Table of Contents
- 1. BJT
- 2. Frequency response of CE Amplifier
- 3. Upper cutoff frequency
- 4. Common emitter amplifier
- 5. Long tailed pair
- 6. Feedback amplifier
- 7. Effects of Negative Feedback
1 BJT
1.1 bias circuit
as temperature changes V_BE varies
1.2 simple bias circuit
feedback circuit potential divider bias circuit
1.3 small signal analysis of CE amplifier
where gain is most important. mid band region. In this region C1, C2, CE would not have much impact and treated as shorted
2 Frequency response of CE Amplifier
Okay guys so if our last lecture we have seen that how we can fix the bias resistances, and also we find out the gain of our amplifier in the mid band range, this is very important for us, because mid band range is on the frequency range over which we want my gains to be constant, so that we can have a nice operation.
At high frequency the frequency response of an amplifier
Carefully choose capacitance, that our lower frequency of interest falls into our mid band range. At the high frequency range the cutoff of this high frequency range is actually defined by parasitic capacitors those capacitors as we already seen, Cπ Cμ etc, that needs different types of mathematical treatment. (next lecture) Today's lecture, we are mainly concerned with defining the boundaries of the mid-band-range, and how we can choose the values of coupling and de-coupling resistors for selecting the lower boundary, to do this we must go for frequency analysis, which is typically done in s-plane (the complex frequency plane), we must do the mathematical treatment in a different way.
Coupling capacitors are used to allow AC signals to pass from one stage to another while blocking DC voltage. Coupling capacitors are used between stages.
2.1 Lower frequency response of a CE amplifier
The same hybrid pi model, to simplify the mathematical equations. The coupling capacitors C1 and C2 will be considered separately from the emitter de-coupling capacitors CE.
Traditional hybrid pi model to a 2 port model of the amplifier.
By frequency response, we want to find out how the gain is varying over the different frequency. With the frequency response we can find out at what frequency gain is falling out, this would the starting point of the mid-band, lower frequency.
\begin{align*} \frac{V_{o}(f)}{V_{i}(f)} \end{align*}We can find out the input voltage and the output voltage and their division as the gain.
Considering the input circuit separately. In frequency analysis everything needs to be in s-plane.
Now we have our input source in terms of S, the impedance of a capacitor is \(\frac{1}{CS}\). The first circuit has a very simple voltage divider structure, and the input voltage is the voltage across \(R_{in}\).
\begin{align*} V_{in}(s) &= \frac{R_{in}V_s(s)}{R_{in}+R_{s}+\frac{1}{C_{1}s}} \\ &= \frac{C_{1}sR_{in}V_s(s)}{1+C_{1}s(R_s+R_{in})} \end{align*}The output side of the circuit. The output voltage is also called by convention the load voltage that is across the load resistance.
\begin{align*} V_{out}(s) &= \frac{R_{I}AV_s(s)}{R_{I}+R_{out}+\frac{1}{C_{2}s}} \\ &= \frac{C_{1}sR_{out}AV_{in}(s)}{1+C_{2}s(R_I+R_{out})} \end{align*}Combining these equations, the amplifier gain is given by:
\begin{align*} A(S) = \frac{V_L}{V_S} = \frac{s\tau_{in}}{1+s\tau_{in}}\frac{s\tau_{out}}{1+s\tau_{out}}A_m \end{align*}Where \(A_m\) is the mid-band gain given by
\begin{align*} A_m = \frac{R_{in}}{R_{in}+R_S}\frac{R_I}{R_I+R_{out}}A \end{align*}and \(\tau_{in}, \tau_{out}\) are given by
\begin{align*} \tau_{in} &= C_1(R_s+R_{in}) \\ \tau_{out }&= C_2(R_{out}+R_{in}) \\ \end{align*}We are simplifying the equation with the time constants.
Looking at the response, we don't have any zeroes. There are two different poles, which are -1/τin and -1/τout.
To find the gain function at a frequency f, replace s by \(2\pi fj\). The magnitude and phase form:
\begin{align*} |A(s)| &= \left|\frac{j2\pi f\tau_{in}}{1+j2\pi f\tau_{in}}\right|\left|\frac{j2\pi f\tau_{out}}{1+j2\pi f\tau_{out}}\right|A_m \\ \angle A(s) &= \angle\left(\frac{j2\pi f\tau_{in}}{1+j2\pi f\tau_{in}}\right)+\angle\left(\frac{j2\pi f\tau_{out}}{1+j2\pi f\tau_{out}}\right) \\ \end{align*}The Clearly at very low frequencies gain rises as f2, while at high frequency it is constant.
\begin{align*} |A(s=2\pi f)|_{f\to0 &}\to |j2\pi f\tau_{in}||j2\pi f\tau_{out}|A_m \to f^2 \\ |A(s=2\pi f)|_{f\to\infty &}\to A_m \to \text{constant} \\ \end{align*}While the frequency becoming high, the gain function becomes constant, where it is approaching the mid-band range.
2.2 The effect of the emitter capacitor CE
The analysis is rather more complicated than for the coupling capacitors.
The best way is to take the expression for the CE amplifier with an unbipassed emitter resistor and replace Re with Ze, i.e. RE in parallel with CE. For simplicity we ignore source impedance Rs (i.e. assume it is zero)
\begin{align*} A_V = \frac{V_C}{V_b}=\frac{-g_{m}R_c}{1+g_{m}Z_{e}(1+\frac{1}{\beta})} \\ \end{align*}where,
\begin{align*} Z_{e} = R_{E} \parallel C_{E} = \frac{R_{E}}{1+R_{E}C_{E}s} \end{align*}The equation can be further simplified into:
\begin{align*} A_V(s) &= \frac{-g_{m}R_c}{1+g_{m}(1+\frac{1}{\beta})\frac{R_{E}}{1+R_{E}C_{E}s}} \\ &= -(g_{m}R)\frac{1+R_{E}C_{E}s}{1+\frac{(1+\beta)R_E}{r_{\pi}}+R_{E}C_{E}S} \end{align*}It can be seen that there is a zero at z and a pole at p introduced by the emitter decoupling capacitor
\begin{align*} z &= -\frac{1}{R_{E}C_{E}} \\ p &= -\frac{1}{R_{E}C_{E}}\left(1+\frac{(1+\beta)R_E}{r_{\pi}}\right) \end{align*}Bode plot showing effect of emitter capacitor break points are at
\begin{align*} \omega_1 = \frac{1}{R_{E}C_{E}} \end{align*}2.3 Choosing Coupling and Emitter Decoupling Capacitors
The coupling and decoupling capacitors must be chosen so that the pole frequencies they introduce are below the lowest frequency of interest.
If ω0 is the lowest frequency of interest this implies
\begin{align*} \omega_0 &> \frac{1}{C_1(r_{\pi}+R_S)} \quad\rightarrow \; C_{1} > \frac{1}{\omega_0(r_{\pi}+R_s)}\\ \omega_0 &> \frac{1}{C_2(R_{C}+R_I)} \quad\rightarrow \; C_{2} > \frac{1}{\omega_0(R_C+R_I)} \\ \omega_0 &> \frac{g_m}{C_E}\qquad\qquad\rightarrow \;\frac{1}{\omega_{0}C_E} \ll \frac{1}{g_m} \end{align*}The 2 coupling capacitors and the emitter decoupling capacitor implies 3 breakpoints. Which means the lower frequency of interest (ω0) needs to be bigger than all 3 of these breakpoints, otherwise, system would be unstable on those break-point points.
2.4 Numerical example
Design an amplifier, by choosing suitable capacitances.
\begin{align*} C_1 &\rightarrow \frac{1}{C_1(R_s+R_{in})} \quad C_2 \rightarrow \frac{1}{C_2(R_c+R_L)} \quad C_E \rightarrow \frac{g_m}{C_E} \\ R_L &= 33k\Omega, \quad R_s = 1k\Omega, \quad f_0 = 10kHz \\ C_1 &> \frac{1}{\omega_0(R_s+R_{in})} \rightarrow C_1 \geq 12nF \\ C_2 &> \frac{1}{\omega_0(R_c+R_L)} \rightarrow C_2 \geq 0.5nF \\ C_E &> \frac{g_m}{\omega_0} \end{align*}3 Upper cutoff frequency
3.1 Model
Need to find out the upper cutoff frequency of the mid-band-range. Analysis done with the high frequency hybrid pi model.
It has been assumed that ro is large and maybe neglected and that the bias resistors have a value ≫ rπ. It's easy to include these if required. Need to find out how the response (gain, Vo/Vs) varies with the frequency. At what frequency we have less gain, to derive the cutoff frequency.
Find out the current equation.
\begin{align*} \frac{V_s}{R_s} &= V_{\pi}sC+\frac{V_{\pi}}{r_{\pi}} + (V_{\pi}-V_L)sC_{\mu} \\ (V_{\pi}-V_L)sC_{\mu} &= g_{m}V_{\pi}+ \frac{V_{L}}{R_L} \end{align*}Solving these equations gives:
\begin{equation*} A_{V} = \frac{V_c}{V_s} = -\frac{g_{m}R_{L}r_{\pi}}{R_S+r_{\pi}}\times\frac{1-sC_{\mu}/g_m}{1+sR_{\pi}(C_{\pi}+(1+g_{m}R_L)C_{\mu})+sR_{L}C_{\mu}+s^{2}R_{\mu}R_{L}C_{\mu}C_{\pi}} \end{equation*}where
\begin{equation*} R_{\pi} = \frac{r_{\pi}R_S}{r_{\pi}+R_S} \end{equation*}We also use an approximation, which is called the miller approximation.
The biggest problem of this model, is the capacitor sitting between the input and output Cμ, that is why the system analysis is very difficult, what miller approximation says, we can so how remove this capacitance in terms of models, with equivalent capacitance which replicate the overall effects. Then the input and the output side are completely decoupled, simplifying the mathematics in the process.
3.2 The Miller effect
Consider an amplifier with a voltage gain of -A 1 and capacitance Cf between its input and output.
The amplifier assumed to be ideal (infinite input impedance, zero output impedance)
\begin{align*} V_{o} &= -AV_{s} \\ V_{C_f} &= V_{s} - (-AV_s) = V_s(1+A) \end{align*}Change the input voltage by \(\Delta v\)
\begin{align*} V_s+\Delta v &\rightarrow -A(V_s+\Delta v) \\ V_{C_f}' &\rightarrow V_s+\Delta v - (-A(V_s+\Delta v)) = V_s(1+A)+\Delta v(1+A) \\ \end{align*}A change \(\Delta v\) at the input changes the voltage across Cf by \(\Delta v(1+A)\).
\begin{align*} V_{c_f}' - V_{c_f} = \Delta v(1+A) \end{align*}Thus the charge supplied at the input is given by
\begin{align*} C &= \frac{Q}{v} \\ \Delta q &= C_{f}(1+A)\Delta v \end{align*}The effect is equivalent to removing Cf and replacing it by a capacitance of value Cin between the input and ground, where
\begin{align*} C_{in} = \frac{\Delta q}{\Delta v} = (1+A)C_{f} \end{align*}3.3 Modified circuit for CE amplifier
From the miller effect:
\begin{align*} C_{in} = \frac{\Delta q}{\Delta v} = (1+A)C_{f} \end{align*}The miller capacitance:
\begin{align*} C_m = (1+g_{m} R_L) C_{\mu} \end{align*}The analysis is much more simple. The effective input impedance is 2 capacitances and 1 resistance.
\begin{align*} Z_{\pi} = \frac{r_{\pi}}{1+r_{\pi}(C_{\pi}+(1+g_{m}R_L)C_{\mu}s)} \end{align*}Vπ fit into this circuit as part of a potential divider configuration:
\begin{align*} V_{\pi} = \frac{Z_{\pi}+V_s}{R_s+Z_{\pi}} \end{align*}Finally, the frequency dependent voltage gain:
\begin{align*} A_V(s) &= \frac{V_o}{V_s} = \frac{-g_{m}^{}R_{L}V_{\pi}}{V_{s}} \\ &= -\frac{g_{m}R_{L}r_{\pi}}{R_{s}+r_{\pi}}\frac{1}{1+R_{\pi}(C_{\pi}+(1+g_{m}R_{L})C_{\mu})s} \end{align*}This is similar to the full model, but does not include the second pole or the zero.
3.4 Upper 3db Bandwidth
Using the miller approximation the frequency response of a CE amplifier has a single higher frequency pole.
\begin{align*} A_{v} (s) = \frac{A_{0}}{1+s\tau} \end{align*}where
\begin{align*} A_{0} = -\frac{g_{m}R_{L}r_{\pi}}{R_{s}+r_{\pi}} \\ \tau = (C_{\pi}+(1+g_{m}R_{L})C_{\mu})R_{\pi} \end{align*}Replacing s by jω gives the frequency response
\begin{align*} A_{v} = \frac{A_{0}}{1+j\omega\tau} \quad\text{hence,} |A_{v}| = \frac{A_{0}}{\sqrt{1+(\omega\tau)^{2}}} \end{align*}therefore, the upper cutoff frequency is:
\begin{equation*} f_{3db} = \frac{1}{2\pi\tau} \end{equation*}3.5 Full small-signal equivalent circuit and the Miller approx
Comparison of the full small-signal equivalent circuit and the Miller approx.
Gain bandwidth product = A Δ f Gain can be sacrificed to make the amplifier work on a bigger range of frequencies.
3.6 Example
frequency range 20Hz - 20kHz Choose the potential divider resistances R1,R2,RC,RE Select the capacitances according to 20Hz, and the upper cutoff frequency satisfy 20kHz. C1,C2,CE Some parameter needs to be assumed
4 Common emitter amplifier
Two other type of amplifier
4.1 Common base amplifier
It looks very much similar to the common emitter amplifier, interms of R1,R2,Rc. The only difference is in the common emitter amplifier, the emitter is grounded, the input is even at the base terminal, the output is taken out of the collector terminal. In the common base amplifier, the output is taken out of the collector X, but the input is actually taken out of the emitter terminal.
The bias requirement is pretty much the same. C1,C2,CB.
Replace the transistor a hybrid pi model, and ground all the resistors around it. Essentially a low frequency hybrid pi model.
Vπ = voltage between emitter and base = VB - Ve = -Ve. Because \(V_{\pi} = -V_{e }\;\rightarrow\; g_{m}V_{\pi} = -g_{m}V_{e}\).
For the other resistance, since we are doing ac analysis, all the internal power source will be grounded. This is a standard technique.
r0 is ignored. Splitting the current source,
Look another interesting concept, what is a current source?  Therefore, a current source can be represented alternatively as a voltage and a resistance component.
Figure 1: Splitting current source
Figure 2: Simplifying the current sources
Thus the CB amplifier generally has a very low input impedance (of order 25Ω @ collector current of 1mA) Usually RE can be neglected since it is much greater than 1/gm.
The analysis for the gain:
\begin{align*} V_{c} &= g_{m}R_{c}V_{e} \\ &\approx \frac{1/g_{m}}{R_s+1/g_m}g_{m}R_{c}V_{s} = \frac{g_mR_c_{}V_{s}}{1+R_{s}g_{m}} \end{align*}The voltage gain of the CB stage is therefore:
\begin{align*} A = \frac{V_{c}}{V_{s}} = \frac{g_{m}R_{c}}{1+R_{s}g_{m}} \end{align*}If Rs is very small the stage voltage gain has the same magnitude as the CE stage.
\begin{align*} A = \frac{g_{m}R_{c}}{1+R_{s}g_{m}} \approx g_mR_{c} \end{align*}Which is the same as the CE gain. The reason why CB amplifier is not widely used, is because the low input impedance it has.
We want our amplifier to have high input impedance so the amplifier can amplifier the full range of the voltage (since less voltage will be dropped in the source impedance). We also want low output impedance, more on this later.
However, the current gain of the stage is of order unity (αf).
The circuit is not widely used but may be useful for high frequency amplifiers since there is no capacitive feedback from output to input.
4.1.1 High frequency model
No capacitor between the input and the output. Which is easier to handle, since no Miller approximation is required.
A potential divider structure. To find out Ve. Vo is Vc.
\begin{align*} V_o = \frac{R_cg_m}{1+R_sg_m}\frac{1}{1+\tau_iS}\frac{1}{1+\tau_oS}V_{s} \end{align*}where
\begin{align*} \tau_i = \frac{R_{sC_{\pi}}}{1+R_sg_m}, \quad \tau_o = R_cC_{\mu} \end{align*}4.2 Common collector amplifier (Emitter Follower)
In the collector side, there is no output resistance. Input comes at the base, the output is at the emitter terminal. This is the difference.
voltage gain, using \(g_{m} = \beta/r_{\pi}\):
\begin{align*} A_v = \frac{V_e}{V_s} = \frac{R_{e(1+\beta)}}{r_{\pi}+R_e(1+\beta)} \end{align*}The term in the denominator is smaller than the term above, this implies that this amplifier provides no voltage gain. However, if we choose RE to be very high, then rπ will become negligible, and the gain becomes 1. Hence, the Emitter Follower, it sends whatever voltage input to the output given that RE is high.
\begin{align*} i_b = \frac{1}{r_{\pi}}(v_s_{-V_e}) = \frac{V_s}{r_{\pi}}(1 - \frac{R_e(1+\beta)}{r_{\pi}+R_E(1+\beta)}) = \frac{V_s}{r_{\pi}R_{E}(1+\beta)}_{} \end{align*}hence,
\begin{align*} R_{in} = \frac{V_s}{i_b} = r_{\pi} + R_E(1+\beta) \end{align*}We can adjust Rin, we can make it very high.
4.2.1 The effect of bias resistors and the source resistors
Define \(R_s'\):
\begin{align*} R_s' = R_{s} \parallel R_{1} \parallel R_{2} \end{align*}Output impedance
\begin{align*} i_o = V_o\left(\frac{g_mr_{\pi}}{R_s+r_{\pi}} + \frac{1}{R_s+r_{\pi}}\right) = \frac{(1+\beta)V_{o}}{R_s+r_{\pi}} \end{align*}the output resistance is given by:
\begin{align*} R_o = \frac{v_o}{i_o} = \frac{R_s' +r_{\pi}}{1++\beta} \end{align*}Now,
\begin{align*} R_{in} = \frac{V_s}{i_b} = r_{\pi} + R_E(1+\beta) \end{align*}By adjusting RE we can get very high input impedance. And we use a small output resistance by making Rs small, if Rs is small, then Rs' is small. and it will also make Ro small.
Although, this amplifier has a gain of unity, by controlling Rs' and other resistors we can set up the input and the output impedance. Basically, this amplifier can perform an impedance matching.
5 Long tailed pair
Both of the emitter side of the transistor are collected into a current source. Define some parameters. Vdin, differential voltage. Vcin, common mode voltage.
\begin{align*} \text{Differential voltage} \quad V_{din} &= V_{B1} - V_{B2} \\ \text{Common mode voltage} \quad V_{cin }&= (V_{B1}+ V_{B2})/2 \end{align*} \begin{align*} I_{C}= \end{align*}5.1 Large signal analysis
where \(V_{T} = kT/q\).
\begin{align*} \frac{I_{c1}}{I_{C2}} &= e^{\frac{V_{B2}-V_{B1}}{V_{T}}} = e^{\frac{V_{din}}{V_{T}}}\\ I_{E }&= -(I_{E1}+I_{E2}) = I_{C1}(1+\frac{1}{\beta_{1}}) + I_{C2}(1+\frac{1}{\beta_{2}}) \\ \therefore \frac{I_{E}}{I_{C1}} &= (1+\frac{1}{\beta_{1}})(1+\frac{I_{C2}}{I_{C1}}) \end{align*}The tail current (IE) is constant. If these two transistors are identical, then β1 = β2 = β. Solving these equations, gives, after some manipulation:
\begin{align*} I_{C1} = \frac{\beta I_{E}}{(1+\beta)(1+exp(-V_{din}/V_{T}))}\\ I_{C2} = \frac{\beta I_{E}}{(1+\beta)(1+exp(+V_{din}/V_{T}))} \end{align*}Either one of them is increasing, or decreasing. The potential between Q1 and Q2 is constant. The maximum value of the ICs are \(\frac{\beta I_{E}}{1+\beta}\).
5.2 DC Analysis
For pure common mode input Vb1 = Vb2. Common mode gain different to differential gain.
5.3 The merits of the Long Tailed Pair
- it provides a differential input (and output if required) with a large differential gain and small common mode gain.
- If the transistors are well matched changes in VBE due to temperature appear as a common mode voltage. Hence the amplification of thermally induced voltages is small. This makes the LTP suitable as a DC amplifier.
- It provides good gain and good thermal stability with no de-coupling capacitor, very useful for IC design.
- Provided the common mode voltage is limited, the circuit may be designed so that neither transistor saturates on overload gives fast recovery.
6 Feedback amplifier
6.1 What is feedback
The concept of NFB is to take some of the output signal from an amplifier and subtract/add it from the input signal.
6.2 Why should we do this?
- To stabilise the gain it reduces the sensitivity to changes in bias current, temperature, etc
- To reduce distortion (reduction in gain)
- To increase bandwidth
- To control input and output impedances
The price
- Reduced gain gain is cheap
- Instability the amplifier may oscillate unless care is taken Stability is important. The issue is essentially the same as stability of control systems and the mathematical techiniques will be considered in more detail
6.3 The ideal feedback equation
A fraction k of the output signal is fed back and subtracted from the input.
\begin{align*} S_{fb} &= kS_{0} \\ S_{\epsilon }&= S_{i} - S_{fb} = S_{i} - kS_{0} \\ S_{O }&= AS_{\epsilon} = A(S_{i}-kS_{o}) \end{align*}Hence, the closed loop gain, which is reduced
\begin{align*} A_{cl} = \frac{S_{o}}{S_{i}} = \frac{A}{1+Ak} \end{align*}6.3.1 terminology
A is called the OPEN LOOP GAIN k is called the FEEDBACK FRACTION T = Ak is called the LOOP GAIN (or, return ratio) Acl is called the CLOSED LOOP GAIN
6.4 Gain sensitivity
Consider the fractional change in closed loop gain for changes in A and k
\begin{align*} \frac{\Delta A_{cl}}{A_{cl}} = (\frac{1+Ak}{A})\frac{\partial A_{cl}}{\partial A}\Delta A + (\frac{1+Ak}{A})\frac{\partial A_{cl}}{\partial k}\Delta k \end{align*}where
\begin{align*} \frac{\partial A_{cl}}{\partial A} = \frac{1}{(1+Ak)^{2}} \frac{\partial A_{cl}}{\partial k} = \frac{-A^{2}}{(1+Ak)^{2}} \end{align*}hence,
\begin{align*} \frac{\Delta A_{cl}}{A_{cl}} = \frac{1}{1+Ak}\frac{\Delta A}{A} - \frac{Ak}{1+Ak}\frac{\Delta k}{k} \end{align*}6.5 Reduction of distortion
Distortion is taken to mean the variation of gain with signal amplitude. Higher linearity means the circuit (amplifier) can perform under a higher range of voltage for inputs and outputs.
6.6 Feedback configuration
| input | output |
|---|---|
| V | I |
| V | V |
| I | V |
| I | I |
6.6.1 Series voltage or Series shunt
Feedback network senses output voltage and subtracts voltage from the input. In this case we have input as voltage and output as voltage. Applying input voltage is applying in series to the actual amplifier input voltage to the amplifier. What quantity we are getting out of the output
6.6.2 Series voltage configuration
Feedback voltage Vfb, output voltage Vo, input voltage Vi.
\begin{align*} A_{cl} &= \frac{V_{o}}{V_{i}} \\ A &= \frac{V_{o}}{V_{e}} \\ V_{o} &= AV \\ V_{i} &= V_{e} + kV_{o} \\ &= V_{e} + AkV_{e} \\ &= V_{e} (1+Ak) \\ \therefore A_{cl} &= \frac{AV_{e}}{V_{e}(1+Ak)} = \frac{A}{1+Ak} \end{align*}The input resistance, Ri
\begin{align*} R_{i} = \frac{V_{i}}{i_{i}} = \frac{V_{e}(1+Ak)}{i_{i}} = r_{i}(1+Ak) \end{align*}The output resistance Ro Ro = \frac{Vo}{io} io = \frac{Vo-AVe}{Vo} = \frac{Vo+AkVo}{ro} Ro = \frac{ro}{1+Ak}
6.6.3 Shunt voltage or Shunt shunt
In this configuration our input impedance of the ideal amplifier is actually reduced by a factor of 1+Ak
\begin{align*} R_{out} &= \frac{V_{o}}{i_{o}} \\ i_{o }&= \frac{V_{o}-Ai_{e}}{r_{o}} = \frac{V_{o} + AkV_{o}}{r_{o}} = \frac{V_{o}(1+Ak)}{r_{o}} \\ i_{e} &= -i_{fb} = -kV_{o}, \quad R_{o} = \frac{V_{o}}{i_{o}} = \frac{r_{o}}{1+Ak}\Omega \end{align*}"k" is actually the conductance.
6.6.4 Series current
Input is voltage and output is current.
\begin{align*} A_{cl} &= \frac{i_{o}}{V_{i}} \\ i_{o} &= AV_{e} \\ V_{i} &= V_{e} + ki_{o} = V_{e}(1+Ak) \\ A_{cl} &= \frac{AV_{e}}{V_{e}(1+Ak)} \end{align*}Input resistance
\begin{align*} R_{i} &= \frac{V_{i}}{i_{i}} \\ V_{i} &= i_{i}r_{i} + ki_{o} = i_{i}r_{i} + kAV_{e} = V_{e} + AK V_{e} = V_{e}(1+Ak) \\ R_{i} &= \frac{V_{i}}{i_{i}} = \frac{V_{e}(1+Ak)}{i_{i}} = \frac{r_{i}}{1+Ak} \end{align*}Output resistance
\begin{align*} R_{o} &= \frac{V_{o}}{i_{o}} \\ i_{o} &= AV_{e}+\frac{V_{o}}{r_{o}} \\ &= -Aki_{o} + \frac{V_{o}}{r_{o}} \\ i_{o} (1+Ak) &= \frac{V_{o}}{r_{o}} \\ V_{o} &= -ki_{o} \end{align*}6.6.5 Shunt current
Current amplifier
\begin{align*} A_{cl} &= \frac{i_{o}}{i_{i}} \\ i_{o} &= Ai_{e} \\ i_{i} &= i_{e} + ki_{o} = i_{e}(1+Ak) \\ A_{cl } &= \frac{Ai_{e}}{i_{e}(1+Ak)} = \frac{A}{1+Ak} \end{align*}Input resistance Rin = \frac{Vi}{ii} = \frac{riie}{ie(1+Ak)} = \frac{ri}{1+Ak} Output resistance
\begin{align*} R_{out} &= \frac{V_{o}}{i_{o}} \\ i_{e} &= -ki_{o} \\ i_{o} &= Ai_{e} + \frac{V_{o}}{r_{o}} \\ &= -Aki_{o} + \frac{V_{o}}{r_{o}}\\ i_{o}(1+Ak) &= \frac{V_{o}}{r_{o}}\\ R_{out} &= \frac{V_{o}}{i_{o}} = r_{o}(1+Ak) \end{align*}6.6.6 Summary
| Configuration | Amplifier | Rin | Rout |
|---|---|---|---|
| Series voltage | voltage | high | low |
| shunt voltage | transimpedance | low | low |
| shunt current | current | low | high |
| series current | transconductrance | high | high |
NOTES: in this analysis: Feedback is ideal, no loading No feed-forward, feedback unilateral Real amplifiers may only approximate these features.
7 Effects of Negative Feedback
7.1 Effect of the negative feedback on bandwidth
Consider an amplifier with a single dominant pole in its open loop response (e.g. an op-amp).
\begin{align*} A(s) = \frac{A_{o}}{1+s/p} \end{align*}The open loop 3db bandwidth is given by \(\omega_{3db} = p\) Suppose that the amplifier is used in a feedback loop with feedback fraction k (assumed real), then the closed loop gain is given by
\begin{align*} A_{cl}(s) = \frac{A(s)}{1+kA(s)} = \frac{A_{0}}{1+kA_{0}}\frac{1}{1+\frac{s}{p(1+kA_{0})}} \end{align*}The gain is reduced by a factor of \(\frac{A_{0}}{1+kA_{0}}\). The bandwidth is extended by the same factor, \(\therefore \omega_{3db} = p(1+kA_{0})\). This is called the gain bandwidth product.
\begin{align*} A\Delta f = \text{constant} \end{align*}For a fully compensated opamp, the gain bandwidth product is the same as the unity gain frequency, fT. Gain A = 1, the bandwidth fT. \(\therefore \Delta f = f_{T}\).
7.2 The stability of Feedback amplifiers
Closed loop gain ACL = \frac{A(s)}{1+k(s)A(s)} = \frac{A(s)}{1+T(s)} The gain will become unstable if \(1+kA = 0\). To avoid unstability, this equation must be unsatisfied, the two conditions:
\begin{align*} | A(s)k(s) | = 1 \\ \angle(Ak) = 180^{\circ} \end{align*}Bode plot, 2 criteria
- The phase margin: The amount by which the phase shift is less than 180ˆ at the unity gain frequency.
- The gain margin: The amount by which the gain is less than 0db at the frequency where the phase shift is 180ˆ
7.3 Op-amp
Most operational amplifier are based on 3 stages
- A differential input stage (LTP)
- A common emitter (or common source) voltage gain stage
- A class B output stage
741, a typical of a bipolar op-amp IC.
7.4 Simplified Estimation of Gain
Assume current gain β of 100
First stage This stage acts as a trans-conductance amplifier providing a current drive ii to Q7. i1 = iC3 - iC2 = gmVdiff typical gm ≈ 40IC
7.5 Compensation
With several stages the basic amplifier will have several poles in its transfer function.
The phase shift will exceed 180ˆ before the amplifier gain has fallen to unity (0dB).
The consequence of this is that with negative feedback the amplifier may become unstable especially with a large feedback fraction (low closed loop gain).
Many applications require a large amount of feedback (unity gain buffer, integrator).
To ensure stability the op-amp is compensated to give a single dominant pole.
They incorporate a capacitor between the second and the first stage. This is referred as the
7.6 Dominant pole compensation.
Suppose the natural poles are all around or over 1MHz and open loop gain is 100dB.
The gain falls at 20dB/decade due to a single pole, therefore the introduced dominant pole must be at about 10Hz to ensure the gain has fallen to 0dB at the frequency of the natural poles.
The dominant pole is obtained using the "Miller Capacitance" added to the second stage. This allows the use of a moderate value capacitance (30pF) can be realised on-chip.
| f | gain |
| 1 | 100 |
| 10 | 80 |
| 100 | 60 |
| 1k | 40 |
| 10k | 20 |
| 100k | 1 |
7.7 slew rate
The small signal bandwidth is not the only factor governing the high frequency response. For large signals there is a maximum rate of change of output voltage - the slew rate.
The origin of the slew rate limitation is that the current available to change the compensation capacitor, C, is limited by the quiescent current of the input stage. The voltage across C is the same as the output voltage, so we can write
\begin{align*} \frac{dv_{out}}{dt} = \frac{i_{1}}{C} \end{align*}The maximum value of i1 is the tail current of the LTP stage. For a 741 the tail current is about 20μ A and C is 30pF, giving a slew rate of 20μ A/30pF, or, 0.67V/μ s.
7.8 Limitation of Op-amp
The consideration of the differential pair of op-amp is the same, in analysis, but the transistors cannot be the same practically.
- Input bias current - need to bias input transistors.
- Less than 200nA for most bipolar op-amps
- Avoided by using JFET (or MOSFET) input transistors.
- Minimised by the use of bias current compensation
- input offset voltage - imbalance leading to non-zero output voltage when vdiff = 0
- Usually < 2mV for general purpose bipolar
- JFET op-amps usually inferior to bipolar
- input offset current - difference in bias currents for the two inputs
- For bias current compensate op-amps the input offset current may exceed the bias current.
Footnotes:
because we are trying to match it with the typical common amplifier, hence the negative sign


Formula 1 drivers pay tribute to Lewis Hamilton's sixth Drivers' Championship
by Bobby VincentLewis Hamilton won his sixth Formula 1 Drivers' title on Sunday when he finished in P2 in the United States. Hamilton has become just the second ever F1 driver to reach this tally, after Michael Schumacher, who is leading the way with seven world titles to his name. Here's what the rest of the F1 grid had to say about Hamilton making even more history, when talking to Formula 1's official YouTube channel.
Charles Leclerc (Ferrari)
"Of course, it's an achievement. One world championship is an achievement, so six is even more (of an achievement)," Leclerc said.
"I think everyone of us drivers in Formula 1 respect Lewis and I definitely do. He's a great champion, and this year, I think the best has won. He's proved on many occasions that he's done an excellent job, so congratulations to him."
Antonio Giovinazzi (Alfa Romeo)
"(It's) unbelievable. Six world championships is a lot. So, congratulations to him!"
Max Verstappen (Red Bull)
"It's very impressive to win six titles. You need to be very consistent for it. He clearly deserves it."
Nico Hulkenberg (Renault)
"Congratulations to him and the team. Once again, they've killed it!"
Pierre Gasly (Toro Rosso)
"For me, he's a legend in Formula 1. (He's) one of the best, if not the best, in the sport. He's proved it again and I think it's not the last time that he's going to win. It's really impressive to see what he's achieved in Formula 1."
Daniel Ricciardo (Renault)
"As a fellow racer, all you can do is really respect what he's done. There will always be people saying, 'great car, great team', all of that but yes maybe. But also, to hold that level week in week out; there's so much more involved in this sport than just driving. There's the circus and for him to hold that level for all these years at the top of the circus; you can only respect that."
Valtteri Bottas (Mercedes)
"He did less mistakes in the season and performed at a better level. There are things I could've done better in some of the races. Mixed feelings but huge congratulations to him. He deserves it."
George Russell (Williams)
"Amazing driver. He puts a huge amount of effort in, obviously. He's had a fantastic car beneath him to achieve this in the last number of years, but he's still got to do the job week in week out, and he's definitely delivered it."
Kimi Raikkonen (Alfa Romeo)
"It's a great achievement. Congratulations."
Romain Grosjean (Haas)
"Really good job. Mercedes are winning races even when they're not supposed to win and I think that's the strengths of that team. They've been amazing in the championship and Lewis has done a good job."
Lando Norris (McLaren)
"I'm happy for him. I don't know what it feels like to win a world championship, so I can't really say it, but I congratulated him after the race. He's the second driver ever to get to six world championships and I'm sure he's going to go on to win more. I'm very happy, he's done an awesome job. He's beat his team-mate in every single season in Formula 1, bar one, or two. So, he's pretty handy."


© 2019
| podocarpus podocarpa podocarpum | footed seed | |||
| pous | pod | noun/m | πουϛ, ποδοϛ foot | |
| o | o | cnct | connective vowel in botanical Latin, usually for Greek words but in some cases, such as color tingeing, for Latin words | |
| carpos | carp | noun/m | καρποϛ fruit | |
| us | adj | Latinizing suffix | ||
| Oxytropis podocarpa Gray Desmodium podocarpum DC. Astragalus podocarpus C. A. Mey. | ||||
| podolepis podolepides | footed scales | |||
| pous | pod | noun/m | πουϛ, ποδοϛ foot | |
| o | o | cnct | connective vowel in botanical Latin, usually for Greek words but in some cases, such as color tingeing, for Latin words | |
| lepis | lepid | noun/f | λεπιϛ scale, shell | |
| Catapyrenium podolepis Breuss | ||||
| podophyllus podophylla podophyllum | sturdily stalked leaf | |||
| pous | pod | noun/m | πουϛ, ποδοϛ foot | |
| o | o | cnct | connective vowel in botanical Latin, usually for Greek words but in some cases, such as color tingeing, for Latin words | |
| phyllon | phyll | noun/n | φυλλον leaf; foliage | |
| us | adj | Latinizing suffix | ||
| Rodgersia podophylla A. Gray Syngonium podophyllum Schott | ||||
| poeticus poetica poeticum | of poets, especially Roman and Greek poets | |||
| poeticus | poetic | adj | ποιητικοϛ poetic | |
| Dodecatheon poeticum Henderson Narcissus poeticus L. |


Problem Name: lastIndexOf
Use the lastIndexOf function to find a ‘Cricket’ member. Start from the 3rd index and print its index.


{% load static %}
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Dr. Virendra Swarup Educational Foundation, Uttar Pradesh. VSEF was established in the year
1989 in the memory of Late Dr. Virendra Swarup, MLC & Chairman of Uttar Pradesh Legislative
Council and an eminent educationalist.
The motto of the school is 'He that seeketh findeth'. This is our guideline. We are seeking the best
in each child encouraging and developing this to make a better individual who in turn will contribute
towards a better society.


Jack Black
This is a tribute
Actor, comedian, and rock star, Jack Black
Some notable moments in Jack Black's illustrious life and career
- 1969 – Born in Santa Monica, California
- 1982 – Appeared in commercial for the video game Pitfall!
- 1994 – Formed comedic rock group Tenacious D with friend Kyle Gass
- 2000 – Break out performance as Barry in High Fidelity
- 2001 – Played Hal Larsen in Shallow Hal
- 2001 – Released the self-titled album Tenacious D
- 2003 – Played Dewey Finn in School of Rock for which he was nominated for a Golden Globe award
- 2005 – Played Carl Denham in King Kong
- 2006 – Played Miles in The Holiday
- 2006 – Played Nacho in Nacho Libre
- 2006 – Released the album The Pick of Destiny
- 2006 – Played Jack "Jables" Black in Tenacious D in The Pick of Destiny
- 2008 – Played the voice of Po in Kung Fu Panda
- 2008 – Played Jeff "Fats" Portnoy in Tropic Thunder
- 2011 – Played the voice of Po in Kung Fu Panda 2
- 2011 – Played Bernie Tiede in Bernie for which he recevied his second Golden Globe nomination
- 2012 – Released the album Rize of the Fenix
- 2015 – Played/voiced multiple parts in Goosebumps
- 2016 – Played the voice of Po in Kung Fu Panda 3
- Currently – Still Rockin'
"Jack Black is the only person I know who can steal a scene from a 25-foot ape."
– Peter Jackson
Learn more about Jack Black by visiting his Wikipedia Page or check out some of the other films he has appeared in on his IMDB Page
Check out the music video for Tribute by Tenacious D below!
Listen to more Tenacious D on Spotify!


How to use the require hook.
One of the ways you can use Babel is through the require hook. The require hook
will bind itself to node's require
and automatically compile files on the
fly. This is equivalent to CoffeeScript's
coffee-script/register.
Not suitable for libraries
The require hook automatically hooks itself into all node requires. This will pollute the global scope and introduce conflicts. If you're writing an application, it's completely fine to use. If, however, you're writing a library then you should compile your library and depend on the babel-runtime.
utility.inlineEnvironmentVariables
usePlease note the when used in conjunction.
$ npm install babel
require("babel/register");
All subsequent files required by node with the extensions .es6
, .es
, .jsx
and .js
will be transformed by Babel. The polyfill is also automatically required.
NOTE: By default all requires to node_modules
will be ignored. You can
override this by passing an ignore regex via:
require("babel/register")({
// This will override `node_modules` ignoring - you can alternatively pass
// an array of strings to be explicitly matched or a regex / glob
ignore: false
});
require("babel/register")({
// Optional ignore regex - if any filenames **do** match this regex then they
// aren't compiled
ignore: /regex/,
// Optional only regex - if any filenames **don't** match this regex then they
// aren't compiled
only: /my_es6_folder/,
// See options above for usage
whitelist: [],
blacklist: [],
// Setting this will remove the currently hooked extensions of .es6, `.es`, `.jsx`
// and .js so you'll have to add them back if you want them to be used again.
extensions: [".es6", ".es", ".jsx", ".js"]
});
By default babel-node
and babel/register
will save to a json cache in your
temporary directory.
This will heavily improve with the startup and compilation of your files. There are however scenarios where you want to change this behaviour and there are environment variables exposed to allow you to do this.
Specify a different cache location.
$ BABEL_CACHE_PATH=/foo/my-cache.json babel-node script.js
Disable the cache.
$ BABEL_DISABLE_CACHE=1 babel-node script.js


Open Boundaries Implementation in MITgcmUV
These are some quick explanatory notes regards implementation of open boundaries
(OB) and the example configuration found in .../verification/exp4/
Input and control of OBs
-
The open boundaries code is used only if the LOGICAL parameter openBondaries
is .TRUE.
-
Four 1-dimensional INTEGER arrays indicate the position of an OB: OB_Jnorth,
OB_Jsouth, OB_Ieast and OB_Iwest.
The values of OB_* indicate the absolute computational coordinate of an
open boundary. eg. OB_Jsouth(3)=4
means that in the column I=3 there is a "southern" OB at tracer point
J=4. Similarly OB_Iwest(2)=1 means that
the tracer point (2,1) is an OB point. This structure for specifying
OBs assumes that there will be only one northern and southern OB per column
and similar in the other direction. An entry of 0 (zero) means there is
now OB in that column/row.
The reference to "tracer points" above introduces an important distinction
between particular boundaries due to the C-grid staggering of variables.
At a northern OB tracer point (i,j) the OB v-velocity point is at (i,j)
while at a southern OB tracer point (i,j) the OB v-velocity point is at
(i,j+1). This is all taken care of in the code but when specifying OB values
it is important to remember the physical coordinates of T,S,U and V are
not the same for a given computational index.
Changing the type of open boundary
The supplied code simply specifies model variables at the appropriate points.
The value to be specified can (and should) be determine using a radiative
condition. A skeleton code is supplied set_obcs.F to illustrate how
this might be done. This is the only routine that needs to be edited.
Edit others at your own peril!
The examples
There are three examples in .../verification/exp4/
-
A zonal channel (solid side walls) with open boundaries at either end:
data.channel
-
A zonal channel with a Gaussian bump and open boundaries at either end:
data.bumpchannel
-
An open domain (OBs on four sides) with a Gaussian bump: data.bump
Each data file has a corresponding topog.* file which is specified through
the variable bathyFile in data.*
The physical problem has a flow of U=25 cm/s specified at all open
boundaries. The details of the experiment can be found in Adcroft's thesis
(Imperial College) and in Adcroft, Hill and Marshall, MWR 1997.
The differences between the supplied data.* files also show how to turn
particular OBs on and off.
Compiling and running the examples
The examples supplied are the barotropic zonal flow over a Gaussian bump
in a channel.
-
Header files: Copy or link SIZE.h, CPP_OPTIONS.h and CP_EEOPTIONS.h from
.../verification/exp4/code/ to .../model/inc/, .../model/inc/ and .../eesupp.inc/
respectively
-
Modified source: Copy set_obcs.F to .../model/src/
Now compile the code as usual:
-
cd bin
-
../tools/genmake -makefile
-
make depend
-
make
To run the code you must first choose a data file to use:
-
cd ../verification/exp4/input/
-
cp data.bump data
-
../../exe/mitgcmuv
Happy Open Boundarying!


| GRIDSEARCH |
Function minimization using grid search.
Synopsis:
[min_x,min_y,X,Y]=gridsearch(Min,Max,Grid,nloops,fun)
[min_x,min_y,X,Y]=gridsearch(Min,Max,Grid,nloops,fun,varargin)
Descritpion:
This function implements the grid search to find the minimum
of a given function y = fun(x), x \in X.
The domain X is discretized to the grid with minimal and
maximal point given by Min and Max respectively. Number of points
of the grid is given by Grid. The Max, Min and Grid are vectors
their entries correspond to individual dimensions of X.
After the mininum is found then the grid search is recursively
repeated with a finer grid. Number of nested loops is given by nloops.
The string fun determines function to be minimized. The function is
called as y=feval(fun,x,varargin{:}).
Input:
Min [dim x 1] Minimum point of the grid.
Max [dim x 1] Maximum point of the grid.
Grid [dim x 1] Number of point in the grid, i.e. grid density.
nloops [1x1] Number of nested loops of the grid search.
fun [string] Identifies the minimized function.
varargin [cell] Additional arguments of the minimized function.
Output:
min_x [dim x 1] Found minimum.
min_y [1x1] min_y = fun(min_x).
X [dim x n] Points which the grid search checked through.
Y [1 x n] Y(i)=fun(X(:,i)).
Example:
[min_x,min_y,X,Y]=gridsearch(0,10,10,3,'sin');
figure; grid on; hold on;
plot(X,Y,'.');
plot(min_x,min_y,'+r');
About: Statistical Pattern Recognition Toolbox
(C) 1999-2003, Written by Vojtech Franc and Vaclav Hlavac
Czech Technical University Prague
Faculty of Electrical Engineering
Center for Machine Perception
Modifications:
18-July-2003, VF
17-July-2003, VF


Golden toilet worth more than a $1 million may break world record
by Ben CostIf Goldfinger had a toilet, it would probably resemble this nearly $1.3 million gem-encrusted, golden john that surfaced at a Shanghai trade fair last week.
The lavish loo is gunning for a Guinness World Record for most diamonds in a toilet seat (yes, that’s a category) with over 40,000 adorning its bulletproof rim — as if all the gold wasn’t ostentatious enough.
The $1,287,652 (£999,497) piece is produced by Coronet, one of Aaron Shum Jewelry’s brands. Shum himself debuted the high-karat crapper Monday at the second China International Import Expo in Shanghai.
Shum didn’t disclose whether or not he had any prospective buyers, although he told the Daily Mail that the bejeweled baño might be better off at a “diamond art museum so that more people can enjoy it.”
'BONELESS' DOUGHNUT PACKAGE HAS TWITTER ASKING WHAT A 'BONE-IN' DOUGHNUT IS
It would fit right in within any esteemed collection. The gilded throne’s rim features more than 40,815 diamonds, clocking in at an eye-popping 334.68 carats in total.
And if that wasn’t James Bond villain enough, the seat’s also bulletproof.
CLICK HERE TO SIGN UP FOR OUR LIFESTYLE NEWSLETTER
Coronet said the gaudy potty is slated for inspection by Guinness this week to determine if it qualifies for a world record. This would mark the 10th record for the jeweler, which previously won titles for most diamonds set in a watch, and most diamonds set in a handbag with a swanky Coca-Cola-inspired purse.
Coronet has also produced a diamond-studded guitar and a pair of pink shoes worth $2 million and $4.28 million, respectively. Although it’s safe to say none are as garish as this gilded toilet.
CLICK HERE TO GET THE FOX NEWS APP
And in the realm of bathroom bling, Coronet’s crapper is not nearly as tasteful as Maurizio Cattelan’s 18-karat solid gold throne.
This content originally appeared in the New York Post.


Gecode::Int::ConstIntView Class Reference
[Integer views]
Inherits Gecode::ConstantViewBase.
Detailed Description
Constant integer view.An constant integer view for an integer provides operations such that behaves as a view assigned to .
Definition at line 736 of file view.icc.
Constructors and initialization | |
| ConstIntView (void) | |
| Default constructor. | |
| ConstIntView (int n) | |
| Initialize with integer value n. | |
| void | init (int n) |
| Initialize with integer value n. | |
Value access | |
| int | min (void) const |
| Return minimum of domain. | |
| int | max (void) const |
| Return maximum of domain. | |
| int | med (void) const |
| Return median of domain. | |
| int | val (void) const |
| Return assigned value (only if assigned). | |
| unsigned int | size (void) const |
| Return size (cardinality) of domain. | |
| unsigned int | width (void) const |
| Return width of domain (distance between maximum and minimum). | |
| unsigned int | regret_min (void) const |
| Return regret of domain minimum (distance to next larger value). | |
| unsigned int | regret_max (void) const |
| Return regret of domain maximum (distance to next smaller value). | |
Domain tests | |
| bool | range (void) const |
| Test whether domain is a range. | |
| bool | assigned (void) const |
| Test whether view is assigned. | |
| bool | in (int n) const |
| Test whether n is contained in domain. | |
| bool | in (double n) const |
| Test whether n is contained in domain. | |
Domain update by value | |
| ModEvent | lq (Space *home, int n) |
| Restrict domain values to be less or equal than n. | |
| ModEvent | lq (Space *home, double n) |
| Restrict domain values to be less or equal than n. | |
| ModEvent | le (Space *home, int n) |
| Restrict domain values to be less than n. | |
| ModEvent | le (Space *home, double n) |
| Restrict domain values to be less than n. | |
| ModEvent | gq (Space *home, int n) |
| Restrict domain values to be greater or equal than n. | |
| ModEvent | gq (Space *home, double n) |
| Restrict domain values to be greater or equal than n. | |
| ModEvent | gr (Space *home, int n) |
| Restrict domain values to be greater than n. | |
| ModEvent | gr (Space *home, double n) |
| Restrict domain values to be greater than n. | |
| ModEvent | nq (Space *home, int n) |
| Restrict domain values to be different from n. | |
| ModEvent | nq (Space *home, double n) |
| Restrict domain values to be different from n. | |
| ModEvent | eq (Space *home, int n) |
| Restrict domain values to be equal to n. | |
| ModEvent | eq (Space *home, double n) |
| Restrict domain values to be equal to n. | |
Domain update by range iterator | |
| template<class I> | |
| ModEvent | narrow (Space *home, I &i) |
| Replace domain by range sequence described by i. | |
| template<class I> | |
| ModEvent | inter (Space *home, I &i) |
| Intersect domain with range sequence described by i. | |
| template<class I> | |
| ModEvent | minus (Space *home, I &i) |
| Remove from domain the range sequence described by i. | |
Dependencies | |
| void | subscribe (Space *home, Propagator *p, PropCond pc, bool process=true) |
| Subscribe propagator p with propagation condition pc to variable. | |
| void | cancel (Space *home, Propagator *p, PropCond pc) |
| Cancel subscription of propagator p with propagation condition pc to view. | |
Cloning | |
| void | update (Space *home, bool share, ConstIntView &x) |
| Update this view to be a clone of view x. | |
Propagator modification events | |
| static ModEvent | pme (const Propagator *p) |
| Return modification event of propagator p for view. | |
| static PropModEvent | pme (ModEvent me) |
| Translate modification event me to propagator modification event for view. | |
View comparison | |
| bool | same (const Int::ConstIntView &x, const Int::ConstIntView &y) |
| Test whether views x and y are the same. | |
| template<class View, unsigned int n> | |
| bool | before (const Int::ConstIntView &x, const Int::ConstIntView &y) |
| Test whether view x is before y (arbitrary order). | |
Protected Attributes | |
| int | x |
Related Functions | |
| (Note that these are not member functions.) | |
| std::ostream & | operator<< (std::ostream &, const Gecode::Int::ConstIntView &) |
| Print constant integer view. |
Constructor & Destructor Documentation
|
|
Default constructor.
Definition at line 31 of file constint.icc. |
|
|
Initialize with integer value n.
Definition at line 33 of file constint.icc. |
Member Function Documentation
|
|
Initialize with integer value n.
Definition at line 35 of file constint.icc. |
|
|
Return minimum of domain.
Definition at line 45 of file constint.icc. |
|
|
Return maximum of domain.
Definition at line 49 of file constint.icc. |
|
|
Return median of domain.
Definition at line 53 of file constint.icc. |
|
|
Return assigned value (only if assigned).
Definition at line 57 of file constint.icc. |
|
|
Return size (cardinality) of domain.
Definition at line 62 of file constint.icc. |
|
|
Return width of domain (distance between maximum and minimum).
Definition at line 66 of file constint.icc. |
|
|
Return regret of domain minimum (distance to next larger value).
Definition at line 70 of file constint.icc. |
|
|
Return regret of domain maximum (distance to next smaller value).
Definition at line 74 of file constint.icc. |
|
|
Test whether domain is a range.
Definition at line 84 of file constint.icc. |
|
|
Test whether view is assigned.
Definition at line 88 of file constint.icc. |
|
|
Test whether n is contained in domain.
Definition at line 93 of file constint.icc. |
|
|
Test whether n is contained in domain.
Definition at line 97 of file constint.icc. |
|
|
Restrict domain values to be less or equal than n.
Definition at line 107 of file constint.icc. |
|
|
Restrict domain values to be less or equal than n.
Definition at line 111 of file constint.icc. |
|
|
Restrict domain values to be less than n.
Definition at line 116 of file constint.icc. |
|
|
Restrict domain values to be less than n.
Definition at line 120 of file constint.icc. |
|
|
Restrict domain values to be greater or equal than n.
Definition at line 125 of file constint.icc. |
|
|
Restrict domain values to be greater or equal than n.
Definition at line 129 of file constint.icc. |
|
|
Restrict domain values to be greater than n.
Definition at line 134 of file constint.icc. |
|
|
Restrict domain values to be greater than n.
Definition at line 138 of file constint.icc. |
|
|
Restrict domain values to be different from n.
Definition at line 143 of file constint.icc. |
|
|
Restrict domain values to be different from n.
Definition at line 147 of file constint.icc. |
|
|
Restrict domain values to be equal to n.
Definition at line 152 of file constint.icc. |
|
|
Restrict domain values to be equal to n.
Definition at line 156 of file constint.icc. |
|
|
Replace domain by range sequence described by i.
Definition at line 168 of file constint.icc. |
|
|
Intersect domain with range sequence described by i.
Definition at line 173 of file constint.icc. |
|
|
Remove from domain the range sequence described by i.
Definition at line 179 of file constint.icc. |
|
|
Return modification event of propagator p for view.
Definition at line 191 of file constint.icc. |
|
|
Translate modification event me to propagator modification event for view.
Definition at line 195 of file constint.icc. |
|
|
Subscribe propagator p with propagation condition pc to variable. In case process is false, the propagator is just subscribed but not processed for execution (this must be used when creating subscriptions during propagation). Definition at line 205 of file constint.icc. |
|
|
Cancel subscription of propagator p with propagation condition pc to view.
Definition at line 207 of file constint.icc. |
|
|
Update this view to be a clone of view x.
Definition at line 216 of file constint.icc. |
Friends And Related Function Documentation
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Test whether views x and y are the same.
Definition at line 295 of file constint.icc. |
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Test whether view x is before y (arbitrary order).
Definition at line 299 of file constint.icc. |
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Print constant integer view.
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Member Data Documentation
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The documentation for this class was generated from the following files:
- gecode/int/view.icc (Revision: 3559)
- gecode/int/view/constint.icc (Revision: 3559)


ionic
flux
An Ionic Application for to practise finding the best word from a set of tiles Uses Angular Flux


This package contains the main classes and abstract interfaces for this
SMT-LIB tool; the tool provides a stand-alone SMT-LIB-compliant command-line
application, an application that responds to SMT-LIB commands over a
network port, and an SMT-LIB Java API. For a tool that encapsulates this functionality as an
Eclipse plug-in, see the companion package org.smtlib.plugin.


packages = [
"sqlite-utils"
]
Sqlite-utils Example
# was ok with https://pyscript.net/2022.06.1/pyscript.js
# you can try last official release: https://pyscript.net/latest/pyscript.js and css
# or current unstable master: https://pyscript.net/unstable/pyscript.js and cssfrom __future__ import print_function, unicode_literals, division # if Python2.7
import sqlite_utils
db = sqlite_utils.Database(memory=True)
print(list(db.query("select 3 * 5")))
print('sqlite-version', list(db.query("select sqlite_version()")))


Department of Mathematics
Indian Institute of Science
Bangalore 560 012
SEMINAR
|
Speaker |
: |
Dr. Praveen Pankajakshan |
| Affiliation | : |
Department of Computer Science, INRIA, Sophia-Antipolis
Cedex, FRANCE |
|
Subject Area |
: |
Mathematics
|
|
Venue |
: |
Lecture Hall - I, Dept of Mathematics
|
|
Time |
: |
4.00 pm
|
|
Date |
: |
August 6,2008 (Wednesday) |
|
Title |
: |
Inverse Problems in Image Processing |
| Abstract | : |
The first half of this talk will focus on the initiatives by the Ariana project in providing image processing tools that aid in solving inverse problems arising in Earth observation and cartography. Some of these problems are related to cartographic updating, land management, and agriculture. For such problems, even in the rare instance when the existence and uniqueness of the solution is guaranteed, the solution is unstable due to the perturbing effects of observation noise. We use two broad classes of techniques to attack these problems: probabilistic models combined with stochastic algorithms, and variational models combined with deterministic algorithms. In addition to applying these techniques to specific cases, the project advances these techniques more generally, through innovative modeling and theoretical analysis, and a comparative study of the two classes. The Bayesian methodology is followed as far as possible and probabilistic models are used for two purposes: to describe the class of images to be expected from any given scene, and to describe prior knowledge about the scene in the absence of the current data. Some of the probabilistic models used are Markov Random Fields (MRF) for texture segmentation, wavelets that sparsely represent the image data for denoising and deconvolution. The use of variational models is mostly to regularize the inverse problems. Attention is focused on the use of such models for image segmentation, in particular texture segmentation; on the theoretical study of the models and their associated algorithms, in particular level set methods; and on the incorporation of prior geometric information concerning the regions sought using higher-order active contour energies. Another interesting problem studied is how to estimate the parameters that appear in the models. For probabilistic models, the problem is easily framed, but is not necessarily easy to solve, particularly in the case when it is necessary to extract simultaneously from the data both the information of interest and the parameters. We shall provide some applications examples from image restoration and denoising, multicamera reconstruction and super resolution, extraction of various complex structures in a scene, 3D modeling, segmentation and classification, and information mining and retrieval from remote sensing image databases. The second half of this talk will address a specific issue related to restoration of fluorescence Confocal Laser Scanning Microscope (CLSM) images with parametric estimation of the acquisition systemís Point Spread Function (PSF). The CLSM is an optical fluorescence microscope that scans a specimen in 3D and uses a pinhole to reject most of the out-of-focus light. However, the quality of the image suffers from two primary physical limitations. The first is due to the diffraction-limited nature of the optical system and the second is due to the reduced amount of light detected by the photomultiplier tube (PMT). These limitations cause blur and photon counting noise respectively. The images can hence benefit from post-processing restoration methods based on deconvolution. An efficient method for parametric blind image deconvolution involves the simultaneous estimation of the specimen 3D distribution of fluorescent sources and the microscope PSF. By using a model for the microscope image acquisition physical process, we reduce the number of free parameters describing the PSF and introduce constraints. The parameters of the PSF may vary during the course of experimentation, and so they have to be estimated directly from the observation data. We also introduce a priori knowledge of the specimen that permits stabilization of the estimation process and favors the convergence. Experiments on simulated data show that the PSF could be estimated with a higher degree of accuracy and those done on real data show very good deconvolution results in comparison to the theoretical microscope PSF model.
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