Sec. 3.7 Differentials 微分 1 :微分(differential)的幾何量化與代數推導 增量(in creme n ):函數y f (x),當x 分量由％變動到x 2時，x x 2 x 1稱為x 的一增量 對應x 的此一增量 x ， y 的增量為 y y 2 y 1 f (x 2) f (x 1) f(% x) f (x 1) 如圖示 幾何說明：現賦予dy 、dx 幾何上的意義，讓來不尼茲符號 说有兩數相除的意思。 如上圖所示(注意微分與增量的關係 ) 3 例 1~2： Let y f (x) x . Find x and y : a) When x cha nges fron2 to2.01. b) Whe n x chan ges fron2 to 1.98. 解： a) x 2.01 2 0.01(新的減舊的) y f(X i x) f(X 1) f(2.01) f(2) (2.01)3 23 0.1206.1 b) x 1.98 2 0.02(新的減舊的) y f(X i x) f(xj f (1.98) f(2) (1.98)3 23 0.237608 定義： 函數 y f(x), x 的微分 (differentia )為 dx x (dy 隨x 與dx 變動) y 的微分 (differentia )為 dy f (x) dx
a )x 的增量為 x AB PR , y 的增量為 y f(x x) f (x) CD RQ b ）如圖過點P 的切線斜率 m f （x ） dy RS dx PR 若令dx x PR ，則dy 可取為如圖示的線段長 RS ； c ） f dy 即 RS RQ d ） 用切線逼近曲線 例(補充)：Find dy if a y x 3 3x 1 b y .. x 2 3x 解： a 矽 —3x 1 3x 2 3 2 dy 3x 3 dx dx dx 例（補 充）： d x 3 3x 1 Find - d x 2 3x 解: d x 3 3x 1 3x 2 3 dx 3x 2 3 # d x 2 3x 2x 3 dx 2x 3 2、近似ApproximatenS :（看圖也有相同結論） f (x) f (x lim x) f(x) f(x x) f(x) x 0 x x f(x x) f(x) f (x) x ??- -(1) (即 f dy 或 RS RQ ) f(x x) f(x) f (x) x ??? -(2) (即 BQ BR RS ) 例題 3： Let y f (x) x 3. a) Find the differential dy of y . b) Use dy to approximate y When x changes fron2to2.01. c) Use dy to approximate y When x changes fron2to 1.98. d) Compare the results of part (b) with those of Example b dy d , x 2 3x dx dx 2x_3_ 2、x 2 3x dy 上丄dx # 2、、x 2 3x
Chapter 1 First-Order ODEs C C h h a a p p t t e e r r 22 S S e e c c o o n n d d --O O r r d d e e r r L L i i n n e e a a r r O O D D E E s s ((二二階階線線性性常常微微分分方方程程式式)) Chapter 3 Higher-Order Linear ODEs Chapter 5 Series Solutions of ODEs Chapter 6 Laplace Transforms ? Ordinary differential equations may be divided into two large classes, linear (線性) and nonlinear (非線性) ODEs. Where nonlinear ODEs are difficult to solve, linear ODEs are much simpler because there are standard methods for solving many of these equations. 22..11 H H o o m m o o g g e e n n e e o o u u s s L L i i n n e e a a r r O O D D E E s s o o f f S S e e c c o o n n d d O O r r d d e e r r ((二二階階線線性性齊齊性性微微分分方方程程式式)) ? A second-order ODE is called linear (線性的) if it can be written as ()()()y p x y q x y r x '''++=. (1) ? 線性：方程式的每一項都不得出現()y x 和其導數(y ', y '',…)的乘積或自乘 In case ()0r x =, the equation is called homogeneous (齊性的). In case ()0r x ≠, the equation is called nonhomogeneous (非齊性的). The functions ()p x and ()q x are called the coefficients of the ODEs. ? Theorem 1 Superposition Principle for the Homogeneous Linear ODE (適用於線性 齊性常微分方程式的疊加原理) If both 1()y x and 2()y x are solutions of the homogeneous linear ODE ()()0y p x y q x y '''++=, (2) then a linear combination (線性組合) of 1y and 2y , say 1122()()c y x c y x +, is also a solution of the differential equation. Proof –
Differential Amplifier Thus far we have used only one of the operational amplifiers inputs to connect to the amplifier, using either the "inverting" or the "non-inverting" input terminal to amplify a single input signal with the other input being connected to ground. But we can also connect signals to both of the inputs at the same time producing another common type of operational amplifier circuit called a Differential Amplifier. Basically, as we saw in the first tutorial about operational amplifiers, all op-amps are "Differential Amplifiers" due to their input configuration. But by connecting one voltage signal onto one input terminal and another voltage signal onto the other input terminal the resultant output voltage will be proportional to the "Difference" between the two input voltage signals of V1 and V2. Then differential amplifiers amplify the difference between two voltages making this type of operational amplifier circuit a Subtractor unlike a summing amplifier which adds or sums together the input voltages. This type of operational amplifier circuit is commonly known as a Differential Amplifier configuration and is shown below: Differential Amplifier By connecting each input inturn to 0v ground we can use superposition to solve for the output voltage Vout. Then the transfer function for a Differential Amplifier circuit is given as:
Noise is classified into two types according to the conduction mode. The first type is differential mode noise which is conducted on the signal (VCC) line and GND line in the opposite direction to each othe. This type of noise is suppressed by installing a filter on the hot (VCC) side on the signal line or power supply line, as mentioned in the preceding chapter. The second type is common mode noise which is conducted on all lines in the same direction. With an AC power supply line, for example, noise is conducted on both lines in the same direction.With a signal cable, noise is conducted on all the lines in the cable in the same direction. Therefore, to suppress this type of noise, EMI suppression filters 4.1. Differential and Common Mode Noise methods are applied. 1.Noise is suppressed by installing an inductor to the signal line and GND line, respectively. 2. A metallic casing is connected to the signal line using a capacitor. Thus, noise is returned to the noise source in the following order; signal/GND lines ¨ capacitor ¨ metallic casing ¨ stray capacitance ¨ noise source.
Real-Time Differential(1) This paper describes the Real –Time Differential Option that can be installed on ASHTECH XⅡreceivers and its two output formats :Ashtech format and RTCM-104version 2.0 format, In connection with the general description of real –time differential, it covers: 1.Major sources of error affecting accuracy 2.Age of range corrections 3.Issue of orbit data of the base and remote receivers A description of the format for range corrections is followed by an example and a few suggestions in case of trouble. This is followed oby a section devoted to RTCM 104 format. Setting receivers for RTCM format is described in the last half of this paper. (Reading and setting receivers for ashtech format was described in connection with Screen 5.) Real-Time Differential in General Real-Time differential GPS involves a base receiver computing the satellite range corrections and transmitting them to the remote receivers. It transmits these corrections in real time to the remote receivers via a telemetry link. Remote receivers apply the corrections to their measured ranges; they use corrected ranges to compute their position. The base receiver determines range, computed by using the accurate position entered in the receiver. (this accurate position must have been previously surveyed using GPS or some other technique.) The remote receivers subtract the received corrections from their measured ranges and use the corrected ranges for position computation. A stand-alone GPS receiver can compute a position of around 25 meters with Selective Availability off and around 100 meters with SA on. Differential GPS can achieve 1-10 meters at the remote receivers even with SA on. The receiver can be designated as the base or remote station .In base mode, the receiver computes the range errors in every cycle. The communication link can be a radio link , telephone line, cellar phone, satellite communication link or any other medium that can transfer digital data. For testing, connect both receivers via a full handshake null modem RS-232 cable via their B ports. Sources of Error The major sources of error affecting the accuracy of GPS range measurements are orbit estimation, satellite clock estimation, ionosphere, and receiver noise in measuring range. The first four are almost totally removed using differential GPS. Their residual error is in the order of 1 millimeter for every kilometer of separation between base and remote receivers. Receiver noise ,the last error in the list , is not correlated between the base and the remote receiver and is not canceled by differential GPS. However in ASHTECH Ⅻreceivers, integrated Doppler is used to smooth the range measurements and reduce the receiver noise. At the instant a satellite is locked, there is also RMS noise affecting the range measurement. This RMS noise is reduced with the square root of n where n is the number of measurements. For example, after 100 seconds of locking to a satellite, the RMS noise in range measurement is reduced by a factor of 10(1 meter of noise is reduced to 0.1meter) . The noise is further reduced over time. If the lock to a satellite is lost , the noise goes back to 1 meter and smoothing starts from the 1-meter level. The loss of lock to a satellite is rare .It typically happens only when the direct path to the satellite is blocked by an object. Total position error (or error-in-position),is a function of the range errors (or error-in –range), multiplied by the PDOP(there –coordinate position dilution of precision). The PDOP is a function of the geometry of the satellite . Ashtech Format The remote receiver needs to know the smooth count of the range corrections .Smooth count is the number of measurements used to compute the current smoothing correction on a given data point .Therefore ,the smooth count for each range correction is also transmitted from base to remote receiver. A range correction with a smooth count of 100 has a noise level of 0.1 meter ,while a range correction with a smooth count of 25 has a noise level of 0.2 m.
Differential Pair 走線注意事項 1>>凡遇到Connector 有另加防ESD的零件時,請依下圖走法 2>>凡Differential Pair由pin腳拉出時,請將斜邊拉至pin腳尖端
3>>Differential Pair的線長差異太多時,tune等長步驟如下: Step1.>>先試著在打VIA的地方適當的繞線 Step2.>>如果線長還是有所差異,請點選delay tune,且在options的Sawtooth Gap中key上1x width
Step3.>>點選需要加長的net ,按mouse右鍵把Single trace mode打勾 Step4.>>將點選的net往外提高一格
Step5.>>先測量tune線的space .length 是否符合S1<2S.及length <3W Step6.>>此例子為4/4/4的Diff pair , S1=2S未符合規範,因此必需再手動將Air gap 往下調至S1<2S 點選Slide>mouse右鍵選Temp group>點選tune線最高線段> mouse右鍵選Complete>再往內拉至S1<2S
Step5.>>再次測量tune線的space .length 是否符合S1<2S.及length <3W 差分信号（Differential Signal）在高速电路设计中的应用越来越广泛，电路中最关键的信号往往都要采用差分结构设计，什么另它这么倍受青睐呢？在PCB设计中又如何能保证其良好的性能呢？带着这两个问题，我们进行下一部分的讨论。何为差分信号？通俗地说，就是驱动端发送两个等值、反相的信号，接收端通过比较这两个电压的差值来判断逻辑状态“0”还是“1”。而承载差分信号的那一对走线就称为差分走线。差分信号和普通的单端信号走线相比，最明显的优势体现在以下三个方面： a.抗干扰能力强，因为两根差分走线之间的耦合很好，当外界存在噪声干扰时，几乎是同时被耦合到两条线上，而接收端关心的只是两信号的差值，所以外界的共模噪声可以被完全抵消。 b.能有效抑制EMI，同样的道理，由于两根信号的极性相反，他们对外辐射的电磁场可以相互抵消，耦合的越紧密，泄放到外界的电磁能量越少。 c.时序定位精确，由于差分信号的开关变化是位于两个信号的交点，而不像普通单端信号依靠高低两个阈值电压判断，因而受工艺，温度的影响小，能降低时序上的误差，同时也更适合于低幅度信号的电路。目前流行的LVDS（low voltage differential signaling）就是指这种小振幅差分信号技术。
Differential Trace Design Rules Truth vs Fiction
Truth vs Fiction
Truth vs Fiction
Truth vs Fiction Array (half of a 360 degree complete cycle), and if propagation time is 6” per nanosecond in FR4, then one degree phase shift equates to 333 mils distance. If we set one degree as our threshold, (which might be too much!) then the corresponding offsets would be: Frequency (MHz) Offset (mils, or thousandth in.) 50 333 500 33 5 GHz 3 Conclusion: The equal length design rule is important IF the signal equal-and-opposite assumption is important. The signal equal-and-opposite assumption is important if we are worried about EMI or if we require discontinuities in the power system grounds between two circuits. Close together rule, part 1: It is generally understood that EMI is related to loop area.2 The loop area is defined as the area between the signal path and its return path. On differential traces, the signal is on one trace and the return is on the other trace. So the loop area is a function of how close the traces are routed together. If we are concerned about EMI, we must route the traces close together. The more closely we route them to each other, the smaller the loop area will be and the less EMI that will be generated. Close together rule, part 2: One of the primary advantages of differential signals is the signal-to-noise ratio improvement that is obtained. Since the signal is one polarity on one trace and the other polarity on the other trace, the resulting signal at the receiving device is twice what the single-ended signal would be. An additional advantage is that the receiving circuits are designed so that they are highly sensitive to the difference in signal level between the two traces, but highly insensitive to shifts in signal level that occur on both traces. This is normally called common mode rejection at the receiving device.
Sec. 3.7 Differentials 微分 1：微分(differential)的幾何量化與代數推導 增量（increment ）： 函數)(x f y =，當x 分量由1x 變動到2x 時，12x x x -=?稱為x 的一增量。 對應x 的此一增量x ?，y 的增量為)()()()(111212x f x x f x f x f y y y -?+=-=-=? 如圖示 例1~2：Let 3)(x x f y ==. Find x ? and y ?： a) When x changes from 2to 01.2. b) When x changes from 2to 98.1. 解： a) 01.0201.2=-=?x (新的減舊的) 1.1206.02)01.2()2()01.2()()(3311=-=-=-?+=?f f x f x x f y b) 02.0298.1-=-=?x (新的減舊的) 237608.02)98.1()2()98.1()()(3311-=-=-=-?+=?f f x f x x f y ＃ 定義：函數)(x f y =， x 的微分（differential ）為x dx ?= y 的微分（differential ）為dx x f dy ?'=)(。 (dy 隨x 與dx 變動) 幾何說明：現賦予dy 、dx 幾何上的意義，讓來不尼茲符號dx dy 有兩數相除的意思。 如上圖所示(注意微分與增量的關係)
a)x 的增量為PR AB x ==?，y 的增量為RQ CD x f x x f y ==-?+=?)()( b) 如圖過點P 的切線斜率dx dy x f m = '=)(PR RS =； 若令PR x dx =?=，則dy 可取為如圖示的線段長RS ； c) dy f ≈?即RQ RS ≈ d) 用切線逼近曲線 例(補充)：Find dy if ()13y 3+-=x x a ()x x b 3y 2+= . 解： ()() 3313 23-=+-=x x x dx d dx dy a ()dx x dy 332-=? ()x x x x x dx d dx dy b 32323 22++=+=dx x x x dy 32322++=? ＃ 例(補充)：Find () ()x x d x x d 31323++- . 解： () ()x x d x x d 31323++-()()3233323322+-=+-=x x dx x dx x ＃ 2、近似Approximations ）： (看圖也有相同結論) x x f x x f x x f x x f x f x ?-?+≈?-?+='→?)()()()(lim )(0 x x f x f x x f ??'≈-?+?)()()(‥‥(1) (即dy f ≈?或RQ RS ≈) x x f x f x x f ??'+≈?+?)()()(‥‥(2) (即RS BR BQ +≈) 例題3：Let 3)(x x f y ==. a) Find the differential dy of y . b) Use dy to approximate y ? When x changes from 2to 01.2. c) Use dy to approximate y ? When x changes from 2to 98.1. d) Compare the results of part (b) with those of Example 2.
1. 微分方程式 (Differential Equation)：一個含有未知函數與此函數的導數(常導數或偏導數)之方程式。 2. 常微分方程式 (Ordinary Differential Equation, ODE )：含有常導數的微分方程式稱為常微分方程式。(例如, x y y cos 4=+'') 3. 偏微分方程式 (Partial Differential Equation, PDE)：含有偏導數的微分方程式 稱為偏微分方程式。(例如, 0222222=??+??+??z u y u x u ) 4. 微分方程式的階(Order)：方程式中最高階(次)導數的階。(例如, x y y cos 4=+''因為最高階導數y ''為二階) 5. n 階常微分方程式：令dx dy y =', 22dx y d y ='', …, n n n dx y d y =)(, 則n 階常微分方程式的一般式可寫成() 0,,,,,)(='''n y y y y x F 6. 微分方程式的解 (Solution)：若一個函數)(x h y =及其導數)(x h ', )(x h '', )()(x h n 代入方程式使得方程式變成一個恆等式, 則)(x h y =即為方程式的一個解。(例如, 32-=x e y 是方程式62=-'y y 的一個解) 7. 微分方程式的解可分成三類：通解、特解與奇異解。 a. 通解 (General Solution)：當解含有任意常數的個數等於微分方程式的階數時, 即為通解。(例如, 32-=x e c y 是方程式62=-'y y 的通解；x c x c y sin cos 21+=是方程式0=+''y y 的通解) b. 特解 (Particular Solution)：通解中的任意常數如果全部或部分指定值, 即稱為特解。(例如, 322-=x e y 是方程式62=-'y y 的一個特解； x y cos =是方程式0=+''y y 的的一個特解) c. 奇異解 (Singular Solution) ：不是透過指定通解的任意常數的值而得 到的解, 即為特解。(例如, ()c x y +=sin 是方程式()0122=-+'y y 的通 解；1=y 是方程式()0122 =-+'y y 的一個奇異解，它不能經由指定c 的值而得到)
DIFFERENTIAL GALOIS THEORY MOSHE KAMENSKY 1.Introduction We are interested in the following kind of statements: proposition1.The integral e?t2dt can not be solved by elementary functions. proposition2.The di?erential equation x +tx=0can not be solved by elemen-tary functions and integration. The aim of the talk is to explain how to make these statements precise and prove them,using di?erential Galois theory.The reference to all of the material here is [vdPS03]. By solving a di?erential equation we mean obtaining a solution via a?nite num-ber of operations of the following kind(starting with a rational function):?Adding a function algebraic over the functions we already have. ?Adding the exponential of(the integral of)a function we have. ?For the second problem,adding the integral of a function we have(alter-natively,allowing logarithms) At each stage,after adding the functions we are allowed to take any rational combinations of them,as well as their derivatives. We?rst explain that the problem is completely algebraic.To this end,we de?ne de?nition3.A di?erential?eld is a?eld K(of characteristic0),endowed with a derivation D:K→K.This means that D is additive and satis?es the Leibniz rule:D(xy)=D(x)y+xD(y)for any x,y∈K.D(x)will also be denoted by x . The set of elements of K satisfying Dx=0is a sub?eld of K called the?eld of constants,and is denoted C K.D is thus a linear operator over C K. Examples of di?erential?elds include the?elds rational functions over the real or complex numbers,and more generally the?eld of meromorphic functions on a region in P1(C),and sub?elds of such?elds,all with the usual derivative. Given a di?erential?eld K,it makes sense to consider di?erential equations with coe?cients in K.Solutions of such equations will generally lie in(di?erential)?eld extensions of K.The?rst operation described above corresponds to forming algebraic?eld extensions of the?eld K(the derivation extends uniquely to such extensions.) Adjoining an integral of a function amounts to passing to a?eld generated by the solution of an equation x =b,where b∈K,or equivalently,by a non-constant solution of x ?(b /b)x =0.Similarly,the exponential of b is obtained as the solution of x ?b x=0.In both cases,as well as in the propositions we wish to prove,we are concerned with linear di?erential equations. 1
School Of Engineering KNE222 Electronic Engineering Differential Amplifiers Introduction: One of the most useful circuits in the field of instrumentation is the differential amplifier. This device is designed to amplify the difference between two voltages and to provide a ground referenced output voltage. Operational amplifiers are frequently used in this application, and a common approach is shown in Figure 1. Figure 1. A Simple Differential Amplifier This circuit amplifies the difference between input signals v 1(t) and v 2(t). It can be analysed using the Principle of Superposition by summing the contributions to v o (t) from v 1(t) and v 2(t) when acting independent of each other. Consider the case where v 1(t) alone is applied, as shown in Figure 2. Figure 2. Differential Amplifier with v 1(t) acting alone. Here v 2(t) has been replaced by its internal impedance , which for a voltage source is zero Ohms . If we define the voltage at the non-inverting input terminal as v + and that at the inverting input as v- then we can write: ) ()(2121R R R t v v +=+ (1)