Periodic Analysis

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

origin	transformer
VCVS	CCCS
CCCS	VCVS
VCCS	VCCS
CCVS	CCVS

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