overview

\(\omega_d\) called damped natural frequency

closed loop frequency response

\[\begin{align} A &= \frac{\frac{A_0}{(1+s/\omega_1)(1+s/\omega_2)}}{1+\beta \frac{A_0} {(1+s/\omega_1)(1+s/\omega_2)}} \\ &= \frac{A_0}{1+A_0 \beta}\frac{1}{\frac{s^2}{\omega_1\omega_2(1+A_0\beta)}+\frac{1/\omega_1+1/\omega_2}{1+A_0\beta}s+1} \\ &\simeq \frac{A_0}{1+A_0 \beta}\frac{1}{\frac{s^2}{\omega_u\omega_2}+\frac{1}{\omega_u}s+1} \\ &= \frac{A_0}{1+A_0 \beta}\frac{\omega_u\omega_2}{s^2+\omega_2s+\omega_u\omega_2} \end{align}\]

That is \(\omega_n = \sqrt{\omega_u\omega_2}\) and \(\zeta = \frac{1}{2}\sqrt{\frac{\omega_2}{\omega_u}}\)

where \(\omega_u\) is the unity gain bandwidth

where \(f_r\) is resonant frequency, \(\zeta\) is damping ratio, \(P_f\) maximum peaking, \(P_t\) is the peak of the first overshoot (step response)

damping factor & phase margin

• phase margin is defined for open loop system

• damping factor (\(\zeta\)) is defined for close loop system

The roughly 90 to 100 times of damping factor (\(\zeta\)​) is phase margin \[ \mathrm{PM} = 90\zeta \sim 100\zeta \] In order to have a good stable system, we want \(\zeta > 0.5\) or phase margin more than \(45^o\)

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.

additional Zero

\[\begin{align} TF &= \frac{s +\omega_z}{s^2+2\zeta \omega_ns+\omega_n^2} \\ &= \frac{\omega _z}{\omega _n^2}\cdot \frac{1+s/\omega _z}{1+s^2/\omega_n^2+2\zeta s/\omega_n} \end{align}\]

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 \(\frac{d A_\text{dB}(\omega)}{d\omega} = 0\) \[ \omega_\text{peak} = \omega_z \sqrt{\sqrt{(\frac{\omega_n}{\omega_z})^4 - 2(\frac{\omega_n}{\omega_z})^2(2\zeta ^2-1)+1} - 1} \]

Settling Time

single-pole

\[ \tau \simeq \frac{1}{\beta \omega_\text{ugb}} \]

two poles

Rise Time

Katsuhiko Ogata, Modern Control Engineering Fifth Edition

For underdamped second order systems, the 0% to 100% rise time is normally used

For \(\text{PM}=70^o\)

• \(\omega_2=3\omega_u\), that is \(\omega_n = 1.7\omega_u\).
• \(\zeta = 0.87\)

Then \[ t_r = \frac{3.1}{\omega_u} \]

Settling Time

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 \(\text{PM}=70^o\), \(\omega_2 = 3\omega_u\), that is \[ t_s \simeq \frac{3}{\omega_u} \space\space \text{, for PM}=70^o \]

For \(\text{PM}=45^o\), \(\omega_2 = \omega_u\), that is \[ t_s \simeq \frac{9.2}{\omega_u} \space\space \text{, for PM}=45^o \]

Above equation is valid only for underdamped, \(\zeta=\frac{1}{2}\sqrt{\frac{\omega_2}{\omega_u}}\lt 1\), that is \(\omega_2\lt 4\omega_u\)

reference

Gene F. Franklin, J. David Powell, and Abbas Emami-Naeini. 2018. Feedback Control of Dynamic Systems (8th Edition) (8th. ed.). Pearson.

Katsuhiko Ogata, Modern Control Engineering, 5th edition

Conversion Relationships

Capacitor

Inductor

Derivation

series or parallel representation

reference

Tank Circuits/Impedances [https://stanford.edu/class/ee133/handouts/lecturenotes/lecture5_tank.pdf]

Resonant Circuits [https://web.ece.ucsb.edu/~long/ece145b/Resonators.pdf]

Series & Parallel Impedance Parameters and Equivalent Circuits [https://assets.testequity.com/te1/Documents/pdf/series-parallel-impedance-parameters-an.pdf]

ES Lecture 35: Non ideal capacitor, Capacitor Q and series RC to parallel RC conversion [https://youtu.be/CJ_2U5pEB4o?si=4j4CWsLSapeu-hBo]

Are AC-Driven Circuits Linear?

\[ f(x_1 + x_2)= f(x_1)+ f(x_2) \]

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. \[ V = RI \]

1. Thus, given the phasor for the current we can directly compute the phasor for the voltage across the resistor.

2. 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 \(i=C\frac{dv}{dt}\)

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 \[ i(t) = C\Re\{\frac{d}{dt}V e^{j\omega t}\} = \Re\{j\omega C V e^{j\omega t}\} \] The phasor representation for \(i\) is \(i(t) = \Re\{Ie^{j\omega t}\}\), that is \(I = j\omega C V\)

1. Thus, given the phasor for the voltage across a capacitor we can directly compute the phasor for the current through the capacitor.

2. 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 \(v=L\frac{di}{dt}\)

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 \[ v(t) = L\Re\{\frac{d}{dt}I e^{j\omega t}\} = \Re\{j\omega L I e^{j\omega t}\} \] The phasor representation for \(v\) is \(v(t) = \Re\{Ve^{j\omega t}\}\), that is \(V = j\omega L I\)

1. Thus, given the phasor for the current we can directly compute the phasor for the voltage across the inductor.

2. 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

\[\begin{align} y(t) &= H(s)e^{st} \\ H(s) &= \int_{-\infty}^{+\infty}h(\tau)e^{-s\tau}d\tau \end{align}\]

where \(h(t)\) is the impulse response of a continuous-time LTI system

convolution integral is used here

\[\begin{align} y[n] &= H(z)z^n \\ H(z) &= \sum_{k=-\infty}^{+\infty}h[k]z^{-k} \end{align}\]

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

General Response Classifications

• zero-input response, zero-state response & complete 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 & 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:

reference

Ken Kundert. Introduction to Phasors. Designer’s Guide Community. September 2011.

How to Perform Linearity Circuit Analysis [https://resources.pcb.cadence.com/blog/2021-how-to-perform-linearity-circuit-analysis]

Stephen P. Boyd. EE102 Lecture 7 Circuit analysis via Laplace transform [https://web.stanford.edu/~boyd/ee102/laplace_ckts.pdf]

Cheng-Kok Koh, EE695K VLSI Interconnect, S-Domain Analysis [https://engineering.purdue.edu/~chengkok/ee695K/lec3c.pdf]

Kenneth R. Demarest, Circuit Analysis using Phasors, Laplace Transforms, and Network Functions [https://people.eecs.ku.edu/~demarest/212/Phasor%20and%20Laplace%20review.pdf]

DeCarlo, R. A., & Lin, P.-M. (2009). Linear circuit analysis : time domain, phasor, and Laplace transform approaches (3rd ed).

Davis, Artice M.. "Linear Circuit Analysis." The Electrical Engineering Handbook - Six Volume Set (1998)

Duane Marcy, Fundamentals of Linear Systems [http://lcs-vc-marcy.syr.edu:8080/Chapter22.html]

Gene F. Franklin, J. David Powell, and Abbas Emami-Naeini. 2018. Feedback Control of Dynamic Systems (8th Edition) (8th. ed.). Pearson.

Data Register, DR:

• Bypass Register, BR
• Boundary Scan Register, BSR

Instruction Register, IR

TAP Controller

• 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

1. For DC-coupled interconnect, this time is of no concern
2. For AC-coupled interconnect, the signal may easily decay partially or completely before it can be captured
3. 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
  1. it decays to \(V_T\) after a few time constants
  2. 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

IEEE 1149.6 Tutorial | Testing AC-coupled and Differential High-speed Nets [https://www.asset-intertech.com/resources/eresources/ieee-11496-tutorial-testing-ac-coupled-and-differential-high-speed-nets/]

Prof. James Chien-Mo Li, Lab of Dependable Systems, National Taiwan University. VLSI Testing [http://cc.ee.ntu.edu.tw/~cmli/VLSItesting/]

K.P. Parker, The Boundary Scan Handbook, 3rd ed., Kluwer Academic, 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.

Effective Switching resistance

https://www.eecis.udel.edu/~vsaxena/courses/ece445/s19/Lecture%20Notes/lec15_ece445.pdf

wire delay

The Elmore Delay

Basic idea: use of mean of \(v'(t)\) to approximate median of \(v'(t)\)

Elmore delay approximates the median of \(h(t)\) by the mean of \(h(t)\)

Distributed RC-Line

Lumped approximations

\(rc\)-models

If your simulator does not support a distributed \(rc\)-model, or if the computational complexity of these models slows down your simulation too much, you can construct a simple yet accurate model yourself by approximating the distributed \(rc\) by a lumped RC network with a limited number of elements

The accuracy of the model is determined by the number of stages. For instance, the error of the \(\Pi -3\) model is less than 3%, which is generally sufficient.

Why use "\(\Pi\) Model"

examples

Wire Inductive Effect

• RC delay increases quadratically with length
• LC delay (speed of light flight time) increases linearly with length

Inductance will only be important to the delay of low-resistance signals such as wide clock lines

wave

Signal propagates over the wire as a wave (rather than diffusing as in \(rc\) only models)

Signal propagates by alternately transferring energy from capacitive to inductive modes

reference

Akio Kitagawa, Analog layout design https://mixsignal.files.wordpress.com/2013/03/analog-layout.pdf

THE WIRE http://bwrcs.eecs.berkeley.edu/Classes/icdesign/ee141_f01/Notes/chapter4.pdf

Anoop Veliyath, Design Engineer, Cadence Design Systems. Accurately Modeling Transmission Line Behavior with an LC Network-based Approach [pdf]

Mark Horowitz. Lecture 2: Wires and Wire Models [pdf]

Neil Weste and David Harris. 2010. CMOS VLSI Design: A Circuits and Systems Perspective (4th. ed.). Addison-Wesley Publishing Company, USA.

Cheng-Kok Koh. EE695K Modeling and Optimization of High Performance Interconnect [lec3a_pdf]

Vishal Saxena. ECE 445 Intro to VLSI Design: Lectures for Spring 2019 https://www.eecis.udel.edu/~vsaxena/courses/ece445/s19/ECE445.htm

Ergodicity & autocorrelation

The most important statistical properties of a random process \(x(t)\) are obtained by applying the expectation operator to some functions of the process itself

• The expectation returns the probability-weighted average of the specific function at that specific time over all possible realizations of the process

In many real practical cases, though, data from many realizations of the process are not available. On the contrary, often only data from one of them are known

ensemble autocorrelation and temporal autocorrelation (time autocorrelation)

LTI Systems on WSS Processes

mean

autocorrelation

deterministic autocorrelation function

why \(\overline{R}_{hh}(\tau) \overset{\Delta}{=} h(\tau)*h(-\tau)\) is autocorrelation ? the proof is as follows:

\[\begin{align} \overline{R}_{hh}(\tau) &= h(\tau)*h(-\tau) \\ &= \int_{-\infty}^{\infty}h(x)h(-(\tau - x))dx \\ &= \int_{-\infty}^{\infty}h(x)h(-\tau + x))dx \\ &=\int_{-\infty}^{\infty}h(x+\tau)h(x))dx \end{align}\]

PSD

Topic 6 Random Processes and Signals [https://www.robots.ox.ac.uk/~dwm/Courses/2TF_2021/N6.pdf]

Alan V. Oppenheim, Introduction To Communication, Control, And Signal Processing [https://ocw.mit.edu/courses/6-011-introduction-to-communication-control-and-signal-processing-spring-2010/a6bddaee5966f6e73450e6fe79ab0566_MIT6_011S10_notes.pdf]

Time Reversal \[ x(-t) \overset{FT}{\longrightarrow} X(-j\omega) \]

if \(x(t)\) is real, then \(X(j\omega)\)​ has conjugate symmetry \[ X(-j\omega) = X^*(j\omega) \]

Random Signals Sampling

sampling autocorrelation sequence

Alan V Oppenheim, Ronald W. Schafer. Discrete-Time Signal Processing, 3rd edition

[https://dsp.stackexchange.com/a/17348/59253]

Noise Aliasing

apply foregoing observation

The Periodogram

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 \(v[n] = w[n]x[n]\)

\[\begin{align} \hat{P}_{ss}(\Omega) &= \frac{|V(e^{j\omega})|^2}{LU} \\ &= \frac{|V(e^{j\omega})|^2}{\sum_{n=0}^{L-1}(w[n])^2} \tag{1}\\ &= \frac{L|V(e^{j\omega})|^2}{\sum_{k=0}^{L-1}(W[k])^2} \tag{2} \end{align}\]

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 \((2)\) \[ \hat{P}_{ss}(\Omega) = T_s\hat{P}_{xx(\omega)} = \frac{T_sL|V(e^{j\omega})|^2}{\sum_{k=0}^{L-1}(W[k])^2} = \frac{|V(e^{j\omega})|^2}{f_{res}\sum_{k=0}^{L-1}(W[k])^2} \]

!! ENBW

Wiener-Khinchin theorem

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 (\(\frac{1}{2\pi}d\omega = df\))

[https://www.robots.ox.ac.uk/~dwm/Courses/2TF_2011/2TF-L5.pdf]

Example

Remember: impulse scaling

\[ \cos(2\pi f_0t) \overset{\mathcal{F}}{\longrightarrow} \frac{1}{2}[\delta(f -f_0)+\delta(f+f_0)] \]

Energy Signal

reference

L.W. Couch, Digital and Analog Communication Systems, 8th Edition, 2013

Alan V Oppenheim, George C. Verghese, Signals, Systems and Inference, 1st edition

PSS and HB Overview

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 \[ \frac{\partial F_{\text{n,k}}}{\partial _{\text{m,l}}} \] where \(n\) and \(m\) are the port indices \((1,N)\), and \(k\) and \(l\) are the harmonic indices \((0,...,K)\)

Selecting the Number of Harmonics and Time Samples

In theory, the waveforms generated in nonlinear analysis have an infinite number of harmonics, so a complete description of the operation of a nonlinear circuit would appear to require current and voltage vectors of infinite dimension.

Fortunately, the magnitudes of frequency components invariably decrease with frequency; otherwise the time wavefroms would represent infinite power.

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.

Shooting Newton

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

Periodic small signal analyses

Analysis in Simulator

1. 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.).
2. 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)
3. 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

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

1. First, analyzing the nonlinear device under large-signal excitatin only, where the harmonic-balance method can be applied
2. Then, the nonlinear elements in the device's equivalent circuit are then linearized to create small-signal, linear, time-varying elements
3. 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 \[ f(V_0+v) = f(V_0)+\frac{d}{dV}f(V)|_{V=V_0}\cdot v+\frac{1}{2}\frac{d^2}{dV^2}f(V)|_{V=V_0}\cdot v^2+... \] 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, \[ i(v) = \frac{d}{dV}f(V)|_{V=V_0}\cdot v \]

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

\(V_0\) 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 \(g(t)=\frac{d}{dV}f(V)|_{V=V_L(t)}\)

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 \(r(t) = \frac{d}{dI}f_R(I)|_{I=I_L(t)}\)

A nonlinear element excited by two tones supports currents and voltages at mixing frequencies \(m\omega_1+n\omega_w\), 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

Conversion Mattrix as Bridge

The frequency-domain currents and voltages in a time-varying circuit element are related by a conversion matrix

The small-signal voltage and current can be expressed in the frequency notation as \[ v'(t) = \sum_{n=-\infty}^{\infty}V_ne^{j\omega_nt} \] and \[ i'(t) = \sum_{n=-\infty}^{\infty}I_ne^{j\omega_nt} \] where \(v'(t)\) and \(i'(t)\) are not Fourier series, which includes only half of the mixing frequencies

The conductance waveform \(g(t)\) can be expressed by its Fourier series \[ g(t)=\sum_{n=-\infty}^{\infty}G_ne^{jn\omega_pt} \] Which is harmonics of large-signal \(\omega_p\)

Then the voltage and current are related by Ohm's law \[ i'(t)=g(t)v'(t) \] Then \[ \sum_{k=-\infty}^{\infty}I_ke^{j\omega_kt}=\sum_{n=-\infty}^{\infty}\sum_{m=-\infty}^{\infty}G_nV_me^{j\omega_{m+n}t} \]

A linear periodically-varying transfer function implements frequency translation

Cyclostationary Noise Analysis

which is referred to as a "periodic noise" or PNoise analysis

White Noise

completely uncorrelated versus time

For white noise the PSD is a constant and the autocorrelation function is an impulse centered at \(0\), \(R_n(t,\tau)=R(t)\delta(\tau)\)

The energy-storage elements cause the noise spectrum to be shaped and the noise to be time-correlated.

This is a general property. It the noise has shape in the frequency domain then the noise is correlated in time, and vice versa.

PNOISE

• In LPTV analysis, noise may up-convert or down-convert by \(N\cdot f_c\) (noise folding)

• PNOISE output is cyclostationary noise

  Described by a collection of PSDs at various sidebands: 0 PSDs at various sidebands: \(0, \pm f_c, \pm2f_c, \pm3f_c\), …

\(f_c\): fundamental frequency of PSS

Simulation of switched-capacitor noise

Periodic steady-state analysis is originally intended to analyze a continuous-time circuit with periodic input signals or excitations.

To simulate a switched-capacitor circuit appropriately, one needs to recognize that the output of a switched capacitor circuit is a discrete-time rather than a continuous-time signal. This discrete-time signal should be treated as the output of the circuit sampled after it has settled to the final value for each sampling period.

There are two techniques that one can use to force the simulator to evaluate the output signal correctly in the manner described

• First, in more recent versions of SpectreRF, PNOISE analysis provides a specialized time-domain analysis method

  By enabling this option, the simulator would analyze noise only at particular time instants

• Second, on older versions of spectreRF, an explicit (ideal) sample-and-hold block can be used similarly to force the simulator to evaluate only the output of the circuit at the correct time instants.

  Recall that a sample-and-hold would impose a zero-order hold on a discrete-time signal; thus, the resulting sinc-shaped response in the frequency domain has to be compensated for

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 \[ H_k(j2\pi f) = \frac{1}{T_s}\int_0^{T_s} H(j2\pi f, t) e^{-j2\pi k f_s t}dt \]

LTI and LTV

Linear and Periodically Time Varying

Gain varies periodically with time

Owing to \(H(j\omega, t) = H(j\omega, t+T_s)\), \(H(j\omega,t)\) can be expanded in a Fourier Series \[ H(j\omega,t) = \sum_k H_k(j\omega)e^{jk\omega _s t} \]

Response of an LPTV System

Harmonic Transfer Functions

Harmonic Transfer Functions can be found using PAC analysis

Summary

Sampled LPTV

Move sampler before summation

Owing to \[ k\omega_sT_sn=kn\cdot2\pi \] We get \[ e^{jk\omega_s T_s n} = e^{jkn\cdot 2\pi} = 1 \]

LPTV system sampled at \(T_s\) is equivalent to LTI system sampled at \(T_s\)

\[ H_{\text{eq}}(j\omega) = \sum_k H_k(j\omega) \]

The result derived above makes intuitive sense due to the following.

When an LPTV system is excited by a tone at \(f\) , the output comprises of tones at frequencies \(f + k f_s\), where \(k\) is an integer. When sampled at \(f_s\), frequency components higher than \(f_s\) are aliased to \(f\) . Thus, if one is only interested in the samples of the system's output, they could as well be produced by a properly chosen LTI filter acting on an input tone at a frequency \(f\) .

Note that the equivalence holds only for samples, and not for the waveforms. \(y(nT_s) = \hat{y}(nT_s)\), but \(y(t)\) need not equal \(\hat{y}(t)\)

Finding the Equivalent Filter

Frequency Domain Approach \(H_{eq}(j\omega)=\sum_k H_k(j\omega)\)

Wasted effort in computing all \(H_k\) first and then adding them up.

We prefer to employ Time Domain Approaches

Reciprocity

Does not work with controlled sources

Interreciprocity

Handle controlled sources

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.

Phillips, Joel R. and Kenneth S. Kundert. "Noise in mixers, oscillators, samplers, and logic: an introduction to cyclostationary noise." Proceedings of the IEEE 2000 Custom Integrated Circuits Conference (Cat. No.00CH37044) (2000): 431-438. [pdf]

Karti Mayaram. ECE 521 Fall 2016 Analog Circuit Simulation: Simulation of Radio Frequency Integrated Circuits [pdf1, pdf2]

The Value Of RF Harmonic Balance Analyses For Analog Verification: Frequency domain periodic large and small signal analyses. [https://semiengineering.com/the-value-of-rf-harmonic-balance-analyses-for-analog-verification/]

Shanthi Pavan, "Demystifying Linear Time Varying Circuits"

S. Pavan and G. C. Temes, "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.

S. Pavan and G. C. Temes, "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.

S. Pavan and R. S. Rajan, "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.

Prof. Shanthi Pavan. Introduction to Time - Varying Electrical Network. [https://youtube.com/playlist?list=PLyqSpQzTE6M8qllAtp9TTODxNfaoxRLp9]

Shanthi Pavan. EE5323: Advanced Electrical Networks (Jan-May. 2015) [https://www.ee.iitm.ac.in/vlsi/courses/ee5323/start]

R. S. Ashwin Kumar. EE698W: Analog circuits for signal processing [https://home.iitk.ac.in/~ashwinrs/2022_EE698W.html]

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.

Beffa, Federico. (2023). Weakly Nonlinear Systems. 10.1007/978-3-031-40681-2.

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

Matt Allen. Introduction to Linear Time Periodic Systems. [https://youtu.be/OCOkEFDQKTI]

Fivel, Oren. "Analysis of Linear Time-Varying & Periodic Systems." arXiv preprint arXiv:2202.00498 (2022).

RF Harmonic Balance Analysis for Nonlinear Circuits [https://resources.pcb.cadence.com/blog/2019-rf-harmonic-balance-analysis-for-nonlinear-circuits]

Steer, Michael. Microwave and RF Design (Third Edition, 2019). NC State University, 2019.

Steer, Michael. Harmonic Balance Analysis of Nonlinear RF Circuits - Case Study Index: CS_AmpHB [link]

Vishal Saxena, "SpectreRF Periodic Analysis" URL:https://www.eecis.udel.edu/~vsaxena/courses/ece614/Handouts/SpectreRF%20Periodic%20Analysis.pdf

Josh Carnes,Peter Kurahashi. "Periodic Analyses of Sampled Systems" URL:https://slideplayer.com/slide/14865977/

Kundert, Ken. (2006). Simulating Switched-Capacitor Filters with SpectreRF. URL:https://designers-guide.org/analysis/sc-filters.pdf

Article (20482538) Title: Why is pnoise sampled(jitter) different than pnoise timeaverage on a driven circuit? URL:https://support.cadence.com/apex/ArticleAttachmentPortal?id=a1O0V000009EStcUAG

Article (20467468) Title: The mathematics behind choosing the upper frequency when simulating pnoise jitter on an oscillator URL:https://support.cadence.com/apex/ArticleAttachmentPortal?id=a1O0V000007MnBUUA0

Andrew Beckett. "Simulating phase noise for PFD-CP". URL:https://groups.google.com/g/comp.cad.cadence/c/NPisXTElx6E/m/XjWxKbbfh2cJ

Dr. Yanghong Huang. MATH44041/64041: Applied Dynamical Systems [https://personalpages.manchester.ac.uk/staff/yanghong.huang/teaching/MATH4041/default.htm]

Jeffrey Wong. Math 563, Spring 2020 Applied computational analysis [https://services.math.duke.edu/~jtwong/math563-2020/main.html]

Jeffrey Wong. Math 353, Fall 2020 Ordinary and Partial Differential Equations [https://services.math.duke.edu/~jtwong/math353-2020/main.html]

Tip of the Week: Please explain in more practical (less theoretical) terms the concept of "oscillator line width." [https://community.cadence.com/cadence_blogs_8/b/rf/posts/please-explain-in-more-practical-less-theoretical-terms-the-concept-of-quot-oscillator-line-width-quot]

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

Dr. Ulrich L. Rohde, Noise Analysis, Then and Today [https://www.microwavejournal.com/articles/29151-noise-analysis-then-and-today]

Nicola Da Dalt and Ali Sheikholeslami. 2018. Understanding Jitter and Phase Noise: A Circuits and Systems Perspective (1st. ed.). Cambridge University Press, USA.

Kester, Walt. (2005). Converting Oscillator Phase Noise to Time Jitter. [https://www.analog.com/media/en/training-seminars/tutorials/MT-008.pdf]

Drakhlis, B.. (2001). Calculate oscillator jitter by using phase-noise analysis: Part 2 of two parts. Microwaves and Rf. 40. 109-119.

Explanation for sampled PXF analysis. [https://community.cadence.com/cadence_technology_forums/f/custom-ic-design/45055/explanation-for-sampled-pxf-analysis/1376140#1376140]

模拟放大器的低噪声设计技术2-3实践 아날로그 증폭기의 저잡음 설계 기법2-3실습 [https://youtu.be/vXLDfEWR31k?si=p2gbwaKoTxtfvrTc]

2.2 How POP Really Works [https://www.simplistechnologies.com/documentation/simplis/ast_02/topics/2_2_how_pop_really_works.htm]

Jaeha Kim. Lecture 11. Mismatch Simulation using PNOISE [https://ocw.snu.ac.kr/sites/default/files/NOTE/7037.pdf]

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

B. Boser,A. Niknejad ,S.Gambin 2011 EECS 240 Topic 6: Noise Analysis [https://mixsignal.files.wordpress.com/2013/06/t06noiseanalysis_simone-1.pdf]

Integration Limits

Y. Zhao and B. Razavi, "Phase Noise Integration Limits for Jitter Calculation,"[https://www.seas.ucla.edu/brweb/papers/Conferences/YZ_ISCAS_22.pdf]

TODO 📅

VCO Phase Noise

pnoise - timeaverage

1. Direct Plot/Pnoise/Phase Noise or

   

2. manually calculate by definition

   

3. output noise with unit dBc

   Direct Plot/Pnoise/Output Noise Units:dBc/Hz and Noise convention: SSB

   

The above method 2 and 3 only apply to timeaveage pnoise simulation,

pnoise - sampled(jitter)/Edge Crossing

Direct Plot/Pnoise/Edge Phase Noise or

Another way, the following equation can also be used for sampled(jitter)/Edge Crossing

1

PhaseNoise(dBc/Hz) = dB20( OutputNoise(V/sqrt(Hz)) / slopeCrossing / Tper*twoPi ) - dB10(2)

where dB10(2) is used to obtain SSB from DSB

Output Noise of sampled(jitter) pnoise

The last section's Output Noise (V**2/Hz) can be obtained by transient noise simulation

The idea is that sample waveform with ideal clock, subtract DC offset, then fft(psd)

• samplesRaw = sample(wv)
• samplePost = samplesRaw - average(samplesRaw)
• Output Noise (V**2/Hz) = psd(samplePost)

Expression:

The computation cost is typically very high, and the accuracy is lesser as compared to PSS/Pnoise

Pnoise Sampled(jitter): Sampled Phase Option

• Identical to noisetype=timedomain in old GUI
• Use model:
  • Sampleds Per Period: number of ponits
  • Add Specific Points: specific time point, still time points

pss beat freq = 5GHz

pnoise sweeptype: absolute, from 100k to 2.5GHz

Transient noise

phase noise from transient noise analysis

1. The Phase Noise function is now available in the Direct Plot form (Results-Direct Plot-Main Form) after Transient Analysis is run
   • Absolute jitter Method
   • Direct Power Spectral Density Method
2. PN phase noise function
   • Absolute jitter Method
   • Direct Power Spectral Density Method

Absolute jitter Method: Phase noise is defined as the power spectral density of the absolute jitter of an input waveform

and absolute jitter method is the default method

In below discussion, we only think about the absolute jitter method

PSD and Phase Noise

• phase noise is single-sideband
• psd is double-sideband
• Then the ratio is 2

By PSS_Pnoise

jee

1

rfEdgePhaseNoise(?result "pnoise_sample_pm0" ?eventList 'nil) + 10 * log10(2)

convert single-sideband phase noise to psd by multiplying 2 or 10 * log10(2)

By trannoise PN function

1

PN(clip(VT("/Out1") 2.60417e-08 0.000400052) "rising" 1.65 ?Tnom (1 / 3.84e+07) ?windowName "Rectangular" ?smooth 1 ?windowSize 15000 ?detrending "None" ?cohGain 1 ?methodType "absJitter")

double-sideband psd

By trannoise psd and abs_jitter function

1

dB10(psd(abs_jitter(clip(VT("/Out1") 2.60417e-08 0.000400052) "rising" 1.65 ?Tnom (1 / 3.84e+07)) 2.60417e-08 0.000400052 15360 ?windowName "Rectangular" ?smooth 1 ?windowSize 15000 ?detrending "None" ?cohGain 1))

double-sideband psd

abs_jitter Y-Unit default is rad

Comparison

PN's result is same with psd's

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 \[ x_{\text{RMS}}=\sqrt{\frac{1}{N^2}\sum_{n=0}^{N-1}|X(k)|^2} \]

Remarks

Cadence Spectre's PN function may call abs_jitter and psd function under the hood.

reference

Article (11514536) Title: How to obtain a phase noise plot from a transient noise analysis URL: https://support.cadence.com/apex/ArticleAttachmentPortal?id=a1Od0000000nb1CEAQ

Article (20500632) Title: How to simulate Random and Deterministic Jitters URL: https://support.cadence.com/apex/ArticleAttachmentPortal?id=a1O3w000009fiXeEAI

Tutorial on Scaling of the Discrete Fourier Transform and the Implied Physical Units of the Spectra of Time-Discrete Signals Jens Ahrens, Carl Andersson, Patrik Höstmad, Wolfgang Kropp URL: https://appliedacousticschalmers.github.io/scaling-of-the-dft/AES2020_eBrief/

PMOS LDO

A large output capacitor solves the sudden load change problem, but the stability decreases because the large output capacitor forces the output pole toward the origin with the role of dominant pole

DC output impedance

DC output impedance of NMOS and PMOS LDO is the same for the same error amplifier gain

\[ R_{\text{out}} = \frac{1}{\beta A_{o1} g_{m2}} \]

The NMOS LDO has a faster response to line transients than the PMOS LDO since it has better (smaller) PSRR

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

Power MOS gain affect on PMOS LDO

DC gain \[ A_{dc} = g_mR_\text{ota} A_2 \] 3dB bandwidth \[ \omega_p = \frac{1}{R_\text{ota}(C_g+A_2C_c)} \] and GBW \[ \omega_u = \frac{g_m}{\frac{C_g}{A_2}+C_c} \]

Feedforward Compensation with \(C_\text{FF}\)

• Improved Noise
• Improved Stability and Transient Response
• Improved PSRR

\[\begin{align} R_1 \parallel \frac{1}{sC_{FF}} &= \frac{R_1}{1+sR_1C_{FF}} \\ Z_o &= \left( R_1\parallel \frac{1}{sC_{FF}}+R_2\right)\parallel \frac{1}{sC_L} \\ &=\frac{R_1+R_2+sR_1R_2C_{FF}}{s^2R_1R_2C_{FF}C_L + s[(R_1+R_2)C_L+R_1C_{FF}]+1} \\ A_{V2} &= g_m Z_o \\ &= g_m \frac{R_1+R_2+sR_1R_2C_{FF}}{s^2R_1R_2C_{FF}C_L + s[(R_1+R_2)C_L+R_1C_{FF}]+1} \\ \beta &= \frac{R_2}{\frac{R_1}{1+sR_1C_{FF}}+R_2} \\ &= \frac{R_2(1+sR_1C_{FF})}{R_1+R_2+sR_1R_2C_{FF}} \\ A_{V2}\beta &= \frac{g_mR_2(1+sR_1C_{FF})}{s^2R_1R_2C_{FF}C_L+s[(R_1+R_2)C_L+R_1C_{FF}]+1} \end{align}\]

That is, adding a \(C_{FF}\) also introduces a zero (\(\omega_z\)) and pole (\(\omega_p\)) into the LDO feedback loop

\[\begin{align} \omega_{po} &= \frac{1}{(R_1+R_2)C_L} \\ \omega_z &= \frac{1}{R_1C_{FF}} \\ \omega_{p} &= \frac{1}{(R_1 \parallel R_2)C_{FF}} \end{align}\]

Application Report SBVA042–July 2014, Pros and Cons of Using a Feedforward Capacitor with a Low-Dropout Regulator [https://www.ti.com/lit/an/sbva042/sbva042.pdf]

LDO Basics: Noise – How a Feed-forward Capacitor Improves System Performance [https://www.ti.com/document-viewer/lit/html/SSZTA13]

LDO Basics: Noise – How a Noise-reduction Pin Improves System Performance [https://www.ti.com/document-viewer/lit/html/SSZTA40]

NMOS Slave LDO

\[\begin{align} \frac{V_g}{V_i} &=\frac{R||\frac{1}{s(C_g+C_{gs})}}{R||\frac{1}{s(C_g+C_{gs})}+\frac{1}{sC_{gd}}} \\ &= \frac{sRC_{gd}}{sR(C+C_{gd}+C_{gs})+1} \end{align}\] \[ \frac{V_g}{V_i} = -\frac{sC_{ds}+g_{ds}}{sC_{gs}+g_m} \]

That is, \[ \omega_{z,d} \simeq \frac{1}{R(\frac{g_m}{g_{ds}}C_{gd}+C)} \]

The calculating PSRR pole approach similar with zero, just \(V_o\) to \(V_i\) zero \[ \omega_{p,d} \simeq \frac{1}{R(\frac{C_s}{g_mR}+C)} \]

DC PSRR \[ \text{PSRR} = \frac{1}{g_mr_o} \]

PSRR @Vgate

KCL at output node

\[ g_m(-V_o\beta A_{E} - V_o) + \frac{V_i - V_o}{r_o} = \frac{V_o}{R_1+R_2} \]

Hence \[ \frac{V_o}{V_i} = \frac{1}{A_E\beta g_mr_o+g_mr_o +\frac{r_o}{R_1+R_2}+1} \approx \frac{1}{A_E\beta g_m r_o} \]

Through feedback loop, we derive \[ V_g = V_o \beta (-A_E) \approx \frac{V_i}{A_E\beta g_m r_o} \beta (-A_E) = -\frac{V_i}{g_mr_o} \]

That is \[ \frac{V_g}{V_i} \approx -\frac{1}{g_mr_o} \]

Due to closed loop, \(V_g\) and \(V_o\) is not source follower

High frequency PSRR

feedback resistor divider noise

assuming \(\text{LG} \gg 1\)

\[\begin{align} I_\text{t} &= \frac{V_\text{ref} - v_\text{n2}}{R_\text{2}} \\ V_\text{o} &= V_\text{ref} +v_\text{n1} + I_\text{t}R_\text{1} \\ \end{align}\]

Then, \[ V_\text{o} = \frac{R_1+R_2}{R_2}V_\text{ref} + v_\text{n1} - \frac{R_1}{R_2}v_\text{n2} \] that is,

\[ v_\text{no}^2 = v_\text{n1}^2 + \left(\frac{R_1}{R_2}\right)^2 v_\text{n2}^2 \]

\[ \text{vno1}^2= \text{vn1}^2+\text{vn2}^2/6^2=16.5758 + 99.45453/6^2 = 19.338425833 \]

Flipped Voltage Follower (FVF)

[https://www.linkedin.com/posts/chembiyan-t-0b34b910_flipped-voltage-follower-fvf-basics-activity-7118482840803020800-qwyX?utm_source=share&utm_medium=member_desktop]

reference

Hinojo, J.M., Martinez, C.I., & Torralba, A.J. (2018). Internally Compensated LDO Regulators for Modern System-on-Chip Design.

Mingoo Seok. ISSCC 2020 tutorial "Basics of Digital Low-Dropout (LDO) Integrated Voltage Regulator"

Pavan Kumar Hanumolu. CICC 2015. "Low Dropout Regulators"

Chen, K. (2016). Power Management Techniques for Integrated Circuit Design.

Morita, B.G. (2014). Understand Low-Dropout Regulator ( LDO ) Concepts to Achieve Optimal Designs.

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.

Operational Transconductance Amplifier II Multi-Stage Designs [https://people.eecs.berkeley.edu/~boser/courses/240B/lectures/M07%20OTA%20II.pdf]

Toshiba, Load Transient Response of LDO and Methods to Improve it Application Note [https://toshiba.semicon-storage.com/info/application_note_en_20210326_AKX00312.pdf?did=66268]

Andrew Beckettover 11 years ago

Two approaches:

1. On the transient options form, there's a field called "infotimes" - specify the times at which you want it to output the dc operating point data. You can then annotate the "transient operating points" from any of these times after the simulation, or access them via the results browser.
2. Or you could get the operating point data to be continuously saved during the transient for selected devices - if so, create a file called (say) "save.scs" (make sure it has a ".scs" suffix), and put: save M1:oppoint or save M*:oppoint sigtype=dev in this file, and then reference the file via Setup->Model Libraries or as a "definition file" on Setup->Simulation Files. With this approach you can then find the operating point data for the selected devices in the results browser and plot it versus time (be cautious of saving too much though because this can generate a lot of data if you're not careful)

Regards,

Andrew.

transient options form

setup

access 1

right-click \(\to\) Annotate \(\to\) Transient Operating Points

access 2

tranOpTimed

save.scs

1

save M0:oppoint