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
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
- 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
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 \[
f(V_0+v) = f(V_0)+\frac{d}{dV}f(V)|_{V=V_0}\cdot
v+\frac{1}{2}\frac{d^2}{dV^2}f(V)|_{V=V_0}\cdot v^2+...
\] 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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![Fig.2 The noise signal, its auto correlation function, and spectral density [2]](/2023/11/10/random/trnoise.jpg)