Damping Factor (\(\zeta\)) is
defined for close loop system
We can analyze open loop system in a better perspective because it is
simpler. So, we always use the loop gain analysis to find the phase
margin and see whether the system is stable or not.
The zeros in the right half of the complex plane are called
nonminimum phase zeros. Systems with
nonminimum phase zeros are called nonminimum phase
systems
Zero close to the real pole attenuates the effect of that
pole on the system response
Zeros Tend to Increase the Overshoot of the
System
Let \(s=j\omega\) and omit factor,
\[
A_\text{dB}(\omega) = 10\log[1+(\frac{\omega}{\omega _z})^2] -
10\log[1+\frac{\omega^4}{\omega_n^4}+\frac{2\omega^2(2\zeta ^2
-1)}{\omega_n^2}]
\] peaking frequency \(\omega_\text{peak}\) can be obtained via
\(\frac{\mathrm{d}
A_\text{dB}(\omega)}{\mathrm{d}\omega} = 0\)\[
\omega_\text{peak} = \omega_z \sqrt{\sqrt{(\frac{\omega_n}{\omega_z})^4
- 2(\frac{\omega_n}{\omega_z})^2(2\zeta ^2-1)+1} - 1}
\]
minimum-phase
system vs Non-minimum phase system
a linear, time-invariant system is said to be minimum-phase if
the system and its inverse are causal and
stable
Systems that are causal and stable whose inverses are
causal and unstable are known
as non-minimum-phase systems
Settling Time
One Pole
we have \[
\tau \approx \left(1 + \frac{R_1}{R_2}\right)\frac{1}{A_0\omega_0}=
\frac{1}{\beta \omega_\text{ugb}}
\]
Two Poles
with open-loop transfer function \(A_{OL}=\frac{A_0}{(1+s/\omega_1)(1+s/\omega_2)}\)
and assuming \(\omega_1\) is dominant
pole, then yield closed-loop transfer function
That is \(\omega_n =
\sqrt{\omega_u\omega_2}\), \(\zeta =
\frac{1}{2}\sqrt{\frac{\omega_2}{\omega_u}}\) , where \(\omega_u\approx \beta A_0 \omega_1\) is the
unity gain bandwidth
Rise Time (0% to
100% )
\[
t_r = \frac{\pi - \beta}{\omega_d}=\frac{\pi -
\arctan\frac{\omega_n\sqrt{1-\zeta^2}}{\zeta\omega_n}}{\omega_n\sqrt{1-\zeta^2}}\approx\frac{\pi
-
\arctan\frac{\sqrt{1-\zeta^2}}{\zeta}}{\sqrt{\omega_u\omega_2}\sqrt{1-\zeta^2}}=\frac{\pi
- \arctan\frac{\sqrt{1-\zeta^2}}{\zeta}}{\omega_u\sqrt{k(1-\zeta^2)}}
\] where \(k =
\frac{\omega_2}{\omega_u}\), is the function of PM
Gene F. Franklin, Feedback Control of Dynamic Systems, 8th
Edition
As we know \[
\zeta \omega_n=\frac{1}{2}\sqrt{\frac{\omega_2}{\omega_u}}\cdot
\sqrt{\omega_u\omega_2}=\frac{1}{2}\omega_2
\]
Then \[
t_s = \frac{9.2}{\omega_2}
\]
For \(\text{PM}=70^o\), \(\omega_2 = 2.75\omega_u\), that is \[
t_s \approx \frac{3.35}{\omega_u}
\]
For \(\text{PM}=45^o\), \(\omega_2 = \omega_u\), that is \[
t_s \approx \frac{9.2}{\omega_u}
\]
Above equation is valid only for underdamped, \(\zeta=\frac{1}{2}\sqrt{\frac{\omega_2}{\omega_u}}\lt
1\), that is \(\omega_2\lt
4\omega_u\)
2 Stage RC filter
High Pass Filter
Since \(1/sC_1+R_1 \gg R_0\)\[
\frac{V_m}{V_i}(s) \approx \frac{R_0}{R_0 + 1/sC_0} =
\frac{sR_0C_0}{1+sR_0C_0}
\]step response of \(V_m\)\[
V_m(t) = e^{-t/R_0C_0}
\] where \(\tau = R_0C_0\)
And \(V_o(s)\) can be expressed as
\[\begin{align}
\frac{V_o}{V_i}(s) & \approx \frac{sR_0C_0}{1+sR_0C_0} \cdot
\frac{sR_1C_1}{1+sR_1C_1} \\
&= \frac{sR_0C_0R_1C_1}{R_0C_0-R_1C_1}\left(\frac{1}{1+sR_1C_1} -
\frac{1}{1+sR_0C_0}\right)
\end{align}\]
Gene F. Franklin, J. David Powell, and Abbas Emami-Naeini. 2018.
Feedback Control of Dynamic Systems (8th Edition) (8th. ed.).
Pearson. [pdf]
Katsuhiko Ogata, Modern Control Engineering, 5th edition [pdf]
C. T. Chuang, "Analysis of the settling behavior of an operational
amplifier," in IEEE Journal of Solid-State Circuits, vol. 17,
no. 1, pp. 74-80, Feb. 1982 [https://sci-hub.se/10.1109/JSSC.1982.1051689]
Often, AC-driven circuits can be mistaken as non-linear as the basis
that determines the linearity of a circuit is the relationship between
the voltage and current.
While an AC signal varies with time, it still exhibits a linear
relationship across elements like resistors, capacitors, and inductors.
Therefore, AC driven circuits are linear.
Phasor
Phasor concept has no real physical significance. It is just a
convenient mathematical tool.
Phasor analysis determines the steady-state response to a linear
circuit driven by sinusoidal sources with frequency \(f\)
If your circuit includes transistors or other nonlinear components,
all is not lost. There is an extension of phasor analysis to nonlinear
circuits called small-signal analysis in which you linearize the
components before performing phasor analysis - AC analyses of SPICE
A sinusoid is characterized by 3 numbers, its amplitude, its phase,
and its frequency. For example \[
v(t) = A\cos(\omega t + \phi) \tag{1}
\] In a circuit there will be many signals but in the case of
phasor analysis they will all have the same frequency. For this reason,
the signals are characterized using only their amplitude and
phase.
The combination of an amplitude and
phase to describe a signal is the
phasor for that signal.
Thus, the phasor for the signal in \((1)\) is \(A\angle \phi\)
In general, phasors are functions of frequency
Often it is preferable to represent a phasor using complex
numbers rather than using amplitude and phase. In this case we
represent the signal as: \[
v(t) = \Re\{Ve^{j\omega t} \} \tag{2}
\] where \(V=Ae^{j\phi}\) is the
phasor.
\((1)\) and \((2)\) are the same
Phasor Model of a Resistor
A linear resistor is defined by the equation \(v = Ri\)
Now, assume that the resistor current is described with the
phasor\(I\). Then \[
i(t) = \Re\{Ie^{j\omega t}\}
\]\(R\) is a real constant, and
so the voltage can be computed to be \[
v(t) = R\Re\{Ie^{j\omega t}\} = \Re\{RIe^{j\omega t}\} =
\Re\{Ve^{j\omega t}\}
\] where \(V\) is the phasor
representation for \(v\), i.e. \[
V = RI
\]
Thus, given the phasor for the current we can directly
compute the phasor for the voltage across the
resistor.
Similarly, given the phasor for the voltage across a
resistor we can compute the phasor for the current through the
resistor using \(I =
\frac{V}{R}\)
Phasor Model of a Capacitor
A linear capacitor is defined by the equation \(i=C\frac{\mathrm{d}v}{\mathrm{d}t}\)
Now, assume that the voltage across the capacitor is described with
the phasor\(V\). Then \[
v(t) = \Re\{ V e^{j\omega t}\}
\]\(C\) is a real constant
\[
i(t) = C\Re\{\frac{\mathrm{d}}{\mathrm{d}t}V e^{j\omega t}\} =
\Re\{j\omega C V e^{j\omega t}\}
\] The phasor representation for \(i\) is \(i(t) =
\Re\{Ie^{j\omega t}\}\), that is \(I =
j\omega C V\)
Thus, given the phasor for the voltage across a
capacitor we can directly compute the phasor for the current
through the capacitor.
Similarly, given the phasor for the current through a
capacitor we can compute the phasor for the voltage across the
capacitor using \(V=\frac{I}{j\omega
C}\)
Phasor Model of an Inductor
A linear inductor is defined by the equation \(v=L\frac{\mathrm{d}i}{\mathrm{d}t}\)
Now, assume that the inductor current is described with the
phasor\(I\). Then \[
i(t) = \Re\{ I e^{j\omega t}\}
\]\(L\) is a real constant, and
so the voltage can be computed to be \[
v(t) = L\Re\{\frac{\mathrm{d}}{\mathrm{d}t}I e^{j\omega t}\} =
\Re\{j\omega L I e^{j\omega t}\}
\] The phasor representation for \(v\) is \(v(t) =
\Re\{Ve^{j\omega t}\}\), that is \(V =
j\omega L I\)
Thus, given the phasor for the current we can directly
compute the phasor for the voltage across the
inductor.
Similarly, given the phasor for the voltage across an inductor we
can compute the phasor for the current through the inductor using \(I=\frac{V}{j\omega L}\)
Impedance and Admittance
Impedance and admittance are generalizations of resistance and
conductance.
They differ from resistance and conductance in that they are complex
and they vary with frequency.
Impedance is defined to be the ratio of the phasor for the
voltage across the component and the current through the component:
\[
Z = \frac{V}{I}
\]
Impedance is a complex value. The real part of the impedance is
referred to as the resistance and the imaginary part is referred to as
the reactance
For a linear component, admittance is defined to be the ratio of the
phasor for the current through the component and the voltage
across the component: \[
Y = \frac{I}{V}
\]
Admittance is a complex value. The real part of the admittance is
referred to as the conductance and the imaginary part is referred to as
the susceptance.
Response to Complex
Exponentials
The response of an LTI system to a complex
exponential input is the same complex
exponential with only a change in amplitude
where \(h(n)\) is the impulse
response of a discrete-time LTI system
convolution sum is used here
The signals of the form \(e^{st}\)
in continuous time and \(z^{n}\) in
discrete time, where \(s\) and \(z\) are complex numbers are
referred to as an eigenfunction of the system, and the
amplitude factor\(H(s)\),
\(H(z)\) is referred to as the system's
eigenvalue
Laplace transform
One of the important applications of the Laplace transform is in the
analysis and characterization of LTI systems, which
stems directly from the convolution property\[
Y(s) = H(s)X(s)
\] where \(X(s)\), \(Y(s)\), and \(H(s)\) are the Laplace transforms
of the input, output, and impulse response of the system,
respectively
From the response of LTI systems to complex exponentials, if the
input to an LTI system is \(x(t) =
e^{st}\), with \(s\) the ROC of
\(H(s)\), then the output will be \(y(t)=H(s)e^{st}\); i.e., \(e^{st}\) is an eigenfunction of
the system with eigenvalue equal to the Laplace
transform of the impulse response.
s-Domain Element Models
Sinusoidal Steady-State
Analysis
Here Sinusoidal means that source excitations have
the form \(V_s\cos(\omega t +\theta)\)
or \(V_s\sin(\omega t+\theta)\)
Steady state mean that all transient behavior of the
stable circuit has died out, i.e., decayed to zero
\(s\)-domain and phasor-domain
Phasor analysis is a technique to find the steady-state
response when the system input is a sinusoid. That is, phasor
analysis is sinusoidal analysis.
Phasor analysis is a powerful technique with which to find the
steady-state portion of the complete response.
Phasor analysis does not find the transient response.
Phasor analysis does not find the complete response.
The beauty of the phasor-domain circuit is that it is described by
algebraic KVL and KCL equations with time-invariant sources, not
differential equations of time
The difference here is that Laplace analysis can also give
us the transient response
The zero-state response is given by \(\mathscr{L^1}[H(s)F(s)]\), for the
arbitrary \(s\)-domain input \(F(s)\)
where \(Z_L(s) = sL\), the inductor
with zero initial current \(i_L(0)=0\)
and \(Z_C(s)=1/sC\) with zero initial
voltage \(v_C(0)=0\)
transient response & steady-state
response
natural response & forced
response
Transfer Functions
and Frequency Response
transfer function
The transfer function\(H(s)\) is the ratio of the Laplace
transform of the output of the system to its input assuming
all zero initial conditions.
frequency response
An immediate consequence of convolution is that an input of
the form \(e^{st}\) results in an
output \[
y(t) = H(s)e^{st}
\] where the specific constant \(s\) may be complex, expressed as \(s = \sigma + j\omega\)
A very common way to use the exponential response of LTIs is
in finding the frequency response i.e. response
to a sinusoid
First, we express the sinusoid as a sum of two
exponential expressions (Euler’s relation): \[
\cos(\omega t) = \frac{1}{2}(e^{j\omega t}+e^{-j\omega t})
\] If we let \(s=j\omega\), then
\(H(-j\omega)=H^*(j\omega)\), in polar
form \(H(j\omega)=Me^{j\phi}\) and
\(H(-j\omega)=Me^{-j\phi}\). \[\begin{align}
y_+(t) & = H(s)e^{st}|_{s=j\omega} = H(j\omega)e^{j\omega t} = M
e^{j(\omega t + \phi)} \\
y_-(t) & = H(s)e^{st}|_{s=-j\omega} = H(-j\omega)e^{-j\omega t} = M
e^{-j(\omega t + \phi)}
\end{align}\]
By superposition, the response to the sum of these two
exponentials, which make up the cosine signal, is the sum of the
responses \[\begin{align}
y(t) &= \frac{1}{2}[H(j\omega)e^{j\omega t} + H(-j\omega)e^{-j\omega
t}] \\
&= \frac{M}{2}[e^{j(\omega t + \phi)} + e^{-j(\omega t + \phi)}] \\
&= M\cos(\omega t + \phi)
\end{align}\]
where \(M = |H(j\omega|\) and \(\phi = \angle H(j\omega)\)
This means if a system represented by the transfer function \(H(s)\) has a sinusoidal input, the
output will be sinusoidal at the same frequency with magnitude
\(M\) and will be shifted in phase by
the angle \(\phi\)
Laplace transform vs.
Fourier transform
Laplace transforms such as \(Y(s)=H(s)U(s)\) can be used to study the
complete response characteristics of systems, including
the transient response—that is, the time response to an
initial condition or suddenly applied signal
This is in contrast to the use of Fourier transforms, which
only take into account the steady-state response
Given a general linear system with transfer function \(H(s)\) and an input signal \(u(t)\), the procedure for determining \(y(t)\) using the Laplace transform
is given by the following steps:
FSM and Shift Register of DR and IR works at the
posedge of the clock
TMS, TDI, TDO and Hold Register of DR and IR changes value at the
negedge of the clock
capture IR 01, the fixed is for easier fault
detection
After power-up, they may not be in sync, but there is a trick. Look
at the state machine and notice that no matter what state you are, if
TMS stays at "1" for five clocks, a TAP controller goes back to
the state "Test-Logic Reset". That's used to synchronize the TAP
controllers.
It is important to note that in a typical Boundary-Scan test, the
time between launching a signal from driver (at the falling edge of test
clock (TCK) in the Update-DR or Update-IR TAP
Controller state) and capturing that signal (at the rising edge of TCK
in the Caputre-DR TAP Controller state) is no less
tha 2.5 TCK cycles
Further, the time between successive launches on a driver is governed
- not only by the TCk rate - but by the amount of serial data shifting
needed to load the next pattern data in the concatenated Boundary-Scan
Registers of the Boundary-Scan chain
Thus the effective test data rate of a driver could be thousands of
the times lower than the TCK rate
For DC-coupled interconnect, this time is of no concern
For AC-coupled interconnect, the signal may easily decay partially
or completely before it can be captured
If only partial decay occurs before capture, that decay will very
likely be completed before the driver produces the next edge
AC-coupling
In general, AC-coupling can distort a signal transmitted across a
channel depending on its frequency.
Figure 5
The high frequency signal is relatively unaffected by the
coupling
The low frequency signal is severely impacted
it decays to \(V_T\) after a few
time constants
its amplitude is double the input amplitude > transient response,
before AC-coupling capacitor: \(-A_p \to
A_p\); after AC-coupling capacitor \(V_T \to V_T+2A_p\) > A key item to note
is that the transitions in the original signal are preserved, although
their start and end points are offset > > compared to where they
were in the high frequency
Test signal implementation
The test data is either the content of the Boundary-Scan Register
Update latch (U) when executing the (DC) EXTEST instruction, or an
"AC Signal" when an AC testing instruction is
loaded into the device.
The AC signal is a test waveform suited for transmission through
AC-coupling
Test signal reception
When an AC testing instruction is loaded, a specialized
test receiver detects transitoins of the AC signal seen at the input and
determines if this represents a logic '0' or '1'
When EXTEST is loaded, the input signal level is detected
and sent to the output of the test receiver to the Boundary-Scan
Register cell
When testing for a shorted capacitor, the test software must
ensure that enough time has passed for the signal to decay before
entering Capture-DR, either by stopping TCk or by spending
additional TCK cycles in the Run-Test/Idle TAP Controller
state
EXTEST_PULSE & EXTEST_TRAIN
The two new AC-test instructions provided by this standard differ
primarily in the number and timing of transitions to provide flexibility
in dealing with the specific dynamic behavior of the channels being
tested
AC Test Signal essentially modulates test data so
that it will propagate through AC-coupled channels, for devices that
contatin AC pins
Tools should use the EXTEST_PULSE instruction unless
there is a specific requirement for the EXTEST_TRAIN
instruction
EXTEST_PULSE
Generate two additional driver transitions and
allows a tester to vary the time between them dependent
on how many TCK cycles the TAP is left in the Run-Test/Idle TAP
Controller state.
This is intended to allow any undesired transient condition to decay
to a DC steady-state value when that will make the
final transition more reliably detectable
The duration in the Run-Test/Idle TAP Controller state
should be at least three times the high-pass coupling
time constant. This allows the first additional transition to
decay away to the DC steady-state value for the
channel, and ensures that the full amplitude of the final
transition is added to or subtracted from that steady-state
value
This establishes a known initial condition for the final
transition and permits reliable specification of the detection
threshold of the test receiver
EXTEST_TRAIN
Generate multiple additional transitions, the
number dependent on how long the TAP is left in the
Run-Test/Idle TAP Controller state
This is intended to allow any undesired transient condition to decay
to an AC steady-state value when that will make the
final transition more reliably detectable
IEEE Std 1149.6-2003
This standard is built on top of IEEE Std 1149.1 using the same Test
Access Port structure and Boundary-Scan architecture.
It adds the concept of a "test receiver" to input pins that are
expected to handle differential and/or AC-coupling
It adds two new instructions that cause drivers to emit AC waveforms
that are processed by test receivers.
JTAG Instruction
Implementation
AC mode hysteresis, detect transistion
DC mode threshold is determined by jtag initial value
reference
IEEE Std 1149.1-2001, IEEE Standard Test Access Port and
Boundary-Scan Architecture, IEEE, 2001
IEEE Std 1149.6-2003, IEEE Standard for BoundaryScan Testing of
Advanced Digital Networks, IEEE, 2003
B. Eklow, K. P. Parker and C. F. Barnhart, "IEEE 1149.6: a
boundary-scan standard for advanced digital networks," in IEEE Design
& Test of Computers, vol. 20, no. 5, pp. 76-83, Sept.-Oct. 2003,
doi: 10.1109/MDT.2003.1232259.
Norbert Wiener proved this theorem for the case of a
deterministic function in 1930; Aleksandr
Khinchin later formulated an analogous result for stationary
stochastic processes and published that probabilistic
analogue in 1934. Albert Einstein explained, without proofs, the idea in
a brief two-page memo in 1914
\(x(t)\), Fourier transform over a
limited period of time \([-T/2, +T/2]\)
, \(X_T(f) = \int_{-T/2}^{T/2}x(t)e^{-j2\pi
ft}dt\)
With Parseval's theorem\[
\int_{-T/2}^{T/2}|x(t)|^2dt = \int_{-\infty}^{\infty}|X_T(f)|^2df
\] So that \[
\frac{1}{T}\int_{-T/2}^{T/2}|x(t)|^2dt =
\int_{-\infty}^{\infty}\frac{1}{T}|X_T(f)|^2df
\]
where the quantity, \(\frac{1}{T}|X_T(f)|^2\) can be interpreted
as distribution of power in the frequency domain
For each \(f\) this quantity is a
random variable, since it is a function of the random process \(x(t)\)
The power spectral density (PSD) \(S_x(f
)\) is defined as the limit of the expectation of the expression
above, for large \(T\): \[
S_x(f) = \lim _{T\to \infty}\mathrm{E}\left[ \frac{1}{T}|X_T(f)|^2
\right]
\]
The Wiener-Khinchin theorem ensures that for well-behaved
wide-sense stationary processes the limit
exists and is equal to the Fourier transform of the
autocorrelation\[\begin{align}
S_x(f) &= \int_{-\infty}^{+\infty}R_x(\tau)e^{-j2\pi f \tau}d\tau \\
R_x(\tau) &= \int_{-\infty}^{+\infty}S_x(f)e^{j2\pi f \tau}df
\end{align}\]
Note: \(S_x(f)\) in
Hz and inverse Fourier Transform in
Hz (\(\frac{1}{2\pi}d\omega =
df\))
\[
\frac{1}{2\pi}F^{-1}\{R_{xx}\}d\omega =
\frac{1}{2\pi}F^{-1}\{R_{xx}\}d(2\pi f T)=T\cdot F^{-1}\{R_{xx}\}df =
P_{xx}(f)df
\] power spectral density of a discrete-time
random process \(\{x(n)\}\) is
given by \[
P_{xx}(f) =T\cdot F^{-1}\{R_{xx}\}
\]
The periodogram is in fact the Fourier transform of the aperiodic
correlation of the windowed data sequence
estimating continuous-time stationary random
signal
The sequence \(x[n]\) is typically
multiplied by a finite-duration window \(w[n]\), since the input to the DFT must be
of finite duration. This produces the finite-length
sequence \(v[n] = w[n]x[n]\)
That is, by \((1)\)\[
\hat{P}_{ss}(\Omega) = T_s\hat{P}_{xx(\omega)} =
\frac{T_s|V(e^{j\omega})|^2}{\sum_{n=0}^{L-1}(w[n])^2}=\frac{|V(e^{j\omega})|^2}{f_{res}L\sum_{n=0}^{L-1}(w[n])^2}
\]
That is, by \((2)\)\[
\hat{P}_{ss}(\Omega) = T_s\hat{P}_{xx(\omega)} =
\frac{T_sL|V(e^{j\omega})|^2}{\sum_{k=0}^{L-1}(W[k])^2} =
\frac{|V(e^{j\omega})|^2}{f_{res}\sum_{k=0}^{L-1}(W[k])^2}
\]
The steady-state response is the response that results after any
transient effects have dissipated.
The large signal solution is the starting point for
small-signal analyses, including periodic AC, periodic transfer
function, periodic noise, periodic stability, and periodic scattering
parameter analyses.
Designers refer periodic steady state analysis in time
domain as "PSS" and corresponding frequency
domain notation as "HB"
Harmonic Balance Analysis
The idea of harmonic balance is to find a set of port voltage
waveforms (or, alternatively, the harmonic voltage components) that give
the same currents in both the linear-network equations and
nonlinear-network equations
that is, the currents satisfy Kirchoff's current law
Define an error function at each harmonic,
\(f_k\), where \[
f_k = I_{\text{LIN}}(k\omega) + I_{\text{NL}}(k\omega)
\] where \(k=0, 1, 2,...,K\)
Note that each \(f_k\) is implicitly
a function of all voltage components \(V(k\omega)\)
Newton
Solution of the Harmonic-Balance Equation
Iterative Process
and Jacobian Formulation
The elements of the Jacobian are the derivatives \[
\frac{\partial F_{\text{n,k}}}{\partial _{V_\text{m,l}}}
\] where \(n\) and \(m\) are the port indices \((1,N)\), and \(k\) and \(l\) are the harmonic indices \((0,...,K)\)
Number of Harmonics & Time
Samples
Initial Estimate
One important property of Newton's method is that its speed and
reliability of convergence depend strongly upon the initial estimate of
the solution vector.
Conversion Matrix Analysis
Large-signal/small-signal analysis, or
conversion matrix analysis, is useful for a large class
of problems wherein a nonlinear device is driven, or
"pumped" by a single large sinusoidal signal; another
signal, much smaller, is applied; and we seek only the linear
response to the small signal.
The most common application of this technique is in the design of
mixers and in nonlinear noise analysis
First, analyzing the nonlinear device under
large-signal excitation only, where the harmonic-balance method can be
applied
Then, the nonlinear elements in the device's
equivalent circuit are then linearized to create
small-signal, linear, time-varying elements
Finally, a small-signal analysis is performed
Element Linearized
Below shows a nonlinear resistive element, which has
the \(I/V\) relationship \(I=f(V)\). It is driven by a
large-signal voltage
Assuming that \(V\) consists of the
sum of a large-signal component \(V_0\)
and a small-signal component \(v\),
with Taylor series\[
f(V_0+v) = f(V_0)+\frac{\mathrm{d}}{\mathrm{d}V}f(V)|_{V=V_0}\cdot
v+\frac{1}{2}\frac{\mathrm{d}^2}{\mathrm{d}V^2}f(V)|_{V=V_0}\cdot
v^2+...
\] The small-signal, incremental current is found by subtracting
the large-signal component of the current \[
i(v)=I(V_0+v)-I(V_0)
\] If \(v \ll V_0\), \(v^2\), \(v^3\),... are negligible. Then, \[
i(v) = \frac{\mathrm{d}}{\mathrm{d}V}f(V)|_{V=V_0}\cdot v
\]
\(V_0\) need not be a DC
quantity; it can be a time-varying large-signal voltage\(V_L(t)\) and that \(v=v(t)\), a function of time. Then \[
i(t)=g(t)v(t)
\] where \(g(t)=\frac{\mathrm{d}}{\mathrm{d}V}f(V)|_{V=V_L(t)}\)
The time-varying conductance \(g(t)\), is the derivative of the element's
\(I/V\) characteristic at the
large-signal voltage
By an analogous derivation, one could have a current-controlled
resistor with the \(V/I\)
characteristic \(V = f_R(I)\) and
obtain the small-signal\(v/i\) relation \[
v(t) = r(t)i(t)
\] where \(r(t) =
\frac{\mathrm{d}}{\mathrm{d}I}f_R(I)|_{I=I_L(t)}\)
A nonlinear element excited by two tones
supports currents and voltages at mixing frequencies \(m\omega_1+n\omega_2\), where \(m\) and \(n\) are integers. If one of those tones,
\(\omega_1\) has such a low
level that it does not generate harmonics and the other is a
large-signal sinusoid at \(\omega_p\),
then the mixing frequencies are \(\omega=\pm\omega_1+n\omega_p\), which shown
in below figure
A more compact representation of the mixing frequencies is \[
\omega_n=\omega_0+n\omega_p
\] which includes only half of the mixing frequencies:
the negative components of the lower sidebands
(LSB)
and the positive components of the upper sidebands
(USB)
For real signal, positive- and negative-frequency components are
complex conjugate pairs
Shooting Newton
TODO 📅
Nonlinearity &
Linear Time-Varying Nature
Nonlinearity Nature
The nonlinearity causes the signal to be replicated at multiples of
the carrier, an effect referred to as harmonic
distortion, and adds a skirt to the signal that increases its
bandwidth, an effect referred to as intermodulation
distortion
It is possible to eliminate the effect of harmonic
distortion with a bandpass filter, however the frequency of the
intermodulation distortion products overlaps the frequency of
the desired signal, and so cannot be completely removed with
filtering.
Time-Varying Linear Nature
linear with respect to \(v_{in}\) and
time-varying
Given \(v_{in}(t)=m(t)\cos (\omega_c
t)\) and LO signal of \(\cos(\omega_{LO} t)\), then \[
v_{out}(t) = \text{LPF}\{m(t)\cos(\omega_c t)\cdot \cos(\omega_{LO} t)\}
\] and \[
v_{out}(t) = m(t)\cos((\omega_c - \omega_{LO})t)
\]
A linear periodically-varying transfer function implements
frequency translation
Linear Time Varying
The response of a relaxed LTV system at a time \(t\) due to an impulse applied at a time
\(t − \tau\) is denoted by \(h(t, \tau)\)
The first argument in the impulse response denotes the time of
observation.
The second argument indicates that the system was excited by an
impulse launched at a time \(\tau\)prior to the time of observation.
Thus, the response of an LTV system not only depends on how long
before the observation time it was excited by the impulse but also on
the observation instant.
The output \(y(t)\) of an initially
relaxed LTV system with impulse response \(h(t, \tau)\) is given by the convolution
integral \[
y(t) = \int_0^{\infty}h(t,\tau)x(t-\tau)d\tau
\] Assuming \(x(t) = e^{j2\pi f
t}\)\[
y(t) = \int_0^{\infty}h(t,\tau)e^{j2\pi f (t-\tau)}d\tau = e^{j2\pi f
t}\int_0^{\infty}h(t,\tau)e^{-j2\pi f\tau}d\tau
\] The (time-varying) frequency response can be
interpreted as \[
H(j2\pi f, t) = \int_0^{\infty}h(t,\tau)e^{-j2\pi f\tau}d\tau
\] Linear Periodically Time-Varying (LPTV) Systems, which is a
special case of an LTV system whose impulse response satisfies \[
h(t, \tau) = h(t+T_s, \tau)
\] In other words, the response to an impulse remains unchanged
if the time at which the output is observed (\(t\)) and the time at which the impulse is
applied (denoted by \(t_1\)) are both
shifted by \(T_s\)\[
H(j2\pi f, t+T_s) = \int_0^{\infty}h(t+T_s,\tau)e^{-j2\pi f\tau}d\tau =
\int_0^{\infty}h(t,\tau)e^{-j2\pi f\tau}d\tau = H(j2\pi f, t)
\]\(H(j2\pi f, t)\) of an LPTV
system is periodic with timeperiod \(T_s\), it can be expanded as a Fourier
series in \(t\), resulting in \[
H(j2\pi f, t) = \sum_{k=-\infty}^{\infty} H_k(j2\pi f)e^{j2\pi f_s k t}
\] The coefficients of the Fourier series \(H_k(j2\pi f)\) are given by \[
H_k(j2\pi f) = \frac{1}{T_s}\int_0^{T_s} H(j2\pi f, t) e^{-j2\pi k f_s
t}dt
\]
Ashwin Kumar, Lecture 8: Basics of periodic steady-state (pss), pac
and pxf simulation demos in Cadence SpectreRF [https://youtu.be/I9zkt1OTWB0]
pss, pac and pxf
LPV analyses start by performing a periodic analysis to compute the
periodic operating point with only the large
clock signal applied (the LO, the clock, the carrier,
etc.).
The circuit is then linearized about this time-varying
operating point (expand about the periodic equilibrium point
with a Taylor series and discard all but the first-order term)
and the small information signal is applied. The
response is calculated using linear time-varying analysis
Versions of this type of small-signal analysis exists for both
harmonic balance and shooting methods
PAC is useful for predicting the output sidebands produced by a
particular input signal
PXF is best at predicting the input images for a particular
output
tran simulation verify higher frequency ripple introduce more jitter
at output
sampled pac result support the opinion of Frank Wiedmann — harmonic 0
(with no additional sidebands) introduce maximum output
reference
K. S. Kundert, "Introduction to RF simulation and its application,"
in IEEE Journal of Solid-State Circuits, vol. 34, no. 9, pp. 1298-1319,
Sept. 1999, doi: 10.1109/4.782091. [pdf]
Stephen Maas, Nonlinear Microwave and RF Circuits, Second Edition ,
Artech, 2003. [pdf]
Karti Mayaram. ECE 521 Fall 2016 Analog Circuit Simulation:
Simulation of Radio Frequency Integrated Circuits [pdf1,
pdf2]
Shanthi Pavan, "Demystifying Linear Time Varying Circuits"
—, "Reciprocity and Inter-Reciprocity: A Tutorial— Part I: Linear
Time-Invariant Networks," in IEEE Transactions on Circuits and Systems
I: Regular Papers, vol. 70, no. 9, pp. 3413-3421, Sept. 2023, doi:
10.1109/TCSI.2023.3276700.
—, "Reciprocity and Inter-Reciprocity: A Tutorial—Part II: Linear
Periodically Time-Varying Networks," in IEEE Transactions on Circuits
and Systems I: Regular Papers, vol. 70, no. 9, pp. 3422-3435, Sept.
2023, doi: 10.1109/TCSI.2023.3294298.
—, "Interreciprocity in Linear Periodically Time-Varying Networks
With Sampled Outputs," in IEEE Transactions on Circuits and Systems II:
Express Briefs, vol. 61, no. 9, pp. 686-690, Sept. 2014, doi:
10.1109/TCSII.2014.2335393.
Piet Vanassche, Georges Gielen, and Willy Sansen. 2009. Systematic
Modeling and Analysis of Telecom Frontends and their Building Blocks
(1st. ed.). Springer Publishing Company, Incorporated.
Wereley, Norman. (1990). Analysis and control of linear periodically
time varying systems.
Hameed, S. (2017). Design and Analysis of Programmable Receiver
Front-Ends Based on LPTV Circuits. UCLA. ProQuest ID:
Hameed_ucla_0031D_15577. Merritt ID: ark:/13030/m5gb6zcz. Retrieved from
https://escholarship.org/uc/item/51q2m7bx
Rubiola, E. (2008). Phase Noise and Frequency Stability in
Oscillators (The Cambridge RF and Microwave Engineering Series).
Cambridge: Cambridge University Press. doi:10.1017/CBO9780511812798
Nicola Da Dalt and Ali Sheikholeslami. 2018. Understanding Jitter and
Phase Noise: A Circuits and Systems Perspective (1st. ed.). Cambridge
University Press, USA.
Hueber, G., & Staszewski, R. B. (Eds.) (2010).
Multi-Mode/Multi-Band RF Transceivers for Wireless Communications:
Advanced Techniques, Architectures, and Trends. John Wiley &
Sons. https://doi.org/10.1002/9780470634455
G. Richmond, "Refclk Fanout Best Practices for 8GT/s and 16GT/s
Systems," PCI-SIG Developers Conference, June 7, 2017
Knowing how input phase noise aliases when
sampled by a PLL
An alternate view of phase noise aliasing during the sampling
process
Instead of mirroring the jitter-transfer function
located below \(F_S/2\) across spectral
boundaries located at integer multiples of \(F_S/2\) (i.e. 50 MHz) as shown in Figure 2
(a)
we could alternatively fold the portion of the Raw Data
curve located above \(F_S/2\)
across these spectrum boundaries to appear below \(F_S/2\) as shown in Figure 2 (b)
Integrating the combined area under each Filtered Data curve
shown in Figure 2 (b) is mathematically equivalent to
integrating the entire Filtered Data curve shown in Figure 2 (a)
Phase Noise Analyzer vs TIE jitter using
Real-time Oscilloscope
Since an oscilloscope observes jitter similar to a real
system, we regard its result as the gold
standard against which other methods may be judged
Flat Phase Noise Extension to twice the clock
frequency
Phase Noise Aliasing
& Integration Limits
These two types of measurements deliver the same rms
jitter of \(f_{CK}\)
both rising and falling: integrated from \(-f_{CK}\) to \(+f_{CK}\)
only the rising (or falling) edges: integrated from \(-f_{CK}/2\) to \(+f_{CK}/2\)
temporal autocorrelation and Wiener-Khinchin
theorem is more appropriate to arise rms value
build the abs_jitter function with seconds as the Y
axis and add the stddev function to determine the Jee
jitter value
or integrate psd
The RMS \(x_{\text{RMS}}\) of a
discrete domain signal \(x(n)\) is
given by \[
x_{\text{RMS}}=\sqrt{\frac{1}{N}\sum_{n=0}^{N-1}|x(n)|^2}
\] Inserting Parseval's theorem given by \[
\sum_{n=0}^{N-1}|x(n)|^2=\frac{1}{N}\sum_{n=0}^{N-1}|X(k)|^2
\] allows for computing the RMS from the spectrum \(X(k)\) as \[
x_{\text{RMS}}=\sqrt{\frac{1}{N^2}\sum_{n=0}^{N-1}|X(k)|^2}
\]
Cadence Spectre's PN function may call
abs_jitter and psd function under the
hood.
Phase Noise in vsource
Suppose pnoise result of one block is shown as below, and the result
is stimulus of following block
First export Output Noise and
Edge Phase Noise， then select noiseModelType
and noisefile respectively
Under vsource (Source type: pulse) with
different amplitude & rising/falling time, simulation result
demonstrate that Edge Phase Noise(dBc) maintain
jitter or phase noise by tweaking voltage noise at edge
under the hoods, however Noise Voltage(V^2/Hz)
maintain voltage noise
In the conclusion, Edge Phase Noise(dBc) is
preferred for phase noise evaluation
notice:
@(#)$CDS: spectre version 21.1.0 64bit 12/01/2023 07:24 (csvcm36c-1) $
@(#)$CDS: virtuoso version ICADVM20.1-64b 10/11/2023 09:26 (cpgbld01) $
SSB Phase Noise (dBc)
Divider PN simulation
Cadence Support. "How to set up pss/pnoise when simulating a driven
circuit or a VCO, both containing dividers"
Modeling
Oscillators with Arbitrary Phase Noise Profiles
TODO 📅
reference
Article (11514536) Title: How to obtain a phase noise plot from a
transient noise analysis
Article (20500632) Title: How to simulate Random and Deterministic
Jitters
Cadence, Application Note: Understanding the relations between
time-average noise (phase-noise) and sampled noise (edge-phase noise or
jitter) in Pnoise analysis
A bleeding resistor (or a dedicated bleeding current
source) is typically used in a Low Dropout (LDO) voltage regulator or a
power supply circuit to ensure stability and proper
operation under light load or no-load conditions [Google AI
Mode]
resistor: affect both DC and AC (small
signal)
current source: affect DC bias only
(assuming infinite output impedance of current source)
The entire idea behind Kelvin connection is to
separatethe nodes that are carrying high
currentsfromthe sensing nodes to
the feedback
Error Amplifier
Using type B amplifier to drive the NMOS power stage will enhance the
NMOS’s PSRR performance
PSRR (Power Supply Rejection
Ratio)
A good PSRR is important when an LDO is used as a sub-regulator in
cascade with a switching regulator
The LDO would need to have a sufficiently high rejection at the
switching frequency of the switching converter to filter out the ripples
at that frequency
Mid-Frequency PSRR
High frequency PSRR
Open-Loop PSRR
Chen, Feng & Lu, Yasu & Mok, Philip. (2022). Transfer
Function Analysis of the Power Supply Rejection Ratio of Low-Dropout
Regulators and the Feed-Forward Ripple Cancellation Scheme. IEEE
Transactions on Circuits and Systems I: Regular Papers. [https://sci-hub.se/10.1109/TCSI.2022.3167860]
neglect the contribution of the voltage regulation
circuits to the PSRR
HW #3 - "Precision Low-Dropout Regulators" Online Course (2025) -
Prof. Yan Lu (Tsinghua University) [https://youtu.be/LXX1Xuhv2kI]
In a PMOS LDO, we want the gate of the power MOS transistor
to track the power supply ripple. This keeps the \(V_{GS}\) relatively constant, which in turn
ensures a stable supply of current through the transistor
DC output impedance
The output impedance of the LDO at DC is known as its load
regulation
DC output impedance of NMOS and PMOS LDO is the
same for the same error amplifier gain
That is, \[
\omega_{z,d} \approx \frac{1}{R(\frac{g_m}{g_{ds}}C_{gd}+C)}
\]
To calculate PSRR pole is similar with above PSRR zero, though \(V_o/V_i=0\), i.e. set \(V_0\)0 potential \[
\omega_{p,d} \approx \frac{1}{R(\frac{C_s}{g_mR}+C)}
\]
H. -S. Kim, "Exploring Ways to Minimize Dropout Voltage for
Energy-Efficient Low-Dropout Regulators: Viable approaches that preserve
performance," in IEEE Solid-State Circuits Magazine, vol. 15, no. 2, pp.
59-68, Spring 2023, doi: 10.1109/MSSC.2023.3262767.
Ali Sheikholeslami, Circuit Intuitions: Voltage Regulators IEEE
Solid-State Circuits Magazine, Vol. 12, Issue 4, to appear, Fall
2020.
A. Deutsch et al., "When are transmission-line effects
important for on-chip interconnections?," in IEEE Transactions on
Microwave Theory and Techniques, vol. 45, no. 10, pp. 1836-1846,
Oct. 1997
—. ISSCC 2007 T3: Dealing with Issues in VLSI Interconnect Scaling,
by Ron Ho
Tony Chan Carusone. ISSCC 2017 T6: Signal Integrity Analysis for Gb/s
Links
Byungsub Kim ISSCC 2022 T11: "Basics of Equalization Techniques:
Channels, Equalization, and Circuits"
Power Wave Equations
Peter J. Pupalaikis (Ciena). DesignCon 2026: Port Referencing in
S-Parameters – Critical Insights You Need to Know
1 2 3 4 5 6 7 8 9 10 11 12 13
Transmission Line Theory │ ▼ v = v⁺ + v⁻, i = (v⁺ - v⁻)/Z0 ← Physical decomposition │ ▼ v⁺ = (v + iZ0)/2, v⁻ = (v - iZ0)/2 ← Solve for forward/backward waves │ ▼ a = v⁺/√Z0, b = v⁻/√Z0 ← Normalize so |a|² = power │ ▼ v = √Z0·(a+b), i = (a-b)/√Z0 ← Invert to recover v and i
Z0 is a chosen reference impedance (typically 50Ω),
which is an arbitrary normalization choice. It does
not have to equal the characteristic impedance of the
transmission line
Any signal \(h(t)\) can be split
into even \(h_e(t)\) and odd \(h_o(t)\) components: Even
component:\(h_e(t) = \frac{h(t) +
h(-t)}{2}\), Odd component:\(h_o(t) = \frac{h(t) - h(-t)}{2}\), then
\(H_e(f) = \mathcal{Re}\{H(f)\} \quad H_o(f) =
j\cdot\mathcal{Im}\{H(f)\}\)
Conjugate symmetry ensures the time-domain signal is
real, but it takes the Hilbert Transform
relations to ensure it is causal\[
H(f) = H_e(f) + H_o(f)
\]
Passivity
causality-passivity
correction
P. Triverio, S. Grivet-Talocia, M. S. Nakhla, F. G. Canavero and R.
Achar, "Stability, Causality, and Passivity in Electrical Interconnect
Models," in IEEE Transactions on Advanced Packaging, vol. 30,
no. 4, pp. 795-808, Nov. 2007 [https://sci-hub.ru/10.1109/TADVP.2007.901567]
S. Sercu, C. Kocuba, J. Nadolny, "Causality Demystified", in
DesignCon 2015, Jan. 2015 [pdf]
Use the rational
function to fit data defined in the frequency domain with an equivalent
Laplace transfer function. Using rational function fitting you can
create simple models for a required accuracy, model order reduction,
zero phase on extrapolation to DC, and causal modeling
system among other advantages
S-parameters by definition require very specific control
over the ports. But this breaks down when we chain multiple devices
together
Cascading with ABCD Matrix
aka. transmission matrix
With load impedance \(v_3=i_3 Z_L\),
cascade transfer function can be derived
Cascading with T-Matrix
aka. scattering transfer parameters,
T-Parameters, transmission
parameters
When you cascade two networks by multiplying their T-matrices, you're
implicitly assuming that port 2 of network A and port 1 of
network B share the same wave definitions
Coelho, C. P., Phillips, J. R., & Silveira, L. M. (n.d.). Robust
rational function approximation algorithm for model generation.
Proceedings 1999 Design Automation Conference (Cat. No. 99CH36361). [https://sci-hub.ru/10.1109/dac.1999.781313]
Cadence IEEE IMS 2023, Introducing the Spectre S-Parameter Quality
Checker and Rational Fit Model Generator
Three fast time-domain system simulation techniques:
single-bit response method (SBR)
double-edge response method (DER)
multiple-edge response method (MER)
Symmetric Rising and Falling
Edges
Single-Bit Response (SBR)
Method
Overlapping portions of a pulse response from neighboring bits are
referred to as intersymbol interference (ISI). A received waveform is
formed by superimposing, in time, the pulse responses of each
bit in the sequence, as illustrated in Figure 9, assuming
symmetric positive and negative pulses are
transmitted for 1s and 0s
To avoid spurious glitches between consecutive ones, rising and
falling edge responses shall be symmetric. This is the
limitation of SBR method.
Let \(p(t)\) be the SBR of the
channel, \(t_s\) be the data sampling
phase, \(T\) be the bit time, \(N_c\) is the number of UI in stored pulse
response and \(b_m\) be the \(m\)th transmitted symbol. The voltage seen
by the receiver's data sampler at the \(m\)th data sample is determined by \[
y_m = \sum_{k=m-N_c+1}^{m}b_kp(t_s+(m-k)T)
\] where \(b_k \in [0, 1]\) and
\(p(t) \ge 0\)
We always prepend \(Nc-1\) 0s in
random bit stream for consistency.
For computation convenient, the pulse need to be positive. For
differential signal and amplitude \(V_{peak}\), the peak to peak is \(-V_{peak}\) to \(+V_{peak}\). After pulse added by \(V_{peak}\), peak to peak is \(0\) to \(+2V_{peak}\).
yy_sum = zeros(OSR*Ns, Ns); for idxBit = 1:Ns bs_split = zeros(1, Ns+Nc-1); bs_split(idxBit) = bs(idxBit); yy = zeros(OSR, Ns); for ii = Nc:Nc+Ns-1 bb = bs_split(ii:-1:ii-Nc+1); yy(:,ii-Nc+1) = sum(bb.*yrps, 2); end yy_cont2 = reshape(yy, [], 1); h = plot(yy_cont2); h.Annotation.LegendInformation.IconDisplayStyle = 'off'; yy_sum(:, idxBit) = yy_cont2; end yy_sum = sum(yy_sum, 2); % merge plot(yy_sum, 'k--'); plot(yy_cont, 'm-.'); grid on; legend('sum', 'syn'); title('merge all single bit'); ylabel('mag'); xlabel('Time (\times Ts)');
The pulse response contain rising and falling edge. The 1 bit
first rise from -1 to 1, then fall to -1; The 0
bit just do nothing for synthesized waveform with the help of
falling edge of 1 bit.
To handle the more general cases, with asymmetric rising and
falling edges, the system response can be constructed in terms
of edge transitions instead of bit responses.
The DER method decomposes the input data pattern, in terms of
rising and falling edge transitions. The system
response can be calculated by superimposing the shifted versions of
the rising and falling edge responses : \[
y_m = \sum_{k=m-N_c+1}^{m}(b_k-b_{k-1})s_k(t_s+(m-k)T) + y_{int}
\] where
\(r(t)\) and \(f(t)\) are the rising and falling edge
responses,respectively. \(V_{high}\)
and \(V_{low}\) are the steady state DC
levels, in response to a constant stream of ones and zeros,
respectively. \(y_{int}\) is the
initial DC state (either \(V_{high}\)
or \(V_{low}\) ).
We always prepend \(Nc\) 0s in
random bit stream for consistency.
T. C. Carusone, "Introduction to Digital I/O: Constraining I/O Power
Consumption in High-Performance Systems," in IEEE Solid-State
Circuits Magazine, vol. 7, no. 4, pp. 14-22, Fall 2015
Oh, Kyung Suk Dan, and Xing Chao Chuck Yuan. High-Speed Signaling:
Jitter Modeling, Analysis, and Budgeting. Prentice Hall, 2011. [pdf]
Ren, Jihong and Kyung Suk Oh. "Multiple Edge Responses for Fast and
Accurate System Simulations." IEEE Transactions on Advanced
Packaging 31 (2008) [https://sci-hub.jp/10.1109/TADVP.2008.2002201]
Shi, Rui. "Off-chip wire distribution and signal analysis." (2008).
[pdf]
X. Chu, W. Guo, J. Wang, F. Wu, Y. Luo and Y. Li, "Fast and Accurate
Estimation of Statistical Eye Diagram for Nonlinear High-Speed Links,"
in IEEE Transactions on Very Large Scale Integration (VLSI) Systems,
vol. 29, no. 7, pp. 1370-1378, July 2021, [https://sci-hub.ru/10.1109/TVLSI.2021.3082208]
Tingting Pang, DesignCon 2025: Fast BER Analysis Technique for
Next Generation Chiplet Simultaneous Bi-Directional Transceiver
L. Avallone, M. Mercandelli, A. Santiccioli, M. P. Kennedy, S.
Levantino and C. Samori, "A Comprehensive Phase Noise Analysis of
Bang-Bang Digital PLLs," in IEEE Transactions on Circuits and Systems I:
Regular Papers, vol. 68, no. 7, pp. 2775-2786, July 2021 [https://sci-hub.st/10.1109/TCSI.2021.3072344]
F/2 is the peak-to-peak amplitude of the
periodic jitter occurring at 1/2 of the data rate.
Even-odd jitter, also known as F/2
jitter, arises from a clock signal's duty cycle not being
perfectly 50%
Even-odd jitter has been referred to as
duty cycle distortion by other Physical Layer
specifications for operation over electrical backplane or twinaxial
copper cable assemblies
There are two primary causes of DCD jitter which are usually
generated within a transmitter
If the data input to a transmitter is theoretically perfect, but if
the transmitter sampling threshold is offset
from its ideal level, then the output of transmitter will have duty
cycle distortion as a function of the slew rate of the data
signal
Another cause of duty cycle distortion can be a
mismatch/asymmetry in rising and falling edge
speeds
Unfortunately, other sources such as ISI almost always exist making
it sometimes difficult to isolate the DCD component. One technique to
test for DCD is to stimulate your system/components with a
repeating 1-0-1-0… data pattern. This
technique will eliminate inter-symbol interference (ISI) jitter and make
viewing the DCD within the spectrum display much easier
Why clock pattern? That's because all symbols experience
same inter-symbol interference, which are
canceled out
Estimating the RMS cycle-to-cycle jitter if all you have available is
the RMS period jitter.
Cycle-to-cycle jitter - The short-term
variation in clock period between adjacent clock cycles. This
jitter measure, abbreviated here as \(J_{CC}\), may be specified as either an RMS
or peak-to-peak quantity.
Period jitter - The short-term variation
in clock period over all measured clock cycles, compared to the
average clock period. This jitter measure, abbreviated here as \(J_{PER}\), may be specified as either an
RMS or peak-to-peak quantity.
Let the variable below represent the variance of a single edge's
timing jitter, i.e. the difference in time of a jittery edge versus an
ideal edge, \(\sigma^2_j\)
If each edge's jitter is independent then the variance of
the period jitter can be written as \[\begin{align}
\sigma^2_\text{jper} &= (\sigma_\text{j(n+1)}-\sigma_\text{j(n)})^2
\\
&=
\sigma_\text{j(n+1)}^2-2\sigma_\text{j(n+1)}\sigma_\text{j(n)})+\sigma_\text{j(n)})^2\\
&= \sigma_\text{j(n+1)}^2+\sigma_\text{j(n)})^2 \\
&=2\sigma^2_j
\end{align}\]
In every cycle-to-cycle measurement we use one
"interior" clock edge twice and therefore we
must account for this
The ratio of the variances is therefore \[
\frac{\sigma^2_\text{jcc}}{\sigma^2_\text{jper}} =
\frac{6\sigma_\text{j}^2} {2\sigma_\text{j}^2}=3
\] Then \[
\sigma_\text{jcc} = \sqrt{3}\sigma_\text{per}
\]
N. Da Dalt, "Tutorial: Jitter: Basic and Advanced Concepts,
Statistics, and Applications," 2012 IEEE International Solid-State
Circuits Conference, San Francisco, CA, USA, 2012 [slides,
transcript
]
![Fig.2 The noise signal, its auto correlation function, and spectral density [2]](/2023/11/10/random/trnoise.jpg)
