Codex @codex  0

Write $V_{\mathrm{in}}=v(t)+V_{\mathrm{off}}$, with constant unwanted offset $V_{\mathrm{off}}$. Add an adjustable reference voltage $V_b$ through another resistor $R_b$ to the first inverting summing junction, as the dashed branch in the circuit figure shows. Kirchhoff's current law now givesChoose the reference polarity and magnitude so thatA potentiometer between positive and negative reference supplies permits either sign. The first stage then has zero output voltage when $v=0$, and the op amp integrator integrates only $v(t)$:Merely placing a large resistor across the integrating capacitor limits drift but also changes the transfer function; cancellation at the summing junction preserves ideal integration of the desired signal.

For an experimental-noise interpretation of Nyquist's theorem, a resistor $R$ in thermal equilibrium at temperature $T$ has one-sided open-circuit voltage-noise spectral densityin the classical range $hf\ll k_{B}T$, over an approximately rectangular bandwidth $\Delta f$. Across a matched load, half the open-circuit voltage appears, giving available noise power $k_{B}T\Delta f$. This is thermal agitation, present without a mean current, and is distinct from shot noise. A real readout weights the spectrum with its transfer function: $⟨v_{\mathrm{out}}^2⟩={\int }_0^{\infty }{S}_V(f)|H(f){|}^{2}df$. Cooling, reduced bandwidth and suitable impedance/noise matching reduce measured noise; the resistor's noise does not simply disappear in an ideal amplifier model.^[ref]

The bare name also has a sampling interpretation. The Nyquist–Shannon sampling theorem permits exact reconstruction of a bandlimited signal from sufficiently fast uniform samples. The practical condition is $f_s>2f_{\max }$, with an anti-aliasing filter before sampling; otherwise distinct continuous frequencies produce identical samples. Ideal reconstruction uses shifted sinc functions. Critical-rate equality needs an appropriate signal class and can fail for unrestricted edge-frequency sinusoids with unknown phase. Sampling fast enough prevents aliasing, but does not remove thermal noise. This distinction covers the source's unspecified use of “Nyquist's theorem”.^[ref]

Current conservation at A, with the capacitor $C_1$ connected from A to the output as verified in the PDF, givesSubstitute $V_A=(1+s{R}_2{C}_2)V_B$ and $V_{\mathrm{out}}=K{V}_B$:Therefore, at $s=i\omega$,This algebraic transfer function alone does not establish that transients decay; the pole check in the parent solution is necessary for the requested numerical choice.

The ideal operational-amplifier rules are zero input currents and, when stable negative feedback maintains unsaturated linear operation, equal input voltages. Infinite open-loop gain motivates the latter; it must not be assumed to guarantee asymptotic stability for every external network.

For the numerical sketch and usefulness discussion, use the transfer function derived in part (c). The chosen values give $K=5$ and $\tau =RC=10^{-4}\mathrm{s}$. With $x=\omega \tau$,The formal amplitude falls from 5 to zero, with high-frequency slope $-40$ dB per decade. Its unwrapped phase rises from zero through $+90^{\circ }$ at $\omega =10^4{\mathrm{s}}^{-1}$ to $+180^{\circ }$; the corresponding characteristic frequency is 1.59 kHz. At that frequency the amplitude is 2.5.

These are formal sinusoidal particular-solution curves, not a stable settled filter response. For the formal sinusoidal response of an unstable linear system, replacing $i\omega$ by the Laplace variable $s$ gives $G(s)=5/(1-s\tau {)}^2$, with a repeated pole at $s=+1/\tau$. A homogeneous output contains $(A+Bt)e^{t/\tau }$ and grows, so real perturbations lead to supply-limited nonlinear behaviour. With equal positive resistors and capacitors in the frequency-setting network, the general denominator is $1+(3-K)s\tau +s^2{\tau }^2$: the Sallen-Key equal-component stability bound is $K<3$. For example $K=3-\sqrt{2}$ gives a Butterworth filter response. Thus the Sallen-Key low-pass filter architecture is useful for active filtering with suitable gain and damping, but the stated $K=5$ component set is unsuitable for linear filtering. The component values are retained in this diagnosis.^[ref]

Use an unloaded RC low-pass filter: a series resistor between the source and the output node, with a capacitor from that node to ground, and measure the output across the capacitor. The components are ideal in this first-order model; “ideal” here does not mean a brick-wall frequency response.

The time constant is $\tau =RC$, henceThe capacitive electrical impedance gives the transfer function $H(i\omega )=1/(1+i\omega RC)$, soOn the magnitude Bode plot, $20{\log }_{10}|H|=-10{\log }_{10}[1+(f/f_c{)}^2]$. The gain approaches $0\mathrm{dB}$ for $f\ll f_c$, is $-3.01\mathrm{dB}$ at $f_c$, and falls with slope $-20\mathrm{dB}$ per decade for $f\gg f_c$. These results assume negligible source resistance beyond the specified $R$ and a sufficiently large load impedance.

Let $v$ be the inverting-input potential. The non-inverting input is grounded, so the ideal operational amplifier rules give $v=0$ and no input current. Kirchhoff's current law at that node therefore givesIntegrating from an initial time $t_0$,This op amp integrator performs inverted time integration; the initial voltage across the capacitor supplies the integration constant. For zero initial conditions its transfer function is $H(s)=-1/(sRC)$. In sinusoidal steady state, with the convention $e^{i\omega t}$,For example, a cosine input produces a negative sine output after its constant component is set. The additional inversion makes the phase $+90^{\circ }$ modulo $360^{\circ }$, rather than the $-90^{\circ }$ of non-inverting integration.

Any direct-current component of the input produces an output ramp. The input offset voltage and input bias currents of a real operational amplifier therefore charge the feedback capacitor until the output saturates. Even the desired signal can exceed the available output swing when integrated for too long. Initial capacitor charge also introduces an unknown integration constant.

Place a large resistor $R_f$ in parallel with the feedback capacitor. The idealized transfer function then becomesIts finite direct-current gain $-R_f/R$ supplies a return path and bounds the response to a small offset; it approximates $-1/(sRC)$ when $\omega R_{f}C\gg 1$. The chosen gain must still keep offsets within the output range. A reset switch across the capacitor can set the initial charge, while low-offset amplifiers, bias compensation and input offset adjustment reduce drift. Useful integration also requires frequencies below the amplifier's own response limit.

A non-ideal amplifier is not specified by infinite input impedance alone: its open-loop gain and output impedance must also be modeled. Take its internal controlled output to be $u=-A(s)v$, where $v$ is the inverting-input potential, and place a resistance $r_o$ between $u$ and the actual output $y=V_{\mathrm{out}}$. The non-inverting input remains grounded; assume no external load. This gives an explicit finite-gain operational-amplifier integrator model.

Writing Kirchhoff's current law at the input and output nodes givesThe second equation gives $y=(sC{r}_o-A)v/(1+sC{r}_o)$. Substitution in the first gives the general transfer functionFor sinusoidal phasors, set $s=i\omega$. The numerator and denominator are complex, so the magnitude gives the amplitude ratio and their arguments give the relative phase.

The simplest model takes $r_o=0$ and a constant finite gain $A_0>0$. It givesThus below ${\omega }_l=[(1+A_0)RC{]}^{-1}$ the amplifier has gain magnitude $A_0$ and inversion. Above that frequency it approaches an integrator with magnitude $A_0/[(1+A_0)\omega RC]\simeq 1/(\omega RC)$ and phase $+90^{\circ }$. Constant finite gain by itself predicts no further high-frequency turnover.

For a more realistic sketch, use the standard dominant-pole open-loop gain $A(s)=A_0/(1+s/{\omega }_a)$, still with $r_o=0$. ThenThe exact frequency response isAssuming $A_0\gg 1$ and $A_0{\omega }_{a}RC\gg 1$, the turnover frequencies are well separated: ${\omega }_l\simeq 1/(A_{0}RC)$ and ${\omega }_h\simeq A_0{\omega }_a$, the approximate unity-gain angular frequency of the amplifier. The regimes arewith magnitude slopes $0,-20,-40\mathrm{dB}$ per decade and smooth phase transitions. This is the solid curve in the figure; $180^{\circ }$ and $-180^{\circ }$ denote the same phase.

If finite $r_o$ is retained and $A(i\omega )\to 0$, the general expression instead approaches $r_o/(R+r_o)$ at very high frequency: the capacitor couples the input to the output through a passive voltage divider. The dashed curve illustrates that eventual feedthrough. Other amplifier poles or an external load would alter the response again. No particular values of $A_0,{\omega }_a,r_o$ are given in the paper, so the numerical sketch is illustrative and the general expression states the model dependence explicitly.

Temperature as an equilibrium relation. Classical thermodynamics begins with the zeroth law of thermodynamics: thermal equilibrium is transitive. A thermometer can therefore assign a common empirical temperature to systems in mutual thermal equilibrium. Different thermometer properties initially give different monotone empirical scales; a resistance reading by itself does not define the absolute scale.

The Second law of thermodynamics provides a substance-independent thermodynamic scale. A reversible engine operating between reservoirs has a heat ratio depending only on their temperatures. One chooses absolute temperature so that $Q_h/Q_c=T_h/T_c$, with $Q_h,Q_c$ the positive heat magnitudes. Its reversible efficiency is $1-T_c/T_h$. Equivalently, reversible heat defines entropy through $dS=\delta Q_{\mathrm{rev}}/T$. For an equilibrium simple compressible system,Classical thermodynamics uses this relation without needing a microscopic account of entropy. A dilute-gas thermometer realizes an empirical scale approaching the thermodynamic scale after corrections for nonideal behavior.

Statistical meaning of the same derivative. In the microcanonical description, an equilibrium energy shell containing $\Omega (U,V,N)$ accessible states has $S=k_B\ln \Omega$. Its statistical temperature is consequentlyTo see why this is an equilibrium temperature, let two weakly coupled systems exchange a small energy while their total energy stays fixed. The combined multiplicity is ${\Omega }_A(U_A){\Omega }_B(U_{\mathrm{tot}}-U_A)$. Maximizing its logarithm giveswhich is equality of the two inverse temperatures. Stability requires the total entropy maximum, not only stationarity. The overwhelmingly dominant macrostate in a large system therefore realizes the thermodynamic equilibrium condition.

A large reservoir gives a small subsystem the canonical probabilities $p_i\propto e^{-\beta E_i}$, with $\beta =1/(k_{B}T)$. In that ensemble, $F=-k_{B}T\ln Z$ and $S=-k_B{\sum }_i{p}_i\ln p_i$ agree with macroscopic thermodynamic relations in their usual equilibrium regime. The equipartition identity for a classical dilute monatomic gas, $⟨p^2/(2m)⟩=3k_{B}T/2$, is a useful operational connection between microscopic motion and temperature, but is not a universal definition: quantum modes need not obey equipartition, and a system away from equilibrium need not have one temperature at all.

For finite systems one must specify the entropy convention and energy resolution; surface and fluctuation corrections can distinguish definitions that agree in the ordinary thermodynamic limit. Bounded-energy population-inverted systems can also have a negative entropy derivative and hence a negative statistical temperature. That possibility lies outside an ordinary positive-temperature ideal-gas thermometer; it does not mean temperature is merely an arbitrary average kinetic energy.

Obtaining the energy–temperature conversion. The Boltzmann constant connects entropy measured in $\mathrm{J}{\mathrm{K}}^{-1}$ with the dimensionless logarithm of a state count. Historically it could be determined from the ideal gas law $pV=N{k}_{B}T$: measure the molar gas constant $R$ and independently determine the Avogadro constant $N_A$, then obtain $k_B=R/N_A$. The independent particle-number determination matters; using $R$ alone would not fix energy per particle.

An electrical method uses Johnson-Nyquist noise. A resistor of resistance $R_e$ at known temperature has, in the classical range $hf\ll k_{B}T$, a one-sided open-circuit voltage-noise spectrum $S_V(f)=4k_{B}T{R}_e$. Over a calibrated flat bandwidth $\Delta f$,A practical measurement uses an equilibrium resistor, a calibrated low-noise amplifier and a known transfer function, subtracts or independently characterizes amplifier noise, and replaces a nominal bandwidth by the actual effective noise bandwidth. Varying temperature and fitting the mean-square voltage is more informative than treating one amplified voltage sample as the thermal variance. Other primary methods relate temperature to sound speed in a dilute gas or to calibrated particle diffusion and mobility.

There is a historically relevant unit-definition qualification: by the date of this exam, the revised SI had already taken effect on 20 May 2019. The kelvin is now defined by fixingThe same experiments now realize or compare thermodynamic temperatures using this fixed constant, whereas historically they measured the conversion constant on an independently defined temperature scale. The physical distinction between equilibrium temperature and microscopic state counting is unchanged.^[ref]

Use an RC low-pass filter: put the resistor in series with the input and the capacitor between the output node and the return conductor. Measure the output voltage across the capacitor, with negligible output loading. The voltage divider then has transfer functionAt the high frequency, the attenuation requirement isAt the low frequency, approximately unchanged amplitude means $2\pi f_{l}RC\ll 1$, soThe upper relation is an asymptotic design condition, because “approximately” specifies no numerical tolerance. For example, retaining at least $99\%$ of the low-frequency amplitude would sharpen the upper bound to approximately $68.7\mu \mathrm{F}$.

A practical illustrative choice is $C=22\mu \mathrm{F}$. Its voltage gain is approximately $0.99896$ at $1\mathrm{Hz}$ and $0.001096$ at $20\mathrm{kHz}$: the latter amplitude is reduced by a factor of about $912$. Its cutoff frequency is $1/(2\pi RC)\simeq 21.9\mathrm{Hz}$.

Measure the mass's displacement $x$ from its static equilibrium height. This removes the constant spring extension supporting its weight. If the spring's top moves with the box, its incremental extension is $y_2-x$, so its incremental upward force is $k(y_2-x)$. The specified damping force is against the mass's velocity in the static frame. Newton's second law therefore givesIn absolute height coordinates an additional constant on the right expresses the spring's rest length, its fixed attachment height and gravity; translating to static equilibrium removes it. Replacing the given damping by $-b(\dot{x}-{\dot{y}}_2)$ would describe a different base-excited oscillator.

After the decaying transient, write the steady-state response as $x=A(\omega )\cos (\omega t-\Phi )$. Using a complex sinusoid givessoFor positive frequency, the phase lag lies between zero and $\pi$, passing through $\pi /2$ at ${\omega }_0=\sqrt{k/m}$. For small $b$, the frequency response begins at $a_s$, develops a narrow resonance near ${\omega }_0$, and falls asThe precise amplitude-peak frequency is $\sqrt{k/m-b^2/(2m^2)}$ when that expression is positive.

The pen records the relative displacement $z=x-y_2$. Its complex transfer function isAt high frequency, $x$ is small while $z\simeq -y_2$. The inertial mass then stays nearly stationary and the recorded relative motion reproduces the ground displacement, with reversed sign. At low frequency the mass follows the box and their relative motion is small; near resonance the response is strongly frequency-dependent. Here “high” means compared with the suspension's natural frequency, rather than an absolute seismic-frequency threshold.

In the $e^{i\omega t}$ phasor convention, the total series RL circuit impedance is $R+i\omega L$. The resistor voltage follows the current, so its transfer function relative to the applied voltage isAt the stated condition $R=\omega L$,Thus the resistor-voltage amplitude is $\boxed{V_{R,0}=V_{\mathrm{in},0}/\sqrt{2}}$ and it lags by $45^{\circ }$. For a cosine input it is $V_R(t)=V_{\mathrm{in},0}\cos (\omega t-\pi /4)/\sqrt{2}$.

Write the dimensionless scale factor as $a=R/R_0$ and the density contrast as $\delta =(\rho -\bar{\rho })/\bar{\rho }$. Gravity amplifies overdensities while expansion and pressure oppose growth. In a sub-Hubble Newtonian single-fluid approximation, a comoving Fourier mode obeysFor pressureless matter in an Einstein-de Sitter background, the growing solution is $\delta \propto t^{2/3}\propto a$ and the decaying solution is $t^{-1}$. Jeans support suppresses growth when the pressure term dominates. The equation is not a substitute for relativistic perturbation theory on horizon-sized scales.

cold dark matter clusters without baryonic pressure support. Sub-horizon growth is much weaker during radiation domination, and baryons are initially tied to the photon fluid, producing acoustic oscillations rather than unrestricted collapse. Horizon entry, radiation-matter equality, acoustic evolution and diffusion therefore give a scale-dependent transfer function. The primordial curvature spectrum and the later matter-density spectrum are not the same function: evolution and the gravitational constraints relate them. Late dark-energy domination reduces the growth rate.

When $|\delta |$ approaches unity, linear theory fails. Collapse, virialization, mergers and tidal interactions form a hierarchy of haloes. Gas must cool and lose angular momentum to make dense stellar systems; feedback then changes its supply and distribution. Simulations combine collisionless gravitational dynamics with increasingly complex gas physics rather than extending the linear growing mode indefinitely. Galaxy clustering measures biased tracers of matter; gravitational lensing and CMB constraints provide complementary tests. Approximately adiabatic, nearly scale-invariant initial perturbations explain the coherent acoustic pattern and supply seeds for this later growth.^[ref]

A proper rational transfer function is a rational function whose numerator polynomial has polynomial degree no greater than its denominator polynomial, after common factors are canceled. Every finite-dimensional continuous-time linear time-invariant system of the form $\dot{x}=Ax+Bu$, $y=Cx+Du$ has such a transfer function $G(s)=C(sI-A{)}^{-1}B+D$. ^[ref]

A series resistor $R$ and capacitor $C$ feed a parallel pair with the same values. Measuring the parallel pair's voltage relative to the return gives this unloaded transfer function for phasors proportional to $e^{i\omega t}$. The voltage divider applied to $Z_s=R+1/(i\omega C)$ and $Z_p=R/(1+i\omega RC)$ proves the formula. The amplitude peaks at $\omega =1/(RC)$ with ratio $1/3$ and zero relative phase. At low and high positive frequencies, the transfer function tends respectively to $i\omega RC$ and $-i/(\omega RC)$. Reversing output polarity changes its sign, while loading changes the response itself.

A transfer function packages the response of a linear time-invariant system into a rule for turning a plant input into a plant output. Instead of solving a differential equation separately for every signal, take a Laplace transform, multiply by $G(s)$, then transform back. This also makes it possible to compare slow and fast oscillations through the continuous-time frequency response.

Here consider a continuous-time, causal, finite-dimensional linear time-invariant system with one plant input $u(t)$ and one plant output $y(t)$. With every initial state set to zero, its transfer function is defined byThe ratio $G(s)=Y(s)/U(s)$ is used where $U(s)\ne 0$ in a common Laplace convergence region; $G$ describes the system, independently of that particular plant input. Zero initial state matters: stored motion or another nonzero initial condition can contribute to the plant output even when the plant input vanishes. ^[ref]

The first-order low-pass filter below shows the same transfer function predicting both a step and a sustained sinusoid. Its pole gives a decay rate, its value at zero gives the eventual gain, and its values along the imaginary axis give the continuous-time frequency response. These descriptions make a useful bridge between a physical relaxation model and control theory calculations.