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 \[ \frac{\partial F_{\text{n,k}}}{\partial _{V_\text{m,l}}} \] 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

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

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 excitation 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, \[ i(v) = \frac{d}{dV}f(V)|_{V=V_0}\cdot v \]

\(V_0\) 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 \(m\omega_1+n\omega_2\), 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

Conversion Matrix 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 due to \(\omega_n=\omega_0+n\omega_p\)

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

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

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