscope. This description is given in Section 2.1 below, where we describe basic

sampling operation and differentiate between large-signal and small-signal regimes in the

sampler. We define the sampler’s impulse response in Section 2.2, and the kickout pulse

in Section 2.3. In Section 2.4, we describe the nose-to-nose technique and define the error

in a nose-to-nose estimate. We limit our discussion to the role of the samplers in the

nose-to-nose calibration, since other possible sources of phase error, such as mismatch

corrections and time-base distortion, are treated elsewhere [2, 3, 5, 12, 13].

2.1 Sampler Theory

Figure 2-1 depicts a simplified schematic diagram of a two-diode balanced

sampling circuit. This type of two-diode sampling circuit is required for performing a

nose-to-nose calibration, since it generates a kickout pulse that is related to its impulse

response. The bias supplies, Vb, place the diodes in a high-impedance reverse-biased state

until the strobe fires. Each time the strobe fires, the strobe pulse forward biases the two

diodes, turning them on and lowering their impedances for a short time. Because the

large-signal strobe current flows in one direction only (shown by solid arrows in Fig. 2-1)

and the circuit is balanced, the effects of the strobe current cancel at the input port of the

sampling circuit, that is, Istrobe + Iinverted strobe = 0 at the input node.

While the diodes are in their low-impedance state, a nonzero voltage at the input

port of the sampling circuit causes a net charge to flow from the input port through the

diodes to the hold capacitors. This small-signal current (shown by dashed arrows in Fig.

2-1) flows in opposite directions in the two diodes, and adds a net charge on the hold

Output

Vb

Vb

Strobe

Inverted

Strobe

Bias

Supplies

Input

50Ω

DC 

Offset

CHold

CHold

+

_

+

+ _

_

VDC Σ

Strobe Current

Small-Signal

Current

Figure 2-1. Simplified schematic of a two-diode sampling circuit.
6

capacitors. The sampler discretizes the average voltage on the two hold capacitors after

the strobe fires. This discretized voltage sample is proportional to the net charge

transferred to the hold capacitor when the strobe fired, and ideally will be proportional to

the voltage at the input port when the strobe fired.

In sequential sampling [18], a periodic input signal—that is, a repetitive train of

identical waveforms—is applied to the input port. The sampling circuit is used to

reconstruct the shape of a portion of the periodic input signal. This is accomplished by

firing the strobe at a time ∆t later than it was fired in the previous cycle of the periodic

input signal, as shown in Fig. 2-2. (Equivalently, the strobe may fire at a time nT ± ∆t

different from its previous firing, where n is an integer and T is the period [repetition

rate] of the input signal. For clarity in this description, we describe this process in terms

of t0 + n∆t, t0 = initial strobe, n = 1, 2, ...). In this way the strobe's firing time slowly

"scans" across the input waveform being sampled. Since each successive discretized

voltage sample corresponds to the input voltage at a time ∆t later than the previous

voltage sample, the shape of the individual waveforms in the periodic input signal can be

reconstructed from the discretized output voltage record.

2.2 Nonideal Sampling and the “Impulse Response”

If the diodes injected charge onto the hold capacitors only at the instant that the

strobe fired, the shape of the reconstructed signal at the output of the sampling circuit

would exactly mirror the shape of the individual waveforms in the periodic input signal.

However the strobe pulses have a finite duration and the diodes do not turn on or off

instantaneously. Hence, charge is injected on the hold capacitors in a nonuniform way

over a finite period that may be significantly longer than the sampling interval ∆t. As a

result, the nonideal response of the sampling circuit alters the reconstructed output, as

shown in Fig. 2-3.

Ideal

Reconstructed

Waveform

∆t 2∆t 3∆t

Periodic Input Signal

∆t

1st

Strobe

2nd

Strobe

3rd

Strobe

4th

Strobe

Figure 2-2. Ideal sequential sampling and reconstruction of the discretized input signal.
7

Ideal

Reconstructed

Signal

∆t

Reconstructed

Signal

(altered by 

sampler 

response)

∆t

Figure 2-3. Ideal sampling compared to sampling altered by the sampler’s response.

In general, practical samplers are neither linear nor time invariant. However, the

process of charge injection is nearly linear for small input signals. As a result, we find

that the reconstructed signal can be described as a convolution of an individual pulse at

the input port with a function we will call the “impulse response” of the sampling circuit.

While this function is clearly not an impulse response in the strict sense, we will call it

that because we expect its properties to be similar to those of the impulse response of a

truly linear time-invariant circuit. That is, we expect the output of the sampling circuit to

be equal to the convolution of the input signal and its impulse response. Equivalently, the

output of the sampling circuit can be represented as the product of a frequency-domain

transfer function and the frequency-domain representation of the input signal. It is the

impulse response or, equivalently, frequency-domain transfer function that we wish to

determine and correct using the nose-to-nose technique.

To test the hypothesis that the output response of the sampling circuit is a

convolution, we inserted the circuit values for the sampler described in Ref. [19] into a

SPICE model of the sampling circuit shown in Fig. 2-1 [16]. This model includes a full

nonlinear diode model and diode-package parasitics. To determine the impulse response

of the sampling circuit, we applied a train of very narrow pulses (0.5 ps rise time, fall

time, and pulse width) to its input. The spectrum corresponding to these pulses is very

broad with roll-off frequencies much higher than the 20 GHz input bandwidth of our

sampler. Then we numerically deconvolved this narrow pulse from the circuit’s response

to the input pulse train, as shown in the curve labeled “Impulse Response” in Fig. 2-4.

To verify that the reconstructed output of our sampling circuit is indeed the

convolution of this impulse response and an arbitrary input signal, we compared

numerical convolutions of the impulse response and a single pulse in the input pulse train

to the output responses of the circuit when excited by this input signal. Figure 2-4 shows

some typical results. In all the cases we tested, we found that the output responses were

nearly identical to the convolutions. This lends credence to the somewhat unusual notion

of impulse response employed in Refs. [2-4] that we use here.
8

2.3 The Kickout Pulse

Rush, Draving, and Kerley [1] noted that when the DC offset voltage (see Fig. 2-

1) for a sampling circuit of this design is nonzero, it creates a train of “kickout” pulses at

its input port with a shape similar to that of the circuit’s impulse response. Figure 2-5

compares a kickout pulse created by our SPICE circuit (solid black line with inverted

triangles) to its impulse response (solid green line). The figure shows that the two signals

are indeed similar.

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

0 0.05 0.10 0.15

5 ps Input Pulse

50 ps Input Pulse

Impulse Response

Response to Input Pulse

Convolution of Input Pulse

and Impulse Response

Time (ns)

N

o

r

m

a

l

i

z

e

d

A

m

p

l

i

t

u

d

eNormalized Response

Time (ns)

-0.5

0

0.5

1.0

1.1

0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15

Κιχκουτ

Ιμπυλσε ρεσπονσε

Ρεχονστρυχτεδ ιμπυλσε ρεσπονσε φρομ ιμ πυλσεκιχκουτ

Time (ns)

N

o

r

m

a

l

i

z

e

d

A

m

p

l

i

t

u

d

e

Kickout

Impulse Response

Reconstructed impulse

response from

impulse*kickout

Normalized Response

Time (ns)

The 5 ps and 50 ps curves

have no deconvolution as

in the impulse response

curve

Figure 2-4. Impulse response of a sampling circuit (solid line with crosses) determined by deconvolving

narrow input pulses from the output of the sampling circuit simulation. The figure also compares the

output response of the sampling circuit (solid lines) when excited with input pulses of 5 ps and 50 ps

duration to convolutions (dashed lines) of the input pulse and the impulse response. The 5 ps and 50 ps

pulses are not deconvolved, as the input pulse is for the Impulse Response curve.

Figure 2-5. Typical results when we compare the kickout pulse, the impulse response, and an estimate of

the impulse response reconstructed from a convolution of the impulse response and the kickout pulse. This

reconstruction is the basis of the nose-to-nose calibration.

Kickout

Impulse response

Reconstructed

impulse response

from impulse ⊗

kickout
9

2.4 Nose-to-Nose Calibration

The nose-to-nose calibration [1-4] exploits the similarity of the kickout pulses and

the impulse response. Figure 2-6 illustrates a simplified nose-to-nose procedure. The idea

is to set the DC offset voltage of one sampling circuit, which we will call B, to a nonzero

value so that it creates a train of kickout pulses at its input port. These pulses are fed into

the input port of a second sampling circuit, which we will call A, operating in its

conventional sampling mode. We will call the output of A the nose-to-nose response of

the two sampling circuits.

As discussed above and as shown in Refs. [2-4, 16] and [17], in the absence of

reflections or mismatches, the response of the entire system is proportional to the

convolution of the time-domain representation of the impulse response of Sampler A and

the kickout of Sampler B. In the frequency domain we have

M KH ( ) ( ) ( ), ω ωω AB B A ∝ (2-1)

where MAB is the Fourier transform of the measured waveform, HA is the Fourier

transform of the impulse response of Sampler A and KB is the Fourier transform of the

kickout pulse emanating from Sampler B. We define the Fourier Transform as

F f t e dt j t ( ) () ω ω = −

−∞

∞

∫ , with ω = 2πf and f the frequency in Hertz. In our numerical

simulations we use the Fast Fourier transform (FFT), a time- and frequency-discretized

version of the Fourier transform.

For identical, impedance-matched samplers, such as those we used in the

simulations described in this report, KA(ω) = KB(ω), HA(ω) = HB(ω). In this case,

Sampler B:

Kickout

Generator

Sampler A:

Detector

Kickout pulse train

Kickout Impulse

Response

DC

Offset

+_

Figure 2-6. The configuration used to implement the simplified nose-to-nose calibration procedure for

identical sampling circuits connected by a reflectionless through connection. The output of Sampler A

is called the nose-to-nose response. This response should be equal to the convolution of the kickout

pulse of Sampler B and the impulse response of Sampler A for small-signal inputs.
10

H CK H

CH K

H

CH K

H

A

est

B A

A

B

A

A

B

B

() () ()

( ) ( )

( )

( ) ( )

( ),

ω ωω

ω

ω

ω

ω

ω

ω

≡

=

=

 (2-2)

where C is a simple proportionality constant required because the amplitudes of the

kickout and impulse response are not identical. However, in practice, no two sampling

circuits are identical. In this case, an estimate of the impulse response of Sampler A can

be found from a set of three measurements of three samplers [3]

H C M M

M

CH K

H

A

est AB AC

BC

A

B

B

( ) () ()

( )

( ) ( )

( ).

ω

ω ω

ω

ω

ω

ω

≡

≅

(2-3)

Note that other combinations of samplers also will produce the estimate for sampler A.

In both eqs (2-2) and (2-3), the term C K

H

B

B

( )

( )

ω

ω expresses the deviation of the

estimate from the actual impulse response. We define an (inverse) correction factor E(ω)

of the Fourier transform of the impulse response estimate as

E H

H

C K

H A

A

est

A

B

B

( ) ( )

( )

( )

( )

ω , ω

ω

ω

ω ≡ ≅ (2-4)

where we chose C such that E(0) = 1, with ω = 0, the DC point. To accurately represent

the impulse response of an oscilloscope, the impulse response estimate generated by a

series of nose-to-nose measurements should be divided by the correction factor E(ω).

This correction factor (assuming linearity; reciprocal, reflectionless connection networks;

and a perfect oscilloscope timebase) can be approximated as the square root of the

Fourier transform of the kickout pulse divided by the sampler’s impulse response. Note

that E(ω) is related to the relative or fractional error since

Relative Error A A

est

A

A

est

A

= − = − H H

H

H

H

1 . (2-5)

If the kickout pulse and impulse response were identical, the correction factor in

eq (2-4) would equal one, and we would expect the nose-to-nose reconstruction to be

equal to the impulse response. However, Fig. 2-5 shows that the kickout pulse and

impulse response of our sampling circuit are not identical. The figure compares these to

the reconstruction of the impulse response from the convolution of the kickout pulse and

impulse response, and shows that the reconstruction is, in fact, an average of the

sampling circuit’s kickout pulse and its impulse response, as predicted by Ref. [2] and

defined by the convolution procedure. Since the kickout pulse and impulse response are
11

not identical (and, in the real world we can’t know each of these independently), the

reconstructed impulse response is somewhat in error. Throughout this work, we use the

correction factor to specify the magnitude error in the nose-to-nose calibration as

Ef C K

H A

A

B

( ) = (2-6)

in decibels, and the phase error as

arg ( ) arg ( ) arg ( ) / Ef Kf Hf { } A BB = [ ] { } − { } 2 (2-7)

in degrees, with f = ω/2π. Again, subscripts A and B refer to two different samplers. As

stated above, we use identical samplers in our modeling work and thus KA(f) = KB(f),

HA(f) = HB(f), and EA(f) = EB(f).

In the simulations that follow, calculations were performed in the time domain in

SPICE and then transformed to the frequency domain using the fast-Fourier transform

(FFT) procedure. The sampler’s impulse response was derived using the method of

Section 2.2 and compared to the FFT of a nose-to-nose derived impulse response (Section

2.3), which generates the correction factor E(f), defined in eq (2-4). The magnitude and

phase components were calculated as in eqs (2-6) and (2-7), the phase component was

detrended (see below) and the two were plotted in graphs, such as the one shown in Fig.

2-7 for the SPICE model of the sampler described in Section 4.2.

Throughout the remainder of this work we will focus on the phase error defined in

eq (2-7), an example of which is shown by the dashed line in Fig. 2-7. Because the

sampler introduces an arbitrary time shift into a signal that passes through it (simulated or

measured), we first subtract this time shift from our error ratio vector data, without loss

of generality. A simple time shift (delay), t0, is equivalent to multiplication by e-j2πt0f

 in the

frequency domain, which introduces a linear slope 2πt0f into the phase error arg{E(f)}.

-4

-3

-2

-1

0

1

2

E | | (dB) or Angle E (degree)

0 5 10 15 20

Frequency (GHz)

Default Sampler Model: 

 Magnitude

 Phase

Figure 2-7. Magnitude and phase component of the correction factor E(f) for one model of the sampling

circuit. The curves represent the difference between the impulse response of the sampler and the nose-tonose derived estimate of the impulse response, defined in eqs (2-6) and (2-7).
12

To determine the value of 2πt0, we numerically differentiated the phase-error

vector with respect to frequency. We chose 2πt0 to correspond to the value of the

differentiated phase-error vector at DC, and subtracted it from the phase error to set its

slope to zero at DC. We then integrated to transform back to the frequency domain. Note

that while this alignment procedure is straightforward for simulated data, it is typically

much more difficult for noisy, measured data since numerical differentiation can

significantly increase the noise content in the phase-error vector.

3. Large- and Small-Signal Analysis of the Sampling Circuit

The diodes in a sampling circuit function as voltage-controlled switches, enabling

charge to flow to the hold capacitors only for the brief instant that the strobe pulse

amplitude exceeds the diode’s threshold voltage. In theory, if the diodes behaved as

perfect switches, the nose-to-nose calibration would be exact [3, 15, 20]. However, all

diodes have an inherent voltage-dependent nonlinear junction capacitance and a

conductance that is an asymmetric function of time. These effects and their interaction

with impedances in the circuit cause the kickout pulse to be somewhat different in shape

from the impulse response, with the nonlinear junction capacitance having greater impact

[15, 20]. As a result, the nose-to-nose calibration in its present form will contain errors.

We devote this section to a description of the diodes and a discussion of the

potential error mechanisms due to the effects discussed above. We describe the way

sampling diodes are modeled in SPICE and how we extract additional important

quantities, such as the diode conductance, from the SPICE simulations. We examine the

reason why the asymmetric diode conductance is not a significant source of error, and

why the nonlinear diode junction capacitance does introduce errors into the nose-to-nose

calibration. This discussion sets the background for the parametric study presented in

Section 5.

3.1 Diode Models

The Schottky-barrier diode models used in our simulations are shown in Fig. 3-1.

These common models are found in, for example, Ref. [21]. Because both the large- and

small-signal regimes are important to our discussion of sampling circuits and the nose-tonose calibration, we discuss each separately.

3.1.1 Large-Signal Model

The large-signal representation in Fig. 3-1(a) models the diode junction as a

voltage-controlled current source, corresponding to the diode’s conduction

characteristics, in parallel with a nonlinear capacitance, corresponding to the junction

capacitance of the diode. The parallel combination is in series with a “spreading

resistance,” Rs, that models the loss behavior of the diode when it is strongly conducting.
13

The large-signal forward-bias current through a Schottky-barrier diode can be

expressed as [21, 22]

IV I qV

nkT j s

j ( ) exp = 



 



 − 



 



1, (3-1)

where Vj is the large-signal voltage across the diode junction (not including the voltage

drop, VRS, across the spreading resistance Rs), Is is the reverse saturation current, q is the

charge on an electron (1.6 x 10-19 C), n is the ideality factor, T is the junction temperature

(K), and k is the Boltzmann constant (1.37 x 10-23 J/K). Equation (3-1) is valid for VD

greater than –5nkT/q, where the diode is mildly reverse biased. SPICE utilizes other

equations for the strong reverse-bias and breakdown regions of the diode [22]. The

junction capacitance is given by

C V

C

V t j j

j0

j

bi

( )

( ) = ,

− 



 



1  φ

γ (3-2)

where Cj0 is the zero-bias junction capacitance. (Note that Cj = Cj0, a constant, when Vj =

0.) This is an important concept in our nose-to-nose error calculations), φbi is the

junction’s built-in potential, and γ is the grading coefficient (γ = 0.5 for Schottky-barrier

diodes and for linearly graded junction in pn-type diodes [23]). We define the largesignal junction capacitance as an incremental quantity C(Vj) = dQd/dVj, after Maas [21].

This definition offers consistency with the large- and small-signal definitions of the

conductive branch of the diode, where a large-signal conductance I/V is not typically

defined.

I(Vj)

+

_

VD

C(Vj)

VRS

Vj

_

+

+

_

(a)

c(t)

Rs

vj(t)

_

+

g(t)

 (b)

Figure 3-1. Models for a Schottky-barrier diode. (a) Large-signal model; (b) Small-signal model.
14

Two other parameters typically included in SPICE diode models and not defined

above are the transit time, τt, and the energy gap, EG. The transit time is the average time

for minority carriers to travel through the narrow region of a short-base diode [22]. It is

typically set to zero in SPICE Schottky-diode models since there is essentially no

minority-carrier transport. The energy gap is used in SPICE to differentiate between

different types of diodes, such as silicon, germanium, and Schottky-barrier diodes. The

value for Schottky diodes in SPICE is typically 0.69 eV [22].

The values of I(Vj) and C(Vj) are determined by the internal constants of the diode

(Rs, n, Is, φbi, and γ) and by the instantaneous large-signal junction voltage, Vj, and the

junction temperature T. In our sampling circuit, the value of Vj at any moment in time

depends primarily on the large-signal excitation (the strobe pulse) and the surrounding

electrical network. Through network analysis, SPICE determines Vj as a function of time,

and from this, the large signal response I(Vj) and C(Vj), both of which are time-varying

functions.

Figure 3-2 shows a typical example of the change in junction capacitance over

time as the strobe fires across the sampling diodes in our SPICE sampling-circuit model.

We increase the diode’s grading coefficient, γ, from zero (solid flat line) to 0.9 (line with

greatest variation). The grading coefficient γ = 0.5 corresponds to a typical Schottkybarrier diode, as mentioned above. The upper and lower bounds of γ = 0 and γ = 0.9 are

included for illustrative purposes only. The case γ = 0 is a nonphysical case where the

diode’s junction capacitance is constant. Cj0 in this example is 0.045 pF, the value of the

flat line. We see that the maximum instantaneous Cj increases to more than twice its

original value when γ = 0.5.

3.1.2 Small-Signal Model

The model in Fig. 3-1(b) is employed when the sampler is responding to a smallsignal input. In this case, the diode junction is represented as a time-varying conductance

in parallel with a time-varying capacitance. The conductance and the capacitance vary as

linear functions of time. The values of g(t) and c(t) are set by the time-varying largesignal junction voltage. A small-signal input voltage, vD(t), will not alter their values, but

will respond to them as to any time-varying, linear circuit element [21].

The diode’s small-signal voltage and current can be related by [15, 21]

i t gtv t dq t

dt

gtv t

dC t v t

dt

gtv t Ct

dv t

dt

v t

dC t

dt

d j

j

j

j

j

j

j

() () () ( )

() () () ()

() () () ( ) ( ) ( ) ,

= +

= +

=+ +

 

where qj(t), the small-signal charge stored in the diode’s junction capacitance, is C(t)vj(t).

(3-3 (a))

(3-3 (b))

(3-3 (c))
15

Calculating these small-signal quantities from our SPICE simulations is made

difficult by the fact that SPICE provides us with only VD, the total large-signal voltage

across the diode (see Fig. 3-1(a)), and ID, the total current through the diode. To find Vj

(refer to Fig. 3-1), we first calculated VRS, the voltage across the spreading resistance, as

the product of ID and Rs. We then subtracted VRS from VD to get Vj. We found I(Vj), the

current in the non-capacitive branch of the large-signal model of Fig. 3-1, using eq (3-1).

At this point we were able to determine the small-signal conductance g(t) =

dI(Vj(t))/dVj(t). The small-signal capacitance was found from eq (3-2). 

We found the small-signal junction voltage in the diode by subtracting two values

of Vj from SPICE simulations with and without a small-signal input applied to the

sampling circuit. The range of values for a “small-signal” input will be defined in Section

4.3.3. With the small-signal junction voltage, the conductance, and the capacitance

250

200

150

100

50

0

C(

j V) (fF)

j

9.05 9.10 9.15 9.20 9.25

Time (ns)

γ=0.0

γ=0.2

γ=0.5

γ=0.9

Figure 3-2. Sampling diodes’ junction capacitance C(Vj) for various values of grading coefficient, γ, as

the strobe fires.

-0.4

-0.2

0.0

0.2

0.4

Various

30.04 30.06 30.08 30.10 30.12 30.14

Time (ns)

Conductance (S/10)

 Nonlinear Junction

γ=0.0

γ=0.5

Small-Signal Junction Voltage (v)

Capacitance (pF)

Figure 3-3. Small-signal junction voltage with (solid) and without (dashed) nonlinear capacitance included in

the SPICE model. Also shown are the diode conductance (dots) and junction capacitance (dash-dots). For the

case with γ = 0.0, the junction capacitance (not shown) is a constant Cj0 = 0.045 pF. The ringing in the curves

is due to the presence of inductive packaging parasitics, not used in the default sampling-circuit model

described in Section 4.
16

known, we could then find the small-signal current through the diode. Typical smallsignal values are plotted in Fig. 3-3. Knowledge of the small-signal response can help us

understand potential errors in the nose-to-nose calibration caused by the diode’s

asymmetric conductance and nonlinear junction capacitance.

3.2 Diode-Related Errors in the Nose-to-Nose Calibration

To illustrate the effect that the asymmetry in the diode conductance function and

nonlinear diode junction capacitance have on the nose-to-nose calibration, we will first

examine a nearly ideal case in which we replace the sampling diodes by ideal switches.

We define an ideal switch as one with infinitely short turn-on and turn-off times, but

which stays on for a finite period. Thus, the conductance of the switch is a rectangular

function of time with a width corresponding to the duration of its closure (This pulse is

rectangular when the RC time constant of the circuit resistance and the hold capacitor is

much longer than the period of the sampling window. This condition is true for the

sampling circuits we investigated.). Because the sampling circuit will respond to any

input signal that arrives while the switch is shut, but will not respond to its input when the

switch is open, the impulse response has the same shape as the switch-closure

conductance function.

+

+

t

Output

Strobe

Inverted

Strobe

Bias

Supplies

50Ω

CHold

CHold

Σ

τ

t

Impulses in

Impulse Response of Sampler

Using Ideal Switches

vout(t)

Strobe

Inverted

Strobe

Bias

Supplies

Input

Port

50Ω

DC 

Offset

CHold

CHold

+

_

+

+

DC

τ

vout(t)

t

Kickout Pulse of Sampler

Using Ideal Switches

(a)

(b)

Input

Port

Figure 3-4. (a) Impulse response and (b) kickout pulse generation for a simple two-diode sampler.
17

We can explain this in terms of sampling-circuit operation as follows: if a pulse

train of Dirac delta functions is used as excitation to the sampling circuit, the impulse

response may be measured directly. The sampling circuit will respond to the delta

functions only when the switch is closed, allowing the hold capacitors to charge, as

shown at the bottom of Fig. 3-4(a). The impulse response will be traced out after many

sampling cycles.

We generate kickout pulses on a second ideal-switch sampler by replacing the

impulse excitation with a DC offset on the bias lines. Each time the strobe fires, a pulse

of the same duration as the switch closure appears at the input port of the sampler, as

shown at the bottom of Fig. 3-4(b). Again, the kick-out corresponds to the response of the

switch as it closes, allows current to flow, and then opens.

If we now connect the input ports of our two ideal-switch samplers nose-to-nose,

the output will be the convolution of the rectangular kickout and the rectangular impulse

response. This convolution, a triangular waveform of duration 2τ, is what we would

measure at the output of the second sampler. A simple deconvolution process gives us τ,

from which we can find the impulse response.

When the ideal switches are replaced by diodes, the impulse response is no longer

a simple rectangular function, but is altered by both the conductance and the junction

capacitance of the diodes, as shown in the right-hand curves of the simulation results

presented in Fig. 3-5. The left-hand curves in this figure illustrate the additional effect the

nonlinear diode junction capacitance has on the kickout pulse. We see that the kickout

pulse rise and fall times are affected by the increase in nonlinear capacitance, while there

is only a minimal change in the impulse response. This broadening of the kickout pulse

can be explained by considering the components of the small-signal current in the

sampling diodes.

Figure 3-5. Kickout pulse and impulse response for increasing nonlinearity of diode junction capacitance.

The impulse response has been shifted in time for clarity. Offset voltage is 0.1 V DC.

1.0

0.8

0.6

0.4

0.2

0.0

-0.2

Normalized Response

0.04 0.06 0.08 0.10 0.12 0.14

Time (ns)

γ = 0.0 

γ = 0.5 

γ = 0.9 

Kickout Impulse Response
18

3.2.1 Diode Conductance

Asymmetry in the sampling diodes’ conductance function can significantly alter

the shape of the kickout pulse compared to that of the impulse response [15], due to the

“time reversal” of the impulse response described in Ref. [3]. This, in turn, can lead to

errors in the nose-to-nose calibration [2, 3, 14]. Using the technique described above, we

plot the time-varying conductance function of the sampling diodes in Fig. 3-6. We see

that the conductance is indeed an asymmetric function of time.

However, because the strobe pulse turns the diodes on quickly, the transition from

a conductance value of 0 S (corresponding to an infinite impedance) to a value greater

than 1 S (corresponding to an impedance less than 1 Ω) is very rapid, approximately one

or two picoseconds. The shape of the conductance function above 1 S will not

significantly affect the loading of the sampling circuit. Thus, the effect of the

conductance asymmetry on the error in the nose-to-nose calibration is minimal. This is

demonstrated in Section 5, where we examine the phase error in the nose-to-nose

calibration with a constant, non-time-varying diode junction capacitance (see Fig. 5-3).

The remaining error is due to the time-varying diode conductance and, potentially, other

effects, but we see that this phase error is minimal.

3.2.2 Nonlinear Diode Junction Capacitance

The influence of the sampling diodes’ nonlinear junction capacitance on the phase

error in the nose-to-nose calibration is illustrated in Fig. 3-3, which shows simulation

results for the kickout sampler configuration with Voffset = 0.1 V. Shown are the smallsignal junction voltage, vj(t), with nonlinear junction capacitance (γ = 0.5, solid curve)

and with constant junction capacitance (γ = 0.0, dashed curve). Also shown are the diode

conductance and junction capacitance. Recall that the case with constant diode junction

capacitance is nonphysical. For the constant capacitance case, vj(t) changes only when the

diode starts to conduct, as shown in the highlighted area of Fig. 3-3. For the case with γ =

0.5, vj(t) follows the nonlinear capacitance curve. As indicated by the last two terms in eq

9

8

7

6

5

4

3

2

1

0

Conductance (S)

9.00 9.02 9.04 9.06 9.08 9.10

Time (ns)

 Cj0 = 0.0125 pF 

 Cj0 = 0.025 pF

 Cj0 = 0.045 pF

 Cj0 = 0.09 pF

Figure 3-6. Diode conductance function for one of the two sampling-circuit diodes calculated from

SPICE simulations for four different values of diode junction capacitance.
19

(3-3(c)), the time-varying quantities C(t) and vj(t) give rise to a small-signal displacement

current. This current causes the kickout pulse to broaden with nonlinear junction

capacitance [20].

The reason that the impulse response does not broaden can be understood by

realizing that the kickout is proportional to the instantaneous charge on the hold

capacitors, while the impulse response is reconstructed from digitized samples, each of

which is generated over one complete sampling cycle. The net charge transferred to the

hold capacitors through the diode capacitance over one complete sampling cycle must be

zero, since no conduction current path exists through the diode capacitance [15].

Following this introduction to the large- and small-signal diode models and the

effects of the nonlinear junction capacitance on the nose-to-nose calibration, we will

introduce the SPICE model in Section 4 that we then used in the parametric studies of

Section 5. SPICE utilizes the large-signal diode model of Fig. 3-1(a), and the

corresponding equations for large-signal current and diode capacitance given in eqs (3-1)

and (3-2), respectively.

4. SPICE Simulations

We used SPICE simulations to parametrically study the effects of varying the

values of sampling-circuitry components on the nose-to-nose calibration phase error.

SPICE is a circuit simulator that uses time-domain differential equations to represent

circuit element input/output relationships. Circuit elements are linked through common

nodes, and network analysis is used to solve the resulting system of equations. As we will

see, SPICE offers a high degree of accuracy in the analysis of circuits for which wellestablished analytic models exist.

We have two primary goals for our SPICE model of the sampling circuitry:

Firstly, we want to simulate the behavior of real samplers used in nose-to-nose

calibrations. We do this by developing a model of the sampling circuit whose nose-tonose derived impulse response compares well to measurements. When our simulated

results are similar to measured results, we expect that our estimate of the contribution of

the sampling circuitry to the total phase error is at least of the same order as the

contribution of physical samplers. As a result, we expect that our parametric studies will

demonstrate realistic sensitivity to variation in component values. The second goal for

our SPICE model is that it be robust enough for parametric study over a wide range of

component values.

In this section, we consider sampling-circuit models derived from two

independent sources. We compare the nose-to-nose-derived impulse response of these

models to nose-to-nose measurements and to each other. We synthesize a “default”

SPICE model from these and then study convergence to determine the range of valid

simulation conditions, such as time step and sampling rate, for this particular model. In

the end, we will have developed a default nose-to-nose simulation that is robust enough

to accurately perform numerical experiments for a broad range of parametric studies.
20

4.1 SPICE Models

Both sampler models that we considered are based on sampling circuits with 20

GHz input bandwidths and are of the two-branch balanced type shown in Fig. 4-1. The

first was described in a 1982 paper by Sedki Riad [19]. This older-style sampling circuit

used discrete components, and so the model includes diode packaging parasitics. The

second model was provided by John Kerley [24], one of the authors on the seminal noseto-nose work [1]. This model incorporates a monolithic diode structure similar to that

found in samplers used in the type of oscilloscope we desire to calibrate using the noseto-nose procedure.

The two sampler models have a similar layout, with a balanced diode

configuration in which the strobe pulse cancels at both the input and output ports

(although in practice the cancellation is not perfect). The strobe source for these samplers

uses a step generator that drives a shorted transmission line in parallel with the sampling

diode circuitry (see Fig. 4-1). A sharp voltage step initially turns the sampling diodes on.

Reflection of the step from the shorted end of the transmission line turns the diodes off at

2τd, where τd is the transmission line delay. We used a two-section ladder network to

implement the transmission line in SPICE. We describe the models of Riad and Kerley in

detail below, highlighting the features that we incorporated into our default samplingcircuit model. A table summarizing sampler model component values is provided at the

end of Section 4.2.

Vb

Strobe

Excitation

Bias

Supplies

DC 

Offset

CHold

+ _

+

_ V0

Output

Σ

Input

50Ω

Rstrobe

Vb

Inverted

Strobe

Excitation

CHold

+ _

Rstrobe

Cp Lp

Lp Cp

τd, Z0 C(Vj)

Rs

I(Vj)

Vj

+

-

Strobe

Inverted

Strobe

Rbias

Rbias

Figure 4-1. Schematic of the type of two-branch sampling circuit used in nose-to-nose measurements and

in our SPICE simulations. The inset shows the large-signal equivalent circuit of the sampling diode.
21

4.1.1 Riad’s Model

Riad [19] developed an equivalent-circuit based model of a sampling head that

used discrete, packaged sampling diodes. Model parameters were extracted from

dimensional and electrical measurements made on subsections of a dissasembled

sampling head. Refer to Fig. 4-1 for definitions of components described below. Because

this model is simple and straightforward to implement, we used many of its features in

our default SPICE model.

Riad specified a reverse bias voltage, Vb, of 1.63 V and bias resistors, Rbias, of

3600 Ω. The strobe source step generator for this model has a source impedance of

approximately 57 Ω, driving an 85.2 Ω transmission line that is shorted at its far end. We

used a transmission line delay, τd, of 7 ps in our default SPICE model (this value was not

given in Riad’s paper). The strobe generator amplitude was specified as 12 V in Riad’s

paper. We implemented this in SPICE by applying 6 V in each branch to give a

differential strobe pulse of 12 V total amplitude. The strobe inputs are labeled “Strobe”

and “Inverted Strobe” in Fig. 4-1. The simplified output circuit consists of a charge

amplifier that sums the charge stored on the hold capacitors during each sampling cycle.

The value of the hold capacitor, Chold, was not given, so we used a value of 4 pF in our

default model. Our 4 pF hold capacitor combined with the 50 Ω input impedance yields a

200 ps RC time constant, much longer than the 10 ps or so aperture time during which the

diodes are forward biased. This means the hold capacitor essentially looks like a short

circuit for the duration of the sampling aperture and thus the hold capacitor’s response

will not affect the sampled data, as desired.

Several important diode circuit parameters derived from measurements were

given in Riad’s paper. These parameters are used in eqs (3-1) and (3-2), and include the

spreading resistance, Rs, the zero-bias junction capacitance, Cj0, and the reverse saturation

current, Is. We used these values directly in our default SPICE model. For other diode

parameters in our SPICE model, we used values for typical Schottky-barrier diodes from

Ref. [21] or [22], including a minority carrier transit time, τt, of zero. These values are

summarized in Table 4-1 at the end of this section. Because discrete diode structures are

used in the sampling circuitry, packaging parasitics are included in the sampler circuit,

given by Cp and Lp in Fig. 4-1.

4.1.2 Kerley’s Model

 This model was based on a SPICE simulation of a sampler with a 20 GHz input

bandwidth that was conducted by Kerley in 1991 [24]. This sampler is similar to the 50

GHz-input-bandwidth sampling head oscilloscope plug-ins used in nose-to-nose

calibrations today [2, 5]. Thus, we expect this model to approximate nose-to-nose

measurements better than the Riad model, at least to the upper frequency of 20 GHz that

we use in this study.

 Kerley’s model is similar to the Riad model with the primary difference in the

bias circuitry, where the bias resistor was replaced with a series combination of a resistor
22

and an inductor, chosen empirically to match measurement. Other differences include: a

lower bias supply voltage in the Kerley model, a higher impedance for the shorted

transmission line used for strobe-pulse shaping, and a somewhat longer delay time for

this transmission line. The hold capacitors were 2 pF [2]. Because the sampler utilized a

monolithic diode structure, packaging parasitics Cp and Lp were presumed negligible and

were not included in the model. The strobe drive circuit for this model was complicated,

with filtering added to reproduce the strobe pulse observed in this type of oscilloscope.

Kerley used three diode model configurations in his experiments, as shown in Fig.

4-2. One used the standard SPICE nonlinear junction capacitance model [see eq (3-2)],

with a zero-bias junction capacitance, Cj0, of 40 fF (from the manufacturer). The second,

for investigation of the effects of nonlinear capacitance on the nose-to-nose calibration,

used a constant junction capacitance [C(Vj) = constant] of 40 fF. The third model split the

junction capacitance between a branch with a fixed value of 15 fF and a branch that

included the nonlinear junction capacitance with Cj0 = 25 fF [see Fig. 4-2(c)]. In his notes

[24], Kerley stated that the third one is probably the most accurate representation of the

diode, based on the manufacturer’s information. In each Kerley diode model, only the

reverse saturation current Is and the spreading resistance, Rs, were specified, again based

on the manufacturer’s specifications. He left the other diode parameters as the SPICE

default [22]. These values are summarized in Table 4-1 at the end of Section 4.2.

4.2 Development of the Default Sampling-Circuit Model

As mentioned above, we had two primary considerations in developing our

default sampling-circuit model: that the model approximate measurements of a nose-tonose derived impulse response well, and that the model be robust enough for parametric

study. We begin by comparing nose-to-nose measurements with the Kerley model.

Results are shown in Fig. 4-3. In this comparison we used the Kerley model rather than

Rs

g(t)

Rs

g(t)

Cj0 =

25fF

Rs

g(t)

Cp =

15fF

Cj0 =

40fF

Cj0 =

40fF

(a) (b) (c)

Figure 4-2. Three diode models used by Kerley in SPICE simulations: (a) standard diode model with

nonlinear junction capacitance, (b) constant junction capacitance equal to the zero-bias junction

capacitance, (c) split-diode model with 15 fF of constant capacitance and 25 fF of nonlinear capacitance.
23

the Riad model since we believe it is essentially the same circuit used in nose-to-nose

measurements, but with an input bandwidth of 20 GHz rather than 50 GHz.

The solid curves in Figs. 4-3(a) and (b) plot the output time responses of two

oscilloscopes connected in the nose-to-nose configuration and are approximately equal to

the convolution of the kickout pulse emanating from the first oscilloscope and impulse

response of the second oscilloscope (see Section 2.4). The “Low” and “High” bandwidth

settings on the oscilloscopes set the input bandwidth by changing the reverse bias on the

diodes. The Low bandwidth setting corresponds to an RF measurement input bandwidth

of 20 GHz, and the High bandwidth setting corresponds to a measured input bandwidth

of 50 GHz. The curves were generated from measurements made at NIST and have been

corrected for time-base distortion and impedance mismatches between samplers due to

external connectors and cables [5]. The dashed curves in Fig. 4-3 correspond to our

SPICE nose-to-nose simulation using the Kerley model with values of bias voltage

chosen by comparison to measurement. Because an ideal isolation network connects the

output of one sampler to the input of a second sampler, mismatch correction is not

necessary in these simulations.

The measurement and simulation do not match precisely, nor should they. The

measurements include effects of additional unknown filtering internal to the oscilloscope,

but external to the sampling circuitry, not represented in the models. However, the

similarity of the curves in Figs. 4-3(a) and (b) give us an indication that our SPICE circuit

configuration is ostensibly correct and that its model parameters are, at least, of the same

order as those found in real sampling circuits. Having addressed the issue of model and

measurement comparability, we next turn to the issue of developing a robust model

suitable for parametric study.

Figure 4-4 compares the Riad model to the Kerley model. For best comparison,

we use the same strobe pulse conditions in the two models, the Kerley diode model

1.0

0.8

0.6

0.4

0.2

0.0

-0.2

-0.4

-0.6

Amplitude (norm.)

0.00 0.05 0.10 0.15 0.20

Time (ns)

"Low Bandwidth" Setting 

 Measurement 

 Kerley Model, 0.4 V Bias 

1.0

0.8

0.6

0.4

0.2

0.0

-0.2

-0.4

-0.6

Amplitude (norm.)

0.00 0.05 0.10 0.15 0.20

Time (ns)

"High Bandwidth" Setting

 Measurement

 Kerley Model, 0.5 V Bias 

(a) (b)

Figure 4-3. Comparison between nose-to-nose measurements and simulations using the Kerley sampling

circuit SPICE model. (a) Low bandwidth setting, corresponding to a measured upper input frequency of 20

GHz for the oscilloscope; (b) High bandwidth setting, corresponding to a measured upper input frequency of

50 GHz. The bandwidth setting changes the reverse bias on the sampling diodes.
24

shown in Fig. 4-2(a) (since this diode model is most similar to the Riad diode model), and

the Riad model without packaging parasitics, to best approximate the monolithic structure

of Kerley’s circuit. We compare simulation results for the kickout pulse and impulse

response in Fig. 4-4(a). In Fig. 4-4(b), we plot the frequency-domain correction factor,

E(f), which tells us how similar the nose-to-nose-derived impulse estimate is to the true

impulse response (see Section 2.4). The dashed curve indicates the error in the phase

component of the nose-to-nose-derived impulse response when compared to the true

impulse response. Figure 4-4(b) shows that even though the results of the Riad and

Kerley models are not identical, they are similar, with approximately 1° difference

between them in the phase component of the correction factor at 20 GHz.

While Fig. 4-4 shows that the two models give similar results, Fig. 4-3 shows that

the Kerley model agrees well with measurement. However, the Kerley sampler is difficult

to simulate because inductors are used in the bias lines. The discharge time for these

inductors is quite long, making the simulation touchy and extremely long. Because we

require a robust model capable of performing over a wide parameter space, for the

remainder of this work we define our “default” sampling circuit as the basic

configuration of the Riad sampler, with two important modifications. First, we do not

include the packaging parasitics, to better approximate the monolithic diode structure of

present-day samplers (discussed below in Section 4.2.1). Second, we use a strobe pulse

whose rise time is faster than the 32 ps specified by Riad, to better approximate 50 GHz

samplers currently used in nose-to-nose measurements. The decision on which strobe

pulse rise time to use is detailed in Section 4.2.2. A summary of our default sampler’s

characteristics can be found in Table 4-1.

The magnitude and phase of E(f) were plotted in Fig. 2-7 for the default SPICE

model of the sampler. For the default sampling-circuit configuration that we examined,

our default magnitude error was approximately 0.25 dB, which is lower than the ~0.6 dB

difference between a swept-sine and a nose-to-nose calibrated oscilloscope impulse

response measured in Ref. [5]. Our default phase error was approximately 3.3° at 20

1.0

0.8

0.6

0.4

0.2

0.0

-0.2

Amplitude (norm.)

0.00 0.05 0.10 0.15 0.20

Time (ns)

 Riad Kickout Pulse

 Riad Impulse Response

 Kerley Kickout Pulse

 Kerley Impulse Response 

-4

-2

0

2

4

E | | (dB) or Angle E (degree)

0 5 10 15 20

Frequency (GHz)

 Riad

 Kerley 

 Magnitude 

 Phase

(a) (b)

Figure 4-4. Comparison of the Riad and Kerley sampling circuits. (a) Time-domain representation of kickout

pulses and impulse responses. (b) Frequency-domain magnitude and phase of the correction factor E(f) for the

two models. The Riad circuit has no packaging parasitics and a strobe pulse with a shorter, 10 ps rise time to

better emulate modern samplers. We used a strobe-shaping transmission-line delay of τD is 7 ps for the Riad

sampler and 8 ps for the Kerley sampler.
25

GHz. This is of the order of the phase difference between an EOS measurement and a

nose-to-nose calibrated oscilloscope measurement given in Ref. [9]. These differences

between measured results and our sampling circuit are reasonable, since the

measurements involved complete oscilloscopes and our model only approximates the

sampling circuitry.

4.2.1 Packaging Parasitics

Figures 4-5(a) and (b) show the change in both the magnitude and phase of the

correction factor, E(f), for three different values of diode grading coefficient, γ (defined

in Section 3.1.1). We see a significant change in the phase component of E(f) with and

without packaging parasitics included in the model, especially for γ = 0.5, the value for

Schottky diodes. To better approximate the monolithic diode circuit of present-day

samplers, our default model does not include packaging parasitics.

-5

-4

-3

-2

-1

0

1

2

|E | (dB) or Angle E (degrees)

0 5 10 15 20

Frequency (GHz)

γ = 0 (Constant Cj) 

γ = 0.5

γ = 0.9

 Magnitude

 Phase

Table 4-1. Parameters of the two models discussed above and our default sampling circuit model.

Diode parameters Circuit parameters

Rs

(Ω)

Is

(pA) n Cj0(fF)

φbi

(V) γ

EG

(eV)

CH

(pF)

Vbias

(V)

Rbias

(Ω)

Strobe

(V)

Strobe

rise

(ps)

Packaging

(pF/nH)

Riad

model 19 26.7 1.08 45 0.7 0.5 0.69 4.0 1.63 3600 12 32 15/8

Kerley

model 10 0.03 1.0 40 1.0 0.5 1.11 2.0 lower R+L ~10 ~20 --

Default

model 19 26.7 1.08 45 0.7 0.5 0.69 4.0 1.63 3600 12 10 --

(a) (b)

Figure 4-5. Results from SPICE simulations of Riad’s sampler model: (a) with, and (b) without packaging

parasitics for three different values of diode grading coefficient, γ. Packaging is used in samplers consisting

of discrete components, such as the Riad model shown in (a). To better approximate monolithic structures,

packaging parasitics are neglected in (b).

-5

-4

-3

-2

-1

0

1

2

|E | (dB) or Angle E (degrees)

0 5 10 15 20

Frequency (GHz)

γ = 0 (Constant C j)

γ = 0.5

γ = 0.9

 Magnitude

 Phase
26

4.2.2 Strobe-Excitation-Pulse Rise Time

As mentioned in Section 4.1, the strobe pulse in our simulations is generated

when a step-pulse (approximated in SPICE by a long-duration trapezoidal pulse) is fed

into a shorted transmission line (see Fig. 4-1). The reflection of this step-pulse from the

shorted end turns the strobe off after twice the delay time of the transmission line. The

rise time of the strobe-excitation pulse determines the strobe pulse rise time. The

sharpness of the strobe’s rising edge helps determine the bandwidth of the sampler. Our

goal is to use an excitation-pulse rise time that gives a diode “aperture time” similar to

the 10 ps specified in the literature [2]. The aperture time is the period during which the

strobe forward-biases the sampling diodes causing them to conduct.

Figure 4-6(a) shows SPICE simulations of the strobe pulse of our default

sampling circuit for four different excitation-pulse rise times (the rising edge of the steppulse shown in Fig. 4-1): 5 ps, 10 ps, 15 ps, and 20 ps. The strobe pulse is measured from

the hold capacitor/diode input node to ground, as shown in Fig. 4-1. Note that the strobe

pulse itself is not trapezoidal because the excitation pulse is altered by the characteristics

of the transmission line and the other impedances in the sampling circuit. In addition, the

amplitude of the strobe pulse is much smaller than the 6 V of the trapezoidal excitation

pulse, due to voltage drops elsewhere in the sampling circuit.

Figure 4-6(b) shows simulation results of the sampling diodes’ small-signal

junction voltage using the same excitation-pulse rise times as in Fig. 4-6(a). The smallsignal junction voltage gives us an indication of the aperture time of the sampling circuit.

During the aperture time, the diodes strongly conduct, their conductance is high, and the

voltage drop across the diode, vj, is near zero. In Fig. 4-6(b), excitation-pulse rise times

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Amplitude (volts)

9.00 9.02 9.04 9.06 9.08 9.10

Time (ns)

 Strobe Excitation 

 Pulse Rise Time 

 5 ps

 10 ps

 15 ps

 20 ps

-0.20

-0.15

-0.10

-0.05

Small-Signal Junction Voltage (v)

9.01 9.02 9.03 9.04 9.05

Time (ns)

 Strobe

 Excitation

 Pulse 

 Rise Time 

 5ps

 10ps

 15ps

 20ps

 Aperture Time 

11ps

6ps

4ps

9ps

(a) (b)

Figure 4-6. Simulations using our default sampling circuit model with several different strobeexcitation-pulse rise-times. (a) Strobe pulse (measured from the sampling diode input to ground). (b)

Small-signal junction voltage across the sampling diodes. The legend corresponds to the excitation

pulse’s rise-time, and the markings at the top are the approximate aperture times. Note the decreased

aperture time and the increased leading edge of the small-signal junction voltage for increasing

excitation-pulse rise-times.
27

are given in the legend and approximate aperture times are indicated at the top of the plot.

We see that the strobe with the 10 ps excitation-pulse rise time gives an aperture time of

approximately 9 ps. Because the 50 GHz oscilloscope used in nose-to-nose measurements

is reported to have a 10 ps aperture time [2], we choose a 10 ps rise time for the strobe

excitation pulse. Note that we could also have chosen the 5 ps rise-time excitation pulse

(corresponding to an 11 ps aperture time).

Figure 4-6(b) shows an interesting effect: as the strobe-excitation-pulse rise time

increases (corresponding to a less-steep strobe pulse leading edge), the rise time of the

small-signal junction voltage also increases. This effect can be explained by considering

the parallel branches of the diode model, shown in the inset of Fig. 4-1. One branch

corresponds to a conduction path, and the other to a charge displacement path through the

diode junction capacitance. Charge builds up on the diode junction for a longer time

before the diode actually conducts for strobe pulses with longer rise times. The resulting

displacement current induces the longer leading edge in the small-signal junction voltage.

As discussed in Section 3.2, the longer the leading edge, the greater the phase error in the

nose-to-nose calibration. The amount of this phase-error increase is quantified in a

parametric study in Section 5-3-2.

4.3 SPICE Simulation Convergence Studies

Using our default sampling-circuit model, we next synthesized a set of default

SPICE simulation settings to use in our parametric studies of the nose-to-nose calibration

procedure. We did this by checking simulation convergence with various simulation

settings. In these convergence studies, we determined simulation settings that allow us to

accurately resolve important characteristics of the sampling-circuit behavior while

maintaining the highest computational efficiency possible. Specifically, our convergence

studies involve:

1. Time step (tstep): ensuring we use a sufficiently small time step to capture

features in the nose-to-nose response to the highest frequency of interest (20

GHz in the present work).

2. Sampling cycle period (tsamp): ensuring that our sampling period—the

periodicity of the strobe firing—is long enough to keep the output of one

sampling cycle from affecting the output of subsequent sampling cycles.

3. Input voltage level (vin): ensuring that the input signal amplitudes are small

enough to be operating in the small-signal regime (where the kickout pulse and

impulse response are not affected by the input amplitude).

4. Input pulse width (tpw): ensuring that our finite-duration input pulses can be

accurately deconvolved from the output signal to generate a valid impulse

response estimate.
28

Figure 4-7 shows how each of the items enumerated above is defined in the

context of impulse-response generation, where we sample short-duration pulses whose

period is slightly different from that of the strobe period. The short-duration pulses are

deconvolved from the output signal to give us an impulse response estimate. If we were

able to input a pulse train of true Dirac delta functions, the deconvolution step would be

unnecessary. Refer to Section 2.2 for more detail on this procedure. The settings tsamp and

tdelay are chosen by the user so that the period of the input signal is slightly different from

that of the sampling cycle (the time between strobe firings). This enables us to

progressively sample the entire impulse response. Kickout pulse generation occurs each

time the strobe fires when there is a DC offset between the sampler’s two branches. This

is a more straightforward process in terms of the simulation parameters listed above and

is not illustrated above.

Note that SPICE uses several nonstandard definitions: tstep is defined as the

maximum of a variable time step. SPICE uses a time step smaller than tstep to resolve

rapidly changing features of a waveform (such as the pulses in Fig. 4-7), and uses tstep in

the more slowly varying parts of the time record. The pulse width tpw is defined as the

width of the top of the trapezoidal input pulse, rather than the width at full-width, halfmaximum of the pulse.

Reconstructed

ImpulseResponse

tstep 2 tstep 3 tstep

tstep

tsamp

Periodic Input Signal (ideally impulses)

vin

tpw

tsamp

Periodic Output: Impulse Response of Sampler

(generated each time strobe fires)

tdelay 2•tdelay 3•tdelay

tdelay

tstep

Figure 4-7. Definition of the simulation settings listed above and their role in generating the impulse

response of the sampling circuit. When the period of the sampling cycle (given by tsamp) is different

from the period of the input signal, the increasing delay enables sampling of the entire impulse response

of the sampler. Note that the periodic output consists of a convolution of the sampler’s impulse

response and the input pulse. When the input pulse is a Dirac delta function, no deconvolution of the

input pulse is necessary.
29

4.3.1 Time Step

Figures 4-8(a) and (b) show kickout pulses and impulse responses, respectively,

simulated using the Riad sampling-circuit model with three different maximum time

steps (tstep) in SPICE. See Table 4-1 for other circuit parameters, and Table 4-2 for other

simulation settings. SPICE allows the user to set the maximum time step only, rather than

a fixed time step, as mentioned above.

Figure 4-8(c) shows the magnitude and phase of the correction factor, E(f), for

nose-to-nose simulations using three values of tstep. We see that simulations with tstep =

0.5 ps are very similar to those with a tstep = 0.25 ps, but that an increase in both

magnitude and phase error occurs when tstep = 1.0 ps. Based on this result, we used either

tstep = 0.5 ps in simulations for samplers with identical, balanced diodes, or tstep = 0.25 ps

for samplers with a diode imbalance (see Section 5.4 for simulations of imbalanced

diodes), where more detail was required for accurate simulations.

1.0

0.8

0.6

0.4

0.2

0.0

-0.2

Normalized Response

0.00 0.05 0.10 0.15 0.20

Time (ns)

 Kickout Pulse 

 t step < 0.25 ps 

 t step < 0.5 ps

 t step < 1.0 ps

1.0

0.8

0.6

0.4

0.2

0.0

Normalized Response

0.00 0.05 0.10 0.15 0.20

Time (ns)

 Impulse Response 

 t step < 0.25 ps 

 t step < 0.5 ps

 t step < 1.0 ps

-4

-3

-2

-1

0

1

E | | (dB) or Angle E (deg)

0 5 10 15 20

Frequency (GHz)

 t step < 0.25 ps 

 t step < 0.5 ps 

 t step < 1.0 ps

 Magnitude 

 Phase

(c)

Figure 4-8. Results of nose-to-nose simulations using the Riad sampling-circuit model with three

different maximum time steps in SPICE. (a) kickout pulses; (b) impulse responses; and (c) magnitude and

phase of E(f). Results for both magnitude and phase are similar for tstep = 0.25 and 0.5 ps, but degradation

is observed for tstep = 1.0 ps.

(a) (b)
30

4.3.2 Sampling-Cycle Period

The sampling-cycle period is the interval between strobe firings, and is called

“tsamp” in Fig. 4-7. One point in the reconstructed output signal is acquired each sampling

cycle. For computational efficiency in the SPICE simulations, we would like to make the

sampling-cycle period as short as possible. However, to ensure that charge stored from

one sampling cycle doesn’t affect the charge collected on subsequent sampling cycles, we

must make the period long enough for the hold capacitors to fully discharge between

strobe firings. This will give us the steady-state solution. In commercial oscilloscopes, a

long delay, on the order of milliseconds, exists between samples. In our SPICE

simulations, we hastened discharge by inserting switches across the hold capacitors and

had them discharge the capacitors in a time frame that did not affect the accuracy of our

sampling.

To make sure that our discharge switches performed properly, we ran several

SPICE simulations with various sampling-cycle periods. Selected results are shown in

Fig. 4-9, where we see obvious degradation in the correction factor when the sampling

period is less than 400 ps. In this case the impulse response duration combined with the

discharge switch closure time is not sufficient to allow proper discharge of the hold

capacitor. The sample periods greater than 400 ps should theoretically be identical, but

the effect of numerical error introduces differences in the phase-error calculations. We

used a 500 ps sampling-cycle period for the simulations in this work since a 500 ps

period gives greater computational efficiency than 750 ps or 1000 ps.

-4

-3

-2

-1

0

1

2

E | | (dB) or Angle E (deg)

0 5 10 15 20

Frequency (GHz)

 t samp = 400 ps

 t samp = 500 ps 

 t samp = 750 ps

 t samp = 1000 ps 

 Magnitude 

 Phase

Figure 4-9. Magnitude and phase of E(f) calculated for four sampling-cycle periods: 400 ps, 500 ps, 750

ps, and 1000 ps. We see that the phase error increases for sampling periods both less than and (to a

lesser extent) greater than 500 ps. A sampling-cycle period of 500 ps was chosen as our default in this

work.
31

4.3.3 Input Voltage Level

One important premise of sampling-circuit operation is that the diode’s largesignal behavior is not affected by the input signal. In the diode model we use, the largesignal strobe fixes the diode’s conductance and junction capacitance, and ensures that the

reconstructed response of the sampler to a small-signal input is linear. To calculate the

impulse response and perform a simulation of the nose-to-nose calibration accurately, we

must limit the input signal amplitude, vin, so that the inputs to the sampler are in this

small-signal regime.

We calculated the correction factor E(f) for various values of vin, and present the

results in Fig. 4-10. We see only minimal change in the magnitude and phase of E(f) for

vin less than 0.3 V. For this reason, we used an input level in subsequent simulations of

0.2 V ( Note that measurements presented in Refs. [3] and [11] use a DC offset level of

0.1 V, and Ref. [13] states that the offset level must be less than 0.25 V for small-signal

operation.). This input refers to either the peak amplitude of the pulses in the pulse train

used for impulse response calculations (vin in Fig. 4-7), or the DC voltage offset used for

kickout generation. Even though kickout pulse and impulse response generation are two

different effects, the small-signal requirement is satisfied for both with vin = 0.2 V.

4.3.4 Input Pulse Width

We next verified that the deconvolution used to calculate the impulse response

was sufficiently accurate. To generate the impulse response, we ideally would use a pulse

train of Dirac delta functions as the input to the sampler. However, any pulsed input in

SPICE has finite width. Therefore, the input pulse must be deconvolved from the output

response in our impulse response simulations. We used trapezoidal pulses such as those

shown in Fig. 4-7 with rise- and fall-times of 0.5 ps and with pulse width, tpw.

-4

-3

-2

-1

0

1

2

E | | (dB) or Angle E (deg)

0 5 10 15 20

Frequency (GHz)

 vin = 0.1 V

 vin = 0.2 V

 vin = 0.3 V

 vin = 0.4 V 

 Magnitude

 Phase

Figure 4-10. Magnitude and phase of E(f) of the nose-to-nose simulations with various input levels, vin. The

similarity of the correction factors indicates that a 0.2 V input signal or DC offset will be well within the

small-signal regime of our sampling circuit model.
32

As shown in Fig. 4-11, there is very little difference in the phase component of

E(f) when using different pulse widths, indicating that our deconvolution process is

adequate. In our SPICE simulations, we used the smallest tpw of 0.5 ps. For input pulse

widths smaller than this, the maximum timestep in the SPICE simulation had to be

decreased, reducing computational efficiency.

4.4 More Details on the SPICE Simulations

Table 4-2 summarizes our default simulation settings. These settings are chosen to

maximize computation efficiency while accurately resolving details in the parametric

studies discussed in Section 5. Additional details on the SPICE simulations are found

below. The SPICE code itself is found in Section 4.5.

4.4.1 Impulse Response Calculations

In our default SPICE simulations, the impulse response was found by performing

a transient simulation over a large number of sampling cycles. A typical impulseresponse simulation covered 85 ns. With a sampling-cycle period of 500 ps, that was 170

cycles. We made the strobe repetition period 501 ps, and the impulse repetition period

Table 4-2. Summary of default simulation settings

Simulation Parameter Value

Maximum time step, tstep 0.5 ps

Sampling-cycle period, tsamp 500 ps

Input voltage level, vin 0.2 V

Input pulse width, tpw 0.5 ps

Figure 4-11. E(f) for the Riad sampler with various input pulse widths, tpw. Very little variation in the

phase component of E(f) is seen. We use an input pulse width of 0.5 ps in our SPICE simulations.

-4

-3

-2

-1

0

1

2

E | | (dB) or Angle E (deg)

0 5 10 15 20

Frequency (GHz)

 t pw = 0.5 ps 

 t pw = 1.0 ps

 t pw = 5.0 ps

 t pw =10.0 ps 

 Magnitude

 Phase
33

500 ps, for a tdelay (see Fig. 4-7) of 1 ps. We found empirically that 1 ps offered sufficient

resolution of the input pulse train for the impulse response calculation.

As mentioned previously, we inserted switches around the hold capacitors to

discharge the capacitors after each sample was taken. This allowed us to use a shorter

sampling period and provided greater computational efficiency than if we had to wait for

the capacitors to discharge on their own, as occurs in physical samplers. The switches in

our default simulations fired 365 ps after the strobe fired. This time delay was found to be

sufficient to enable complete reconstruction of the impulse response. See Fig. 4-12 for

plots of the time record of a typical impulse response simulation.

4.4.2 Kickout Pulse Calculations

The kickout pulse was generated in a second SPICE transient simulation with a

much shorter time record. We applied an offset DC voltage to the bias lines, and

performed a simulation over a small number of sampling cycles. In a typical calculation,

we specified a transient simulation running from 9.0 ns to 9.35 ns. The delay in starting

the kickout record gave the DC voltages in the sampler a chance to settle before we

started collecting data. Again, the switch fired to discharge the hold capacitors 365 ps

after the strobe fired. See Fig. 4-13.

4.4.3 Nose-to-Nose Calculation and Error

Once the impulse time record and kickout pulse were generated and saved to a

data file (the impulse response time record is typically 5 to 10 Mb), the two files were

imported into an external analysis program. This program first sampled the impulseresponse time record periodically at a user-specified point in time relative to the strobe

pulse. We used a sample time of 0.26 ns after the strobe fires, a bit more than halfway

30

20

10

0

-10

Impulse Response Amplitude (mV)

0 20 40 60 80

Time (ns)

-1.0

-0.5

0.0

0.5

1.0

Impulse Response Amplitude (mv)

20.0 20.5 21.0 21.5 22.0

Time (ns)

 Discharge Switch Fires 

 Strobe Fires 

 Sample Taken 

 0.26 ns 

 After Strobe 

(a) (b)

Figure 4-12. Plots of the time record directly resulting from the SPICE simulations of the impulse response.

(a) The entire time record. Note the impulse response can be seen from ~10 ns to 30 ns. (b) A close-up of the

same time record. The large pulses are the strobe firing, the small plateaus correspond to the charge stored on

the hold capacitors in response to the input pulse train of very short pulses, and the rapid drop-offs of the

plateaus correspond to the discharging of the hold capacitors by the switches.
34

between strobe pulses, as shown in Fig. 4-12(b). The program then deconvolved the input

pulse from the impulse response time record to find the actual impulse response of the

sampler. This operation was done in the frequency domain, where deconvolution

becomes division of two frequency responses. The program then multiplied the

frequency-domain representations of the kickout pulse and impulse response to form the

nose-to-nose response (multiplication in the frequency domain is equivalent to

convolution in the time domain). Finally, the program divided the square root of the noseto-nose response by the impulse response to form the error term, as defined in eq (2-4).

The magnitude and phase components of the error term were then saved to output files.

The phase was detrended according to the process described in Section 2.4 using a

second external program.

0.10

0.08

0.06

0.04

0.02

0.00

Kickout Amplitude (v)

9.0 9.1 9.2 9.3 9.4 9.5 9.6

Time (ns)

 Kickout #1 

 Kickout #2 

 Switch Fires Here 

Figure 4-13. One complete cycle of the kickout pulse. The wavy line indicates that the hold capacitor

discharge switches have fired. The thicker dashed line shows the length of a typical kickout pulse

simulation.
35

Appendix for Section 4. SPICE Code

The actual SPICE code used in our simulations follows. To run this code, we

concatenate the “Sampler.txt” file with either the “impulse.txt” or the “kickout.txt.” The

code runs in SPICE3.

Sampler.txt:

Sampler

** A spice model for the HP-1430A sampler.

** From Sedki M. Riad, "Modeling of the HP-1430A feedthrough wide-band (28-ps)

** sampling head," IEEE Trans. Instrumentation and Measurement, vol. IM-31,

** no. 2, pp. 110-115, June 1982.

**

** This spice model includes transmission lines to shape the strobe pulse

** and to allow the strobe impedance to be included.

** It also includes switches to hasten the hold capacitor discharge time.

**

** To run:

** The program runs on SPICE3.

** You need to concatenate this Sampler.txt file with either Impulse.txt or Kickout.txt.

** For example, to generate the kickout pulse, on a Unix machine, type

** rm Spice.txt

** cat Sampler.txt kickout.txt > Spice.txt

** spice3 Spice.txt

** run

** set nopage

** print v(1) > kick7.dat

** quit

**

** More examples can be found in the file spice.commands.txt

**

**

.OPTIONS NOPAGE

** This is the sampler circuit.

** The format is: SAMPLER [input] [strobe] [output] [kickoutbias] [dischargeswitch] [ground]

**

.SUBCKT SAMPLER 23 25 30 33 70 0

**

** This is the termination impedance of the sampler.

Rterm 23 0 50

**

** The sampling diodes. For the asymmetric case, increase the area of one diode,

** decrease the other, and run two SPICE simulations with + and - offset voltages

** for the kickout pulse. Only one impulse response simulation.

D1 23 24 DIODE_SAMPLE9 1.0

** Second diode is connected between 26 and 23

D2 26 23 DIODE_SAMPLE9 1.0

**

** The diode bias supply

** The hold capacitors discharge throught these diode bias resistors.

** Adjusting them changes the hold time.

** These resistors also change the tail of the sampler impulse response.

** Smaller values of resistor reduce the tail.

rbias1 24 31 3600

vbias1 31 33 1.63

rbias2 26 32 3600

vbias2 32 33 -1.63

**

** Voltage controlled switches discharge the hold capacitors.

** They use the model SWMODEL.

** The last two parameters are the controlling voltage nodes.

S3 24 31 70 0 SWMODEL OFF

S4 26 32 70 0 SWMODEL OFF

**
36

**

** The hold capacitors. These values are not defined in Riad's paper.

Chold1 24 10 4pF

Chold2 26 11 4pF

**

** The source impedance of the strobe. Riad thinks this is 57 ohms total.

** So each should be 57/2 ohms = 28.5 ohms.

Rsource1 25 10 28.5

Rsource2 27 11 28.5

*Rsource1 25 10 50

*Rsource2 27 11 50

**

** A distributed LC model for the transmission lines that shape the strobe voltage.

** Riad says that these have Z0=85.2 ohms and Td=22.2ps.

** To maintain consistency with previous calculations, we will use Td=7ps.

** The inductance per unit lenght: L=Z0/c (c=3E8 m/s=speed of light)

** The capacitance per unit lenght: C=1/(Z0*c)

** The total length: l=TD*c (Model values: L'=l*L/6, C'=l*C/2)

Ltrans1 10 42 0.1nH

Ltrans2 11 43 0.1nH

Ctrans1 42 43 0.041pF

Ltrans3 42 44 0.1nH

Ltrans4 43 45 0.1nH

Ctrans2 44 45 0.041pF

Ltrans5 44 46 0.1nH

Ltrans6 45 47 0.1nH

** These switchs terminate the transmission line with its characteristic impedance

** during the hold capacitor discharge cycle.

Strans1 46 0 70 0 SWMODELTR OFF

Strans2 47 0 70 0 SWMODELTR OFF

** These switches discharge the hold capacitors and keep the transmission lines

** from ringing when the strobe turns off.

S5 10 0 70 0 SWMODEL OFF

S6 11 0 70 0 SWMODEL OFF

**

** The voltage strobe is connected to 25.

** This is the inverted strobe.

EX2 27 0 0 25 1.0

**

** These amplifiers sample and add the voltage injected into the capacitors.

E1 29 33 24 0 0.5

E2 30 29 26 0 0.5

**

.ENDS SAMPLER

**

** Below is half of the strobe circuit. It is set to sample every ps.

** The sampling period is 0.5 ns.

** Setting the voltage to 6V here gives a total strobe voltage of 12V.

** The strobe turns on, then stays on until we discharge the hold capacitors.

** Then the strobe turns off.

** Format is PULSE([V start] [V pulse] [time delay] [rise time] [fall time] [pulse width] [period])

.SUBCKT STROBE 40 41

vstrobe 40 41 DC 0 PULSE(0 6 0NS 10PS 1PS 356PS 0.501NS)

.ENDS STROBE

**

** These are the diode parameters.

** Diode model for a Schottky barrier diode. RS is and Cj can be up in the sampler

.model DIODE_SAMPLE9 D (IS=2.67e-11 N=1.08 VJ=0.7 M=0.5 EG=0.69 CJO=0.045pF RS=19)

** These are the switch parameters.

.model SWMODEL SW (VT=1 VH=0 RON=0.1 ROFF=100000)

.model SWMODELTR SW (VT=1 VH=0 RON=42.6 ROFF=0.1)

.model SWMODEL1 SW (VT=1 VH=0 RON=100000 ROFF=.01)

.model SWMODEL2 SW (VT=1 VH=0 RON=.01 ROFF=100000)

**
37

impulse.txt **

************************************************************************************

** Type "print v(4) > time.txt" to save the impulse response. *

************************************************************************************

**

** Put in a delay in the input impulse so that it starts well after the first sampling pulse ends.

** This is needed because the sampler responds to signals at its input during the duration of the

** sampling pulse and even slightly after the sampling pulse has ended.

** This offset is t_0(=0.06ns in this case).

** Do not make vin too large, as a large vin will turn the diodes on.

** Format is PULSE([V start] [V pulse] [time delay] [rise time] [fall time] [pulse width] [period])

vin 1 0 DC 0 PULSE(0 0.2 0.06NS 0.5PS 0.5PS 0.5PS 0.5NS)

rin 1 2 50

** This is the control for the discharge switches for the hold capacitors.

** Format is PULSE([V start] [V pulse] [time delay] [rise time] [fall time] [pulse width] [period])

vswitch 6 0 DC 0 PULSE(0 5 365PS 1PS 1PS 20PS 0.501NS)

** One half of the strobe source

XS1 0 3 STROBE

** Sampler format is: SAMPLER [input] [strobe] [output] [kickoutbias] [dischargeswitch] [ground]

X1 2 3 4 0 6 0 SAMPLER

** The simulation: Format is .tran [tstep] [tstop] [tstart] [tmax]

** tstep = tmax = max time step (make smaller to smooth out plots)

.tran 0.0005ns 85ns 0 0.0005ns

.end

kickout.txt **

********************************************************************************************

** For symmetric diodes case: type “print v(1) > kick.txt” to save the kickout waveform. *

** For asymmetric case: type “print v(1) > kick.txt” to save the first kickout waveform. *

** Then change the voltage in vcap to -0.2, rerun, and type “print v(1) > kick1.txt” to *

** save the second kickout waveform. *

********************************************************************************************

** Check the kickout pulses to see when they settle down. Then

** choose a suitable delay. This same delay should probably used for the other signals too.

**

** The time delay of the switch should be greater than the point where

** the actual voltage sample is taken (externally)

vcap 5 0 0.2

** One half of the strobe source

XS1 0 3 STROBE

** Format is PULSE([V start] [V pulse] [time delay] [rise time] [fall time] [pulse width] [period])

vswitch 6 0 DC 0 PULSE(0 5 0.365NS 1PS 1PS 20PS 0.501NS)

rin 1 0 50

** Sampler format is: SAMPLER [input] [strobe] [output] [kickoutbias] [dischargeswitch] [ground]

X1 1 3 4 5 6 0 SAMPLER

** The simulation: Format is .tran [tstep] [tstop] [tstart] [tmax]

.tran 0.0005ns 9.35ns 9ns 0.0005ns

**

spice.commands.txt

This file gives the actual commands to generate the kickout pulse or impulse response in SPICE. One can copy and paste

these commands directly onto the command line of a computer capable of running SPICE3.

** Impulse response simulations

rm Spice.txt

cat Sampler.txt impulse.txt > Spice.txt

spice3 Spice.txt

run

set nopage

print v(4) > time7.dat

exit
38

vi time7.dat

4d

:wq

** Kickout pulse simulations

rm Spice.txt

cat Sampler.txt kickout.txt > Spice.txt

spice3 Spice.txt

run

set nopage

print v(1) > kick7.dat

quit

vi kick7.dat

4d

:wq
39

5. Parametric Studies Using SPICE

In this section, we investigate the sensitivity of the nose-to-nose calibration to

variation in sampling-circuit component values. We systematically varied individual

circuit element values in our default sampling-circuit model and noted changes in the

magnitude and phase of the correction factor, E(f). These sets of simulations clearly

demonstrate which sampling-circuit elements dominate the phase error in the nose-tonose calibration and how sensitive the phase error is to variation in component values.

As mentioned in Section 1, this type of study would be practically impossible to

conduct experimentally due to fabrication and measurement limitations. In addition, we

do not know the specifics of the actual sampling-circuit component values. However,

based on the sampling circuits described in Refs. [19] and [24], we can estimate a

realistic range of values for this particular sampling-circuit configuration. Using the

parametric studies presented in this section as a foundation, in Section 6 we quantify the

sensitivity of the phase error by generating a combined uncertainty statement for a given

range of parameter values.

The parametric studies below used the default sampler model developed in

Section 4.2 (a model based on Ref. [19], with no packaging parasitics and faster strobeexcitation-pulse rise time). For each model parameter, we describe the role it plays in the

sampling circuit, the range of parameter values that we chose, and show graphical

simulation results of the corresponding correction factor, E(f). E(f) (defined in Section

2.4) describes how different our nose-to-nose-derived estimate of the sampler’s impulse

response is from the true impulse response for a given model. The dashed curve in each

plot represents the error in the phase component of the impulse-response estimate when

compared to the impulse response derived using the method described in Section 2.2.

Because the nose-to-nose calibration is theoretically exact for ideal samplers (see

Section 3.2 and Refs. [3, 4, 15, 20]), we find that the errors in the calibration depend on

the nonidealities of real sampling circuits, such as using a shorted transmission line to

shape the strobe pulse, and using diodes with nonlinear junction capacitance as switches.

We will see that the diode’s nonlinear junction capacitance and its interaction with other

circuit elements is especially important. As will be shown, some of these interactions are

quite complicated. While we provide analytic descriptions for some of these interactions,

we must rely on the numerical techniques for the bulk of the analysis.

5.1 Sampling Diode Model Parameters

5.1.1 Description

Schottky-barrier diodes are used in the samplers we studied, and their operation

can be modeled quite accurately with a few important parameters [21]. Equations relating

these parameters to the diode’s large-signal behavior are used in SPICE [22], as discussed

in detail in Section 3.1. We repeat relevant equations here for convenience. Specifically,

the large-signal current through the diode is given by
40

IV I qV

nkT ( ) exp j S

j = 



 



 − 



 



1, (5-1)

where Vj is the large-signal voltage across the diode junction (not including the drop

across the spreading resistance Rs), Is is the reverse saturation current, q is the charge on

an electron, n is the ideality factor, k is the Boltzmann constant, and T is the temperature

(K). The diode junction capacitance is:

C V

C

V t j j

j0

j

bi

( )

( ) =

− 



 



1  φ

γ , (5-2)

where Cj0 is the zero-voltage junction capacitance (note that Cj = Cj0, a constant, when Vj

= 0), φbi is the junction’s built-in potential, and γ is the grading coefficient (γ = 0.5 for

Schottky-barrier diodes [21]).

We divide our parametric studies of the diode model parameters into two

categories: those involving all diode parameters except junction capacitance and those

involving C(Vj). We do this because variation of most diode parameters has little effect

on the phase error in the nose-to-nose estimation of the impulse response, while the

nonlinearity associated with the diode’s junction capacitance has a significant effect. We

include the built-in potential, φbi, of the diode junction in the first category since it is

shown to have little effect on the phase error. We also include a section with results for

Kerley’s split diode model discussed in Section 4.1.2 and shown in Fig. 4-2(c).

Diode parameters for our default sampler model (Table 4-1) are reproduced below

in Table 5-1 for convenience. Parametric studies were conducted to cover the range of

both the Riad (default) and the Kerley models, and to go somewhat outside this range.

5.1.2 Simulation Results

5.1.2.1 Diode Model Parameters Except Junction Capacitance

Figures 5-1(a) through (d) show changes in the phase component of E(f) as we

varied diode parameters other than those related to junction capacitance. We see that

(c) (d)

Table 5-1. Diode model parameters.

Rs (Ω) Is

(pA) n Cj0(fF)

φbi

(V) γ

EG

(eV)

τt

(ps)

Riad

model 19 26.7 1.08 45 0.7 0.5 0.69 0

Kerley

model 10 0.03 1.0 40 1.0 0.5 1.11 0

Default

model 19 26.7 1.08 45 0.7 0.5 0.69 0
41

these diode parameters change the phase error of the nose-to-nose estimation by a

maximum of approximately 1°. We found that varying the energy gap, EG, has negligible

effect on the phase error, so do not include it here. As mentioned in Section 3.1, SPICE

uses EG to differentiate between different types of diodes.

5.1.2.2 Parameters Involving Diode Junction Capacitance

Figure 5-2 shows that the correction factor, E(f), increases as the zero-voltage

junction capacitance, Cj0, increases. As shown in eq (5-2), increasing Cj0 results in

stronger nonlinearity of the diode junction capacitance, C(Vj). The corresponding

increase in the phase error is explained in Section 3.2 and Refs. [15, 20]. Note that the

variation in Cj0 in this example is more extreme than we may expect in a real Schottkybarrier diode. Component variation is typically no more than ±25 % (~0.034 pF to ~0.056

pF), and is generally closer to ±10 %, according to the data sheet for a GaAs Schottkybarrier diode with a SPICE model similar to our default model [25].

Figure 5-3 shows the effect on the correction factor, E(f), for parametric variation

of the diode’s grading coefficient, γ. We see that E(f) is positively correlated to the value

of γ. According to eq (5-2), the larger the grading coefficient, the higher the degree of

-5

-4

-3

-2

-1

0

1

2

E| | (dB) or Angle E (deg)

0 5 10 15 20

Frequency (GHz)

 N = 1.0

 N = 1.08 (default) 

 N = 1.18

 N = 1.28

 N = 1.5

-5

-4

-3

-2

-1

0

1

2

E| | (dB) or Angle E (deg)

0 5 10 15 20

Frequency (GHz)

 Rs = 5 Ω

 Rs = 10 Ω

 Rs = 19 Ω (default) 

 Rs = 25 Ω

-5

-4

-3

-2

-1

0

1

2

E| | (dB) or Angle E (deg)

0 5 10 15 20

Frequency (GHz)

 Is = 2.67e-11 (default) 

 Is = 2.0e-10

 Is = 2.67e-13

 Is = 2.67e-15

(a) (b)

-5

-4

-3

-2

-1

0

1

2

E| | (dB) or Angle E (deg)

0 5 10 15 20

Frequency (GHz)

φbi = 0.7 V (default) 

φbi = 0.5 V

φbi = 1.0 V

(c) (d)

Figure 5-1. Phase of the correction factor, E(f), of nose-to-nose simulations as we vary diode parameters:

(a) Ideality factor, n; (b) Built-in potential, φbi; (c) Saturation current, Is; (d) Spreading resistance, Rs. The

maximum change in the phase component of E(f) is approximately 1° for this set of parametric studies.
42

nonlinearity in the diode junction capacitance. γ = 0.5 corresponds to the default sampler

model.

This simulation highlights an important conclusion that the error in the nose-tonose calibration (magnitude and phase) is essentially zero when the junction capacitance

is constant and equal to Cj0 (for γ = 0). However, constant junction capacitance is

physically impossible for real diodes.

5.1.2.3 Split Diode Model

In Kerley’s sampler model [24], three diode configurations were discussed—one

with constant junction capacitance, one with the standard SPICE nonlinear junction

capacitance, and one with a split-diode capacitance model, as shown in Fig. 4-2. Based

on the above discussion of diode junction capacitance, we expect that the constantjunction-capacitance model will have zero phase error and that the split diode model will

have a phase error somewhat lower than that of the default diode model. These results are

shown in Fig. 5-4.

-8

-6

-4

-2

0

2

E| | (dB) or Angle E (deg)

0 5 10 15 20

Frequency (GHz)

 Cj0 = 0.0125 pF

 Cj0 = 0.025 pF

 Cj0 = 0.045 pF (default) 

 Cj0 = 0.09 pF

 Magnitude

 Phase

Figure 5-3. E(f) for various values of grading coefficient, γ. Note that the error in the nose-to-nose

calibration is nearly zero when the junction capacitance is constant (for γ = 0.0, a physical impossibility).

Figure 5-2. E(f) for different values of zero-voltage diode junction capacitance. Cj0 = 0.045 pF in Riad’s

model, Cj0 = 0.040 pF in John Kerley’s model, and Cj0 = 0.025 pF in the split-diode model in Section 4.1.2.

Other model parameter values are for the default diode model specified in Table 5-1. Note that the

maximum variation in Cj0 is closer to 25 % for a Schottky-barrier diode similar to our default model [25].

-5

-4

-3

-2

-1

0

1

2

E| | (dB) or Angle E (deg)

0 5 10 15 20

Frequency (GHz)

Strobe Generator Impedance 

 67 Ω

 62 Ω

 58 Ω (default) 

 52 Ω

 47 Ω

 Magnitude

 Phase
43

The default sampler model has the most conservative (highest) estimate of E(f).

The diode manufacturer has stated that the split model is probably the most accurate [24],

so we expect that the error in real nose-to-nose calibrations due to the sampling circuitry

will be somewhat smaller than that predicted using our default model.

5.2 Hold Capacitors

5.2.1 Description

The hold capacitors collect and store charge in an amount that is ideally

proportional to the input signal during each sampling cycle, as discussed in Section 2.1.

Charge is also transferred to the hold capacitors by the strobe pulse each sampling cycle,

although the voltages induced by this charge ideally cancel at the output amplifier (since

they are of opposite polarity). Here we investigate how charging the hold capacitors

during each sampling cycle affects the nose-to-nose calibration by changing the

impedance presented to the sampling diodes.

As discussed in Refs. [2] and [15], the RC time constant associated with the

strobe generator impedance and hold capacitor is generally much longer than the aperture

time of the sampling circuit. Hence, the hold capacitor appears almost as a short to the

diode circuit during the aperture time. For example, in our default sampler model, the RC

time constant of one branch of the 57 Ω strobe generator and the 4 pF hold capacitor is

(28.5) × (4e-12) = 114 ps, while the aperture time is approximately 10 ps [2]. Therefore,

in the default sampler model, we expect to see little effect on the circuit’s impedance due

to charging of the hold capacitor.

Figure 5-4. E(f) for the three diode models shown in Fig. 4-2.

-4

-2

0

2

4

|E | (dB) or Angle E (deg)

0 5 10 15 20

Frequency (GHz)

 Constant Diode Capacitance

 Default Diode Model 

 Split Diode Model

 Magnitude

 Phase
44

The hold capacitor value in Kerley’s model of the sampler is given as 2 pF, the

value also given by Verspecht and Rush in Ref. [2]. We conducted simulations with hold

capacitor values ranging from 1 pF to 5 pF, with 4 pF our default model value.

5.2.2 Simulation Results

As expected, in Fig. 5-5 we see little effect on the phase error for the various CH

values, with the overall increase in phase error less than half a degree at 20 GHz for the

smallest value of hold capacitor, 1 pF.

5.3 Strobe-Generator and Strobe Pulse-Shaping Circuitry

In our parametric studies, the strobe-pulse circuitry is seen to have a significant

effect on the phase error in the nose-to-nose calibration. Many of these effects can be

linked to interaction of the strobe and the sampling diode’s nonlinear junction

capacitance. As discussed in Section 3.2, the phase error introduced into the nose-to-nose

calibration by the nonlinear junction capacitance arises from the requirement for charge

conservation as the diodes turn on and off. The strobe pulse will naturally have an impact

on this transition.

As a result, the way that we model the strobe pulse, including its rise time,

amplitude, and the strobe-generator impedance, will affect the operation of the sampling

diodes, and thus affect E(f). That is, in the physical sampler, the strobe pulse and input

pulses utilize separate ground schemes, so the direct effect of the strobe generator

impedance on the small-signal behavior of the sampler is difficult to accurately model. In

the following subsections we look at the effect of these strobe pulse-related parameters

on the estimated error in the nose-to-nose calibration.

5.3.1 Ideal Source and Transmission Line Pulse Shaping

5.3.1.1 Description

In the original nose-to-nose analyses [2-4] and in our earlier work [16], the strobe

generator was modeled as an ideal voltage source with source impedance of 0 Ω. In Ref.

Figure 5-5. E(f) as a function of hold capacitor value.

-5

-4

-3

-2

-1

0

1

2

E| | (dB) or Angle E (deg)

0 5 10 15 20

Frequency (GHz)

Hold Capacitor Values

1 pF

 2 pF

 3 pF

 4 pF (default) 

 5 pF

 Magnitude

 Phase
45

[16], this ideal strobe pulse is trapezoidal, with rise- and fall-times and pulse width

specified as SPICE parameters. However, the sampling circuits described in Riad [19]

and in Kerley [24] characterize the strobe using a step generator with a given source

impedance that drives a shorted transmission line. Reflection of the leading edge of this

step pulse off the shorted end of the transmission line turns the strobe pulse off.

To implement this transmission-line-based strobe generator in our default

sampling-circuit model, we used a strobe excitation pulse consisting of a long-duration

ideal trapezoidal pulse, effectively a step pulse. This step pulse excites a shorted

transmission line which we modeled with lumped elements. In our model and in Ref.

[19], the transmission line has a characteristic impedance Z0 of 85.2 Ω.

The strobe is applied differentially across the two sampling-circuit branches with

the input port at zero potential, as shown in Fig. 4-1. To model this, we applied two

strobe excitation pulses, one positive and one negative, across the two sampler branches.

The strobe excitation pulse generator had a 57 Ω source impedance (Rstrobe = 28.5 Ω in

Fig. 4-1), and the strobe excitation amplitude was 12 V (Strobe Excitation and Inverted

Strobe Excitation equal ±6 V respectively in Fig. 4-1). Below we examine the effects on

the error in the nose-to-nose calibration of using the realistic, transmission-line-shaped

strobe pulse rather than the perfect trapezoidal pulse of Ref. [16].

We performed simulations using the two types of strobe pulses. We used two

different strobe-pulse rise times to see whether the effect of rise time on the phase error

was similar for both types of sources. The transmission line strobe source was excited

with step pulses having both 10 and 15 ps rise times (our default sampling-circuit model

has an excitation-pulse rise time of 10 ps). The ideal-source strobe pulses had 15 and 20

ps rise times. We chose these rise time values to give approximately the same dV/dt for

the two types of sources. Theoretically the pulse width is not critical as long as the

sampling diodes are forward biased long enough to induce a charge proportional to the

input signal on the hold capacitors.

-4

-2

0

2

4

|E | (dB) or AngleE (deg)

0 5 10 15 20

Frequency (GHz)

 Ideal Source, tr=15ps 

 Ideal Source, tr=20ps

 TX Line Source, tr=10ps 

 TX Line Source, tr=15ps

 Magnitude 

 Phase

4

3

2

1

0

Amplitude (v)

9.00 9.02 9.04 9.06 9.08 9.10 9.12

Time (ns)

Strobe Pulse

 Ideal Source, tr=15ps 

 Ideal Source, tr=20ps 

 TX Line, tr=10ps (default) 

 TX Line, tr=15ps

(a) (b)

Figure 5-6. Comparison of strobe pulses formed using ideal pulse sources and realistic, transmission-line

pulse shaping. (a) Strobe pulses in the time domain; (b) E(f) associated with these strobe pulses.
46

5.3.1.2 Simulation Results

Figure 5-6(a) compares strobe pulses formed using the SPICE ideal pulse source

to those using the shorted transmission line described above. The correction factor, E(f),

associated with these different strobe sources is shown in Fig. 5-6(b). We see an

increased phase error with the transmission-line-shaped strobe. This increase in error can

be attributed, at least in part, to the interaction of the reactances of the realistic strobe

source (the transmission line is a ladder network consisting of inductors and capacitors)

with the diode’s nonlinear junction capacitance, rather than the difference in shape

between the two types of strobes. The key to this argument is shown in Fig. 5-4, where

we see essentially zero magnitude and phase error for a constant diode junction

capacitance, even when the transmission line based strobe circuitry is used.

5.3.2 Strobe-Pulse Rise Time

5.3.2.1 Description

In this subsection, we demonstrate that the strobe-pulse rise time has a significant

effect on the phase error in our simulations of the nose-to-nose calibration. When the

strobe has a longer rise time, the diode junction capacitance must maintain charge

conservation for a longer time before the diode conducts, as discussed in Section 3.2.

This causes the small-signal junction voltage to change instantaneously, inducing a

corresponding change in the kickout pulse before and after the diode conducts.

The 10 ps rise time of the default model value was chosen to approximate the 50

GHz input bandwidth of the sampling heads used in nose-to-nose measurements, as

opposed to the longer rise times in the Riad (32 ps) and Kerley models. These models

were based on sampling circuits with 20 GHz input bandwidths. In the following, we

chose strobe-excitation-pulse rise time values ranging from 5 to 20 ps. The excitation

pulse incident on the transmission line described creates the strobe pulse.

5.3.2.2 Simulation Results

Figure 5-7(a) shows the strobe pulse for various strobe-excitation- pulse rise time

(tr) values. This figure is reproduced from Section 4.2.2 [Fig. 4-6(a)], where we described

how we chose the strobe pulse rise time for our default model. Note that the default

sampling-circuit model has a strobe excitation-pulse rise time of 10 ps, while the rise

time of the strobe pulse itself will be somewhat different from excitation- pulse rise time

due to interaction of the transmission line impedances with other circuitry.

An increase in the phase error with longer strobe-pulse rise time is shown in Fig.

5-7(b). To explain this increase in error, we turn to the time domain. Figure 5-8 shows a

plot of kickout pulses and impulse responses with different strobe-excitation-pulse rise

times. Two things are noteworthy in Fig. 5-8: first, the full-width, half-maximum

(FWHM) width of the kickout pulse or impulse response is narrower with a longer strobe
47

pulse rise time. This is because the diode’s aperture time is shorter when the strobe pulse

rise time is longer [see Fig. 4-6(b)] for the sampling-circuit configuration we use.

The second item of interest is that, as the strobe-excitation-pulse rise time

increases, the effect of the nonlinear capacitance on the kickout pulse becomes more

pronounced. This effect, caused by charge conservation, as discussed in Section 3.2, is to

broaden the rising edge and to introduce more undershoot on the falling edge of the

kickout pulse with respect to the impulse response of the sampler.

5.3.3 Strobe-Generator Impedance

5.3.3.1 Description

The impedance of the strobe generator is modeled as a resistance in series with

the strobe source, called Rstrobe in Fig. 4-1. As mentioned above, our SPICE model uses

two sources to represent the strobe generator. We use a value for Rstrobe equal to half the

1.0

0.8

0.6

0.4

0.2

0.0

Amplitude (norm.)

0.00 0.05 0.10 0.15 0.20 0.25

Time (ns)

tr values:

 5ps

 10 ps

 15 ps 

 20 ps

 FWHM ~18ps ~15ps ~13ps ~12ps

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Amplitude (volts)

9.00 9.02 9.04 9.06 9.08 9.10

Time (ns)

 Strobe Excitation 

 Pulse Rise Time 

 5 ps

 10 ps

 15 ps

 20 ps

-6

-4

-2

0

2

E| | (dB) or Angle E (deg)

0 5 10 15 20

Frequency (GHz)

 Strobe Excitation 

 Pulse Rise Time

 tr = 5ps

 tr = 10ps (default) 

 tr = 15ps

 tr = 20ps

 Magnitude

 Phase

(a) (b)

Figure 5-7. (a) Strobe pulses and (b) E(f) for different strobe-excitation-pulse rise times.

Figure 5-8. Kickout pulses (solid lines with symbols) and impulse responses (dashed lines) for several

different strobe-pulse rise times. Note that the kickout becomes less similar to the impulse response for

longer rise times. The time axis has been shifted to better illustrate the differences.
48

total generator impedance in each branch. In the physical sampler, the strobe pulse and

input pulses utilize separate ground schemes. The model presented here is an

approximation based on the characteristic impedances of waveguiding elements

associated with the strobe source.

The default value of strobe-generator impedance is 57 Ω, which is distributed

across the two sampling-circuit branches as lumped resistors of 28.5 Ω each. We chose

strobe generator impedances ranging from 47 Ω to 67 Ω, corresponding to values of

Rstrobe ranging from 23.5 Ω to 33.5 Ω.

5.3.3.2 Simulation Results

Figure 5-9 shows that the strobe-generator impedance can have a significant

effect on the phase component of the phase error. The reason why the phase error

increases for smaller values of Rstrobe is not so obvious, especially when we look at the

time domain representation of the kickout pulse (solid lines) and impulse response

1.0

0.8

0.6

0.4

0.2

0.0

Amplitude (norm.)

0.05 0.10 0.15 0.20 0.25

Time (ns)

 Strobe

 Generator 

 Impedance 

 47 Ω

 58 Ω

 67 Ω

0.14

0.12

0.10

0.08

0.06

0.04

0.02

0.00

-0.02

Amplitude (v)

9.00 9.02 9.04 9.06 9.08 9.10

Time (ns)

 Strobe

 Generator

 Impedance 

 47 Ω

 58 Ω

 67 Ω

(a) (b)

Figure 5-10. Kickout pulses (solid) and impulse responses (dashed) for different values of strobe-generator

impedance. (a) Time axis shifted for better resolution; (b) Non-time-shifted, non-normalized kickout

pulses.

Figure 5-9. Effect of changing the strobe-generator impedance on E(f).

-5

-4

-3

-2

-1

0

1

2

|E | (dB) or Angle E (deg)

0 5 10 15 20

Frequency (GHz)

Strobe Generator Impedance 

 67 Ω

 62 Ω

 58 Ω (default) 

 52 Ω

 47 Ω

 Magnitude

 Phase
49

(dashed lines) in Fig. 5-10(a). The shapes of the impulse response and kickout pulse look

similar for all strobe-generator impedance values in this figure. The time axis has been

shifted in Fig. 5-10(a) for better resolution of the kickout pulse and impulse responses.

Figure 5-10(b) is the non-time-shifted, non-normalized representation of the

kickout pulses. Here we see that, for smaller values of Rstrobe, the undershoot in the

kickout pulse occurs later in time relative to the main lobe. The effect on the phase error

of small error pulses such as the undershoot pulse is magnified when those pulses occur

delayed in time relative to the main pulse. This effect is discussed in detail in the

appendix of this section, where we demonstrate that a phase-error term proportional to

the cube of the delay (t1 in the appendix) will be generated for error pulses that are

similar to the undershoot pulses shown in Fig. 5-10. Thus, even a small pulse will have a

significant effect on the phase error when the delay is long.

5.3.4 Strobe-Excitation Pulse Amplitude

5.3.4.1 Description

As discussed above, the strobe pulse in our SPICE model is implemented by the

reflection of an excitation step pulse off of a shorted transmission line. In our model

implementation, the excitation-pulse generator consists of two sources. One generates a

positive-going step pulse applied to the anode of one sampling diode, and the other

generates a negative-going step pulse applied to the cathode of the other diode, as shown

in Fig. 4-1. The amplitude of each source is half of the total specified strobe voltage.

The total strobe-pulse amplitude specified in the Riad sampler model was 12 V,

and the strobe pulse amplitude in the Kerley sampler model was somewhat lower.

Therefore, we chose a range of values from 8 V to 16 V for this parametric study.

5.3.4.2 Simulation Results

Figure 5-11 shows the effect on the correction factor E(f) when we changed the

strobe excitation pulse amplitude in our nose-to-nose simulations. We see that when the

-12

-10

-8

-6

-4

-2

0

2

4

E| | (dB) or Angle E (deg)

0 5 10 15 20

Frequency (GHz)

Strobe Excitation

Pulse Amplitude

 8 V

 10 V

 12 V (default) 

 14 V

 16 V

 Magnitude 

 Phase

Figure 5-11. E(f) calculated for various values of strobe excitation pulse amplitude.
50

strobe is large enough to turn on the diodes strongly (10 V to 16 V), the phase error

changes by at most 1° from the default model value. The 8 V case does not turn the

sampling diodes on sufficiently (seen by the large increase in both magnitude and phase

of E(f)) and is not included in the uncertainty calculation of Section 6.

Figure 5-11 shows that the relative spread in phase error is not as great as it was

for variation in the strobe-excitation pulse rise time, shown in Fig. 5-7(b). The reason

becomes evident when we compare strobe pulses as they appear across the sampling

diodes for various amplitudes of the strobe-excitation-pulse in Fig. 5-12(a). We see that

dv/dt (corresponding to the strobe rise time) is more similar at the diode forward-bias

voltage of ~0.6 V for both large and small excitation-pulse amplitudes than it is when we

vary the strobe rise time, shown in Fig. 5-7(a). These graphs use the same time axis scale.

Because dv/dt is greater for larger amplitude strobes, as shown in Fig. 5-12(a), we

might expect that the phase error would be smaller for these cases. We saw this was the

case when we varied strobe-excitation-pulse rise times in Section 5.3.2 above. However,

neglecting the 8 V case, Fig. 5-11 shows that the phase component of the error is, in fact,

slightly larger for faster strobe-pulse rise times. We see why this is the case when we

look at Fig. 5-12(b). Here we see that the undershoot on the kickout pulse occurs latest in

time and with the largest amplitude for the 16 V strobe-excitation pulse. As discussed in

Section 5.3.3 above, the greater the time delay corresponding to an error pulse (for

example, the undershoot pulse), the greater the phase error. See the appendix to this

section for more information.

5.4 Diode Imbalance

5.4.1 Description

The two-diode sampler of Fig. 4-1 is designed so that the common-mode positiveand negative-going components of the strobe pulse cancel at the sampler input port and at

the output amplifier. However, if the two diodes—or any other pairs of components in the

sampler’s two branches—are not perfectly matched, the strobe will leave a residual

2.0

1.5

1.0

0.5

0.0

-0.5

-1.0

-1.5

Amplitude (v)

9.00 9.02 9.04 9.06 9.08 9.10

Time (ns)

 Strobe Excitation 

 Pulse Amplitude 

 8 V

 10 V

 12 V

 14 V

 16 V

0.12

0.10

0.08

0.06

0.04

0.02

0.00

-0.02

Amplitude (v)

9.00 9.02 9.04 9.06 9.08 9.10

Time (ns)

 Strobe Excitation 

 Pulse Amplitude

 8 V

 10 V

 12 V

 14 V

 16 V

(a) (b)

Figure 5-12. Variations in strobe-excitation-pulse amplitude. (a) Strobe pulses; (b) Kickout pulses.
51

charge imbalance on the hold capacitors. This results in an offset due to the strobe pulse

at the input port and at the output amplifier and affects the kickout pulse generation

process.

In practice, no two components can be perfectly matched, so one performs two

nose-to-nose measurements, one with a positive bias offset (V0 in Fig. 4-1) and one with a

negative bias offset, in an attempt to eliminate this residual common-mode signal [3-5,

12]. The kickout pulses resulting from the positive and negative offsets are averaged as

k t

kt kt

( ) () () = pos neg −

2 , (5-3)

where k(t)pos is the kickout pulse arising from a positive offset voltage and k(t)neg arises

from a negative voltage offset.

Diode imbalance was implemented in our parametric studies by changing the area

parameter of the SPICE diode model. Values of 2 to 10 % were chosen to correspond to

specifications for GaAs diodes with similar SPICE models [25], where 12.5 % variation

in Cj0 is given as typical. A 12.5 % variation in Cj0 corresponds to a diode area imbalance

of 12.5 % because area and capacitance are directly proportional by C = εA/d, where A is

the area of the diode depletion region, d is its thickness, and ε is the dielectric constant of

the depletion region material. The area parameter in SPICE also changes the saturation

current (with area) and the spreading resistance (as 1/area).

5.4.2 Simulation Results

Figure 5-13 shows simulation results where we increase the area parameter for

one diode junction by one-half of the nominal change in area, while decreasing the area

parameter of the other by the same amount. We see that the process of averaging the

kickout pulses generated by negative and positive offset voltages results in very little

change in the phase error of the nose-to-nose calibration due to sampler circuitry.

Because the effect of diode imbalance is negligible, it is not considered in the uncertainty

calculations of Section 6.

Figure 5-14(a) shows the effect of imbalanced diodes on kickout pulse generation.

We see that with increasing diode imbalance, the negative- and positive-offset-generated

pulses show more discrepancy between each other. However, in Fig. 5-14(b), we see that

the averaging process given in eq (5-3) results in kickout pulses that are very similar to

the symmetric case.
52

5.5 Diode Bias Level

5.5.1 Description

The level of reverse bias on the sampling diodes will determine both how strongly

the diodes conduct and their conduction time, when the strobe pulse forward-biases them

into conduction. This is how we change the bandwidth of the oscilloscope. Intuitively,

one may not expect these parameters to change the phase error of the nose-to-nose

calibration. However, this parametric study indicates otherwise. As we saw in Sections

5.3.3 and 5.3.4, when the delay time in the undershoot of the kickout pulse increases, so

does the phase error. We see this effect again here as we explore variation of the bias

level around its nominal value.

-5

-4

-3

-2

-1

0

1

2

E| | (dB) or Angle E (deg)

0 5 10 15 20

Frequency (GHz)

 Symmetric Sampler 

 4% Diode Imbalance 

 2% Diode Imbalance

 10% Diode Imbalance 

 Magnitude 

 Phase

0.14

0.12

0.10

0.08

0.06

0.04

0.02

0.00

-0.02

Amplitude (v)

9.00 9.05 9.10 9.15 9.20 9.25 9.30

Time (ns)

 Symmetric 

 2% Imbalance

 4% Imbalance 

 10% Imbalance 

 Positive Offset 

 Negative Offset 

0.10

0.08

0.06

0.04

0.02

0.00

Amplitude (v)

9.00 9.02 9.04 9.06 9.08 9.10

Time (ns)

 Symmetric 

 2% Imbalance 

 4% Imbalance

 10% Imbalance 

(a) (b)

Figure 5-14. Kickout pulses with diode imbalance. (a) Dashed lines show the kickout pulses arising from

negative and positive offset voltages (the inverse value of the negative-going kickout pulses has been

taken). Solid lines show the average of these calculated according to eq (5-3). The time axis is shifted for

better resolution of imbalanced diode cases. (b) Averaged kickout pulses (the solid lines from Fig. 5-14(a))

with no time-axis shift. The averaged kickout pulses are similar to the symmetric case.

Figure 5-13. Change in E(f) with diode imbalance.
53

The value of the bias supplies is specified as 1.63 V in our default sampler model.

We varied the default value by ±20 % in this parametric study. We chose a limit of 20 %

based on simulations of the Kerley sampler model presented in Section 4.2. There we

showed that the change in bias that best matched high- and low- bandwidth mode

measurements was 20 %. We estimate that the variation in bias level from sampler to

sampler in high-bandwidth mode will be less than this.

5.5.2 Simulation Results

Figure 5-15 shows that the phase component of the correction factor, E(f),

increases as the reverse bias on the sampling diodes decreases, corresponding to stronger

conduction and a longer conduction time for the diodes.

Figure 5-16 shows time-domain plots of the strobe pulse and the kickout pulse for

three of the bias levels given in Fig. 5-15. We can again recognize the increase in error in

the time domain by observing a delay in the undershoot at the falling edge of the kickout

pulse. The appendix explains why the phase error increases with time delay of an error

pulse such as the undershoot of the kickout pulse.

-5

-4

-3

-2

-1

0

1

2

E| | (dB) or Angle E (deg)

0 5 10 15 20

Frequency (GHz)

 Bias Levels

 1.956 V (+20%)

 1.793 V (+10%)

 1.630 V (default) 

 1.467 V (-10%)

 1.304 V (-20%)

 Magnitude

 Phase

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Amplitude (v)

9.00 9.02 9.04 9.06 9.08 9.10

Time (ns)

 default +20% 

1.63 V (default) 

 default -20%

Figure 5-15. E(f) for several bias levels up to ±20 of the default model value.

0.14

0.12

0.10

0.08

0.06

0.04

0.02

0.00

-0.02

Amplitude (v)

9.00 9.02 9.04 9.06 9.08 9.10

Time (ns)

 default + 20%

 1.63 V (default) 

 default -20% 

(a) (b)

Figure 5-16. (a) Strobe pulses and; (b) Kickout pulses corresponding to three different bias levels.
54

5.6 Input Network Impedance

5.6.1 Description

We use mismatch correction to correct for the effects of cables and connectors

joining the two oscilloscopes together in nose-to-nose calibration measurements. In our

simulations, the first sampler (the kickout generator) drives an ideal 50 Ω load and the

second sampler is driven by an ideal 50 Ω source. This ideal isolation network between

the two sampling circuits simulates an infinite delay and eliminates the need for any

mismatch correction.

However, between the input port of the oscilloscope and the input to the sampler

lies an unknown network of filters. The errors introduced by the input filter network

discussed in this section are different from the effects that require mismatch correction

discussed above. Here we examine potential effects of this unknown network on the

sampler response of the nose-to-nose calibration. We introduce additional impedances on

the input port of each sampler, and consider the response of the network-plus-sampler

combination.

The nominal impedance at the input port of the sampling circuits used in

oscilloscopes is given as 50 Ω by both Riad [19] and Kerley [2], [24]. It is assumed to be

purely resistive. Here we implement both purely resistive (25 Ω to 100 Ω) and mixed

resistive/reactive input networks. The reactances are implemented as either a capacitor in

parallel with the input port or an inductor in series with the input port. We chose

reactances ranging in value from on the order of the diode junction capacitance (C = 100

pF and L = 0.1 nH) to reactances on the order of the hold capacitors (C = 1 pF and L = 1

nH).

5.6.2 Simulation Results

As shown in Fig. 5-17(a), if the input network is purely resistive, the correction

factor E(f) is not adversely affected. However, reactances that approach the value of the

hold capacitor placed at the input to the sampler may affect the correction factor of the

nose-to-nose calibration, as shown in Fig. 5-17(b). This latter case may be explained by

the fact that the input network begins to store charge in a manner similar to that of the

hold capacitors, but the balanced strobe signal is not cancelled at the output amplifier as it

is with the two-branch configuration of the hold capacitors.

Based on the simulations in this section, purely resistive input networks appear to

have a negligible effect, and only reactances that approach that of the hold capacitor have

an impact on the phase error in the nose-to-nose calibration. We assume that the

oscilloscopes we use are manufactured to avoid the problem of highly reactive filter

components; therefore, this unknown filter network is not included in the sensitivity

analysis in the next section.
55

-6

-4

-2

0

2

4

6

|E | (dB) or Angle E (deg)

0 5 10 15 20

Frequency (GHz)

 K = 50 Ω, H = 50 Ω

 K = 50 Ω, H = 100 Ω

 K = 100 Ω, H = 100 Ω

 K = 100 Ω, H = 50 Ω

 K = 25 Ω, H = 25 Ω

 K = 50 Ω, H = 25 Ω

 Magnitude

 Phase

-15

-10

-5

0

5

10

15

E| | (dB) or Angle E (deg)

0 5 10 15 20

Frequency (GHz)

 Rin = 50 Ω

 No Input Reactance 

 Cin = 0.1 pf

 Cin = 1.0 pf

 Lin = 0.1 nH

 Lin= 1.0 nH

 Magnitude

 Phase

(a) (b)

Figure 5-17. E(f) for (a) resistive input networks; (b) reactive input networks.
56

Appendix for Section 5. Error Pulses

Consider a main pulse followed by a small error pulse in the time domain:

The Fourier transform pair corresponding to these pulses may be given by

xt xt t X j X j j t () ( ) ( ) ( ) − −↔ − − α ωα ω ω

1 1 e . A(1)

For small α and small values of ωt1, we can use a Taylor series expansion of the form

e x ± x x x=± + ± + 1

2 3

2 3

! ! K . A(2)

With F equal to the right hand side of A(1)

F Xj e

Xj j t jt jt

X j t j t t

j t = −

≅ −− + − +

≅ −− − +

− ( )( )

( )[ ( ( )( ) )]

( )[( ( ) ) ( ( ) )] .

ω α

ωαω

ω ω

ω αα

ω

α ω

ω

ω 1

1 1

2 6

1

2 6

1

1

1

2

1

3

1

2

1

1

3

K A(3)

In this case we end up with a magnitude error of 1

2

1

2

− − α α( ) ωt

, A(4)

and a phase error of αω α

ω

t

t

1

1

3

6

+ ( ) . A(5)

The first term of eq A(5) corresponds to a linear phase error, which we detrend from the

phase component of the error ratio using the method described in Section 2.4. The second term in

eq A(5) corresponds to a phase error term, whose magnitude grows as the cube of the time delay

of the error pulse relative to the main pulse. Thus, even small error pulses can contribute

significantly to the phase error when delayed relative to the main pulse. For main and error

pulses that do not have the same shape, that is, for pulses not of the form X(jω) – αX(jω), we

would expect to see a second-order phase-error term as well. Because our system is not linear,

time-invariant, or passive, the phase function is not required to be an odd function.

t=0 t=t1

time

main

pulse

error

pulse
57

6. Sensitivity Analysis and Standard Uncertainty

In the previous section, we demonstrated that the phase error in the nose-to-nose

calibration is sensitive to variation in sampling-circuit parameter values. We will now

quantify this phase-error sensitivity in two different ways. In Section 6.1, we rank the

circuit parameters discussed in Section 5 with respect to impact on the phase error in the

default sampling circuit. We vary each parameter by the same amount (10 %), and note

the corresponding change in the phase error. The parameters causing a larger change in

phase error are ranked higher. Using this procedure, we see the relative effect that

variation (or uncertainty) in the value of each circuit parameter has on the phase error of

the calibration. This information can be used in a variety of ways: for example, in

determining which circuit elements should be modeled most carefully in sampling-circuit

simulations, and which circuit elements should be considered in sampling-circuit designs

for improved nose-to-nose calibrations.

In Section 6.2, we calculate the standard uncertainty in the phase error due to

combined component value variation. The standard uncertainty provides us with an

important piece of information: it represents our best estimate of a range of phase-error

values due to the sampling circuitry for a real nose-to-nose calibration. Throughout this

report our goal has been to accurately model the real samplers found in the type of

oscilloscope used in nose-to-nose calibrations. However, uncertainty exists both in the

values of some circuit parameters and in the way some circuit elements are modeled. By

assuming a significant level of uncertainty for each individual circuit parameter, we hope

to cover the expected range of phase errors due to the sampling-circuit mechanism in a

real nose-to-nose calibration. Finally, in Section 6.3, we summarize, draw conclusions,

and provide notes for future work.

6.1 Ranking the Contributions to the Phase-Error Uncertainty

In this section, we calculate the uncertainty (or variation) in the phase error

arising from uncertainty (or variation) in the value of each circuit parameter discussed in

the previous section. We do this by finding the change in the default phase-error value for

a given variation in circuit parameter value. The parameters leading to greater phase error

changes are ranked higher.

We define the uncertainty in the phase error due to parametric variation as the

standard deviation in the phase error with respect to variation or uncertainty in a given

parameter. According to Refs. [26], [27], and [28], we should calculate the standard

deviation for each quantity that contributes to uncertainty using Type A evaluation and/or

Type B evaluation. Type A evaluation corresponds to determination of uncertainty using

statistical means based on repeated observation. Type A evaluation of uncertainty is most

often used when repeated measurements, made under identical conditions, are available.

Type B evaluation of uncertainty corresponds to uncertainty that is not measured

statistically. We use Type B evaluation in the present work, since we are basing our

calculations on individual simulations of our sampling-circuit model.
58

We express the uncertainty in the phase error corresponding to each parameter in

our default sampling-circuit model as [26]

u u j

j

φ α j

α ∂φ

∂α ( ) = , (6-1)

where

αj = circuit parameter corresponding to parameter number j,

uα j

= fractional uncertainty (variation) in αj (10 % of the values given in Table 4-1),

φ = phase error in the nose-to-nose calibration simulations (with a default value of ~3.3°

at 20 GHz),

∂φ

∂α j

= change in the phase error due to variation in circuit parameter value αj, and

uφ α j ( )= uncertainty in the phase error corresponding to the uncertainty uα j

.

To calculate the uncertainties expressed in eq (6-1), we performed simulations of

our default model where we chose a value for the fractional uncertainty uα j

for each

model parameter αj studied in Section 5. For comparison purposes, we chose the

fractional uncertainty in each of the model parameters as the same value, ±10 % of αj.

We selected a value of 10 % based on manufacturer’s data sheets for sampling diodes

that are similar to those we expect to find in the type of oscilloscopes used in nose-tonose calibrations [25]. In these data sheets, the maximum variation in junction

capacitance is approximately 10 %, so we chose this as an upper limit for our study.

For each parameter, we calculated a minimum of three sets of phase errors: the

default case, one (or more) case(s) with αj increased, and one (or more) case(s) with αj

decreased. We fitted a straight line to these phase errors at 20 GHz (the highest frequency

considered in our default model) and numerically calculated the derivative [∂φ/∂αj] of

this line. We fitted the change in phase error with a straight line since our typical change

in error was expected to be small (Note that for some parameters this small-value

approximation does not hold, and some additional error is introduced into the uncertainty

calculation.). We then multiplied the derivative (a constant value) by the fractional

uncertainty uα j

and found the uncertainty in the phase error uφ(αj) due to this particular

parameter.

Table 6-1 indicates that the strobe generator impedance (Section 5.3.3) has the

most significant effect on the phase error for a fractional uncertainty of 10 % of αj,

followed closely by nonlinear diode junction capacitance (Section 5.1). Previously, we

mentioned that because the strobe pulse and input pulses utilize separate ground schemes

in the physical sampler, the effect of the strobe generator impedance on the small-signal

behavior of the sampler is difficult to accurately model. The high ranking of the strobe

generator impedance simply indicates that the strobe generator model must be considered

carefully when predicting the phase error of the nose-to-nose calibration.
59

Table 6-1. Uncertainty in the phase error in the nose-to-nose calibration at 20 GHz for the listed samplingcircuit parameter values, using the methodology discussed in Section 6.1. Contributions are ranked highest

to lowest.

Parameter αj

Value of

αj

uα j

= 0.1αj

∂φ

∂α j

uφ α j ( )

Strobe generator

impedance

28.5 Ω /

branch 2.9 Ω 0.12°/Ω 0.35°

Cj0 0.045 pF 0.0045 pF –70.0°/pF 0.32°

γ 0.5 0.05 –5.20°/unit 0.26°

Strobe-pulse rise

time 10 ps 1 ps –0.23°/ps 0.23°

Strobe amplitude 12 V 1.2 V –0.16°/V 0.20°

Bias value 1.63 V 0.163 V +1.15°/V 0.19°

N 1.08 0.108 –1.67°/unit 0.18°

CH 4 pF 0.4 pF –0.36°/pF 0.14°

φbi 0.7 V 0.07 V +0.82°/V 0.06°

Embedding

impedance 50 Ω 5 Ω –0.011°/Ω 0.05°

Isat 20 pA 2 pA +2.97e9°/A 0.01°

Rs 10 Ω --- --- not significant

source of error

Eg 0.69 eV --- --- not significant

source of error
60

6.2 Estimate of Standard Uncertainty Due to Combined Sampling Circuitry

Variation

In this section, we generate a statement of standard uncertainty using the method

described in Section 4.3.8 of both Refs. [26] and [28], and Section 4.6 of Ref. [27]. This

method is a Type B evaluation based on predicting the upper and lower bounds of

uncertainty of a quantity of interest by varying all parameters on which it depends to the

“fullest extent possible” [27]. Assuming that the quantity of interest will take a value

within the upper and lower bounds with equal probability, we find the standard deviation

of the corresponding rectangular distribution and use this value as the standard

uncertainty.

To implement this procedure, we first estimated an upper and lower bound on the

phase error in the default sampler model. We performed simulations that incorporated

combined parametric variation, that is, we varied all of the circuit parameters

simultaneously in such a way as to increase the phase error, and then varied all of the

circuit parameters to decrease the phase error. As shown in Table 6-1, increasing some

parameters increases the phase error (a positive value for [∂φ/∂αj]), while increasing

some parameters decreases the phase error (a negative value for [∂φ/∂αj]).

Figure 6-1 shows the results of our combined parametric simulations with all

circuit parameters αj changed simultaneously by 10 % in such a way as to increase or

decrease the phase error. We see the typical default sampler phase-error value of 3.32° at

20 GHz. We see an increase of 3.64° (to 6.96°) for the upper bound, and a decrease of

1.21° (to 2.11°) for the lower bound. Note that the change in the phase error is not

symmetric about the default value. References [26] and [28] address such a case in

Section 4.3.8, and suggest using a standard uncertainty of

u

a

φ = 3

, (6-2)

-8

-6

-4

-2

0

2

E| | (dB) or Angle E (deg)

0 5 10 15 20

Frequency (GHz)

 Default 

 Lower Bound: 10% 

 Upper Bound: 10%

 Magnitude 

 Phase

Figure 6-1. Simulations of upper and lower phase-error bounds with all parameters changed by 10%.
61

where 2a is the difference between the lower bound and the upper bound, and a is given

by

a

b b = − + +

2 , (6-3)

with b+ the positive offset, and b- the negative offset. Note that eq (6-2) corresponds to

the standard deviation of a rectangular distribution from the lower bound to the upper

bound, and eq (6-3) effectively shifts the mean to the middle of this distribution. The

standard uncertainty uφ(α) is then given by

u

b b

φ( ) α ( )

(. . )

. ,

= +

= +

=

+ −

2

2

12

3 64 1 21

12

1 4

o o

o

(6-4)

where α represents the combined set of circuit parameters. Our estimate of the phase

error for our default sampling-circuit model is then 3.32° ±1.4° at 20 GHz, where 1.4° is

the numerical value of standard uncertainty uφ(α), and not a confidence interval.

This method of finding uncertainty was used in place of a calculation of the

combined uncertainty, that would be found by appropriately combining the fractional

uncertainties uα jcalculated in the previous subsection. To generate such a combined

uncertainty statement using the method described in Section 5 of Ref. [26], we would

need to know the correlation between the individual contributions to the overall

uncertainty. For example, we would need to know whether a change in the grading

coefficient of the sampling diode would affect the value of the diode’s junction

capacitance, or whether an increase in the strobe pulse rise time would lead to an increase

in the strobe amplitude. We do not have this information explicitly, since much of it

depends on the fabrication and design of the sampling circuit. However, we do expect

some correlation to exist between certain circuit parameters, and so cannot ignore

potential correlation effects.

If we were to assume no correlation between the circuit parameters, the

uncertainties uφ(αj) listed in Table 6-1 would add in a root-sum-of-squares (RSS) sense to

give the combined uncertainty. If we were to assume complete correlation, the

uncertainties listed in Table 6-1 would add directly. We expect that the combined

uncertainty for the phase error in the sampling circuit lies somewhere between these two.

Note that if we combined the uncertainties calculated in Section 6.1 linearly (complete

correlation), uφ(α) would be 2.96°, and if we combined the uncertainties in the RSS

sense, uφ(α) would be 0.98°. Our estimate lies between these two extremes.
62

We expect that the phase error in our default model of 3.32° ±1.4° at 20 GHz is

higher than that of real sampling circuits simply because our model is based on

conservative approximations and assumptions. For example, our model of the sampling

diode uses a single term for the zero-bias junction capacitance Cj0. In Section 5.1, a Split

Diode Model was discussed where the total capacitance was distributed between a lower

Cj0 value and a linear, parasitic packaging capacitance. We mentioned that this model is

probably a more accurate diode model than the one used in our default sampler. The

phase error in that model is around 2°, lower than our default sampler phase error. Thus,

while our default model provides an estimate of sampler behavior, we know it may be

enhanced by further study.

6.3 Summary and Conclusions

We performed a sensitivity study of the effect of sampling-circuit parameter-value

variation on the phase error in the nose-to-nose calibration. We first defined a way to

express the difference between our nose-to-nose-derived estimate of the impulse response

of the sampler and the actual impulse response. We then developed a SPICE model of the

sampling circuit as a mechanism to explore the effects of parametric variation. We

performed simulations of two sampling circuits connected nose-to-nose, calculated an

estimate of the sampler’s response, and determined the phase error in our estimate. We

then observed the change in the phase error as we individually varied each circuit

parameter. We ranked the circuit parameters in terms of most significant impact on the

phase error for a given parametric variation (10 % of the default value), and found the

standard uncertainty of the phase error with respect to this parametric variation.

For our default sampling-circuit model, we calculated a phase error of 3.32°

±1.4°. By assuming a significant level of uncertainty for each individual circuit

parameter, our intent has been to cover the expected range of phase errors in a real noseto-nose calibration. This provides us with a first-cut estimate of the contribution of the

sampling circuitry to the phase error in the nose-to-nose calibration, since, as mentioned

in Section 1.1, this quantity cannot be measured directly.

Our parametric studies indicate that both the diode nonlinear junction capacitance

and the strobe generator impedance have a significant influence on the total phase error.

Thus, while overall refinement of the sampling-circuit model will provide a more

accurate estimate of the phase error, we expect that refining the diode model and the

strobe generator model in particular will help to more accurately determine both the

expected phase error due to the sampling circuitry and the uncertainty in that phase error.

———————————

The author is grateful to the following people for help with models, theory, and

discussion of the nose-to-nose calibration: Dylan Williams, Don DeGroot, Jan Verspecht,

Marc Vanden Bossche, Frans Verbeyst, John Kerley, Paul Hale, Tracy Clement, Jim

Randa.
63

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