同步发电机--- 异步发电机/电动机 ---感应发电机/电动机

这些都是一类。
他们的原理是利用变化的磁场（旋转磁场）产生变化的电场  （电流电压），或者是导体在磁场中相对旋转运动（其实磁场并不旋转，只是定子中的磁场随时间大小变化，看起来像旋转一样，因此就可以理解为相对运动）。
比如说同步发电机： 先是在转子上装上直流励磁，然后外力带动旋转起来（我们先假设靠近励磁电流电线的地方磁场比较大）。然后我们来看定子上的某一点a，当励磁电线还没转到a 的时候，a旁边的磁场很弱的；慢慢的，转子转啊转，励磁电线现在转到了离a最近，那么a旁边的磁场由弱转强，变化的磁场产生电流（当然这个电流方向是为了 削弱变强的磁场滴。。。a可没那么傻），这不就是交流电么？？ 或者说是，转子磁场在转动，定子虽然不动，但是看起来就好象是和磁场的相反方向跑，定子导体就会产生电流。这个交流电是由转子磁场产生的，也叫做同步磁场用来产生定子电流，因此转子的速度和同步磁场的速度总是一致的！


再比如说异步电动机吧：
先是在定子上通上交流电，众所周知，交流电本身就是一个旋转的磁场， 还是假设这个磁场某个地方特别强。 让我们来看看转子（注意，转子是短路线圈，就是用来产生电流的）上某一点b，本来b旁边的磁场很弱的，随着时间的推移，强磁场终于出现啦！那b可不愿意啦，立马产生电流去抵抗！！！ok，这个电流现在在磁场下，收到了磁场力，自然转动起来了(当然你得先推它一把，然后它的磁场力就是拉着他运动的)，这不就是电动机了吗！！当然这个转子的速度要比定子的磁场慢一点的，因为转子产生的磁场是和定子磁场方向相反的，所以这个转子磁场拖着转子让它转的稍微慢一点（或者说是它抵消了一部分定子同步磁场，所以受到的洛仑兹电场力也稍微小一点，速度慢一点）。

再比如说异步发电机吧：
结构跟异步电动机一模一样，就是外力好大，加在转子上（比如风什么的），然后转子转动速度不仅不比定子磁场慢，反而比同步磁场快，这时候相对运动的方向不就换了呗，然后转子电流方向也换了，产生的磁场方向也不是和同步磁场方向相反了，而是相同了，这样就加强了同步磁场，从而定子上的电流更大了。。。这不就发电了么？？？

Inductors and capacitors in Power Systems

Inductors and capacitors are very common in power systems. Normally people say inductive loads consume reactive power and capacitive loads produce reactive power. But I have different thought on this. My opinion is that there is no such a reactive power thing, but created simply for calculation convenience. Why??

Inductors and capacitors never consume power, but they can store energy!

Inductors store energy in the form of magnetic field, while capacitors store energy in electric field.

Magnetic field can be generated due to three reasons: (1) moving electric charges (which is current) (2) time-changing electric field (3) natural magnet.


If we put a charged inductor (with current in it) short-circuited, it will contain the energy in it forever (if it is superconductor)

If we put a charged conductor (with non-zero voltage) open-circuited, it will contain the energy forever (if it is the ideal capacitor).

 

Remember, for inductors, larger current means larger magnetic field, therefore more energy stored!
For capacitors, large voltage means larger electric field, therefore more energy stored!

(1) Understanding of reactive power
Based on the above analysis, my opinion is there is no such a reactive power in power system!!! The concept is only for calculation convenience.
Why? We see that inductors and capacitors do not consume energy, they basically return those stored energy sooner or later to power systems. But why we feel there are losses, because there are time difference!!! What we lose here is the time value!

If we only inductive loads in systems, then it will borrow system energy, and return in next cycle when the current is decreased. If a load need 2S power at a time, but the system gives it S-Q at time t1 and S+Q at time t2 for compensation. You think it is enough? No! The lost value here is TIME!  It is like somebody borrows money from you, and returns to you next month, but how about interest (time value, opportunity cost)??  Inductive load: current lagging voltage because u=L*di/dt, if i is sin(theta), u then is cos(theta)=sin(theta+90), which means u leading i 90.

In our textbooks, we know that inductors absorb reactive power, and capacitors produce reactive power. This statement gives us misunderstanding that more capacitors can decrease the reactive power loss! Absolutely not!!  Since there is no reactive power, if a system contains conductive loads, the current will lead the voltage, they are still not on the same boat, time value is still lost! So not: the more the better!

But why we are in need of capacitors in power system, because the power system nature is inductive (store power first and then return). So if capacitors return power first and then store, they will perform like nothing, the power returned and stored cancel!!


When system load is very light, the capacitance> inductance, current is small, therefore, system voltage will increase (more energy stored in electric fields); when system load is heavy, current is large, the capacitance <  inductance, current is large and voltage will drop, because u=di/dt, voltage drop increase!

Ok, then, let's summarize. When system is inductive, more energy stored in magnetic field rather than electric field, voltage drops a lot; when system is capacitive, more energy stored in electric field, voltage increase a lot!

Why eigenvalue analysis?

Eigenvalue of Jacobian Matrix is used in stability analysis. So here we encounter three concepts, "eigenvalue", "Jacobian" and "stability". I will start with "stability", since the other two are the tools to achieve this final goal: answer whether the system is stable or not at some point?

(1) Stability
Stability analysis is to find out the answer to that whether or not the system is stable at some operating point. As time goes to infinity, if the system is stable, then it will achieve a equilibrium or the system state will asymptotically approach a fixed point.

There are stable and non-stable equilibriums, imagine a ball in a bowl, or on top of a bowl. we ask: can small disturbance to this point stir something big happen? If it cannot, it literally means the system can still go back to this equilibrium (bowl bottom point); On the contrary, if it indeed enlarges the deviations and have no intention to stop the trend, eventually lead to another system state, then it is a non-stable equilibrium (bowl top point).

How to determine the stability of an equilibrium?

• Step 1: Find the equilibrium point, let x(t+1)=x(t), or dx/dt=0!
• Step 2: Linearization, [f(x)-f(a)]/[x-a]=f'(x)|a=f'(a) or Jacobian Matrix
• Step 3: Check the value of f'(a) or eigenvalue of Jacobian Matrix
• Step 4: final judge: if f'(a)>1, unstable; f'(a)<1, stable; Jacobian matrix, eigenvalue with positive real part, unstable; with negative real part, stable;

Now I will give a simple example.
For a dynamic system
xt+1=f(xt)
Now explain why at a particular point c, f'(c)>1, it is unstable; f'(c)<1, stable.
Look at the figure below:

 There are two equilibriums, a and b, a is stable and b is unstable. Why? If we move a a little bit larger to blue xt , we get xt+1<xt,  meaning it moves back (the other direction).

a-------------------------------------------------> xt
                                 xt+1<---------------------

But, if we move b a little larger to red xt, then at the next time step, xt+1>xt, even larger!!! It will finally explode x|t=infinity.

b---------------------------------------------->xt
                                                             ------------------------->xt+1

 Note f'(a)<1, f'(b)>1 (we know that g'(x)=1)
OK now. We have proved it graphically. Now we will prove it in mathematical form.
 As an equilibrium, xt+1=xt, meaning f(xt)=xt=g(xt). So the intersection points of the two curves are the equilibriums.

f(a)=a=g(a);
f(b)=b=g(b);

we move a to xt (xt>a), then we need to prove
xt+1<xt ==> f(xt)<xt ==> f(xt)<g(xt) ==>f(xt)-f(a)<g(xt)-g(a) ==> [f(xt)-f(a)]/[xt-a]<[g(xt)-g(a)]/[xt-a]==>f'(a)<g'(a)

We know that g'(a)=1, so if f'(a)<1, the reverse induction is valid, and hence xt+1<xt, go back, stable!!

Same induction applies to point b. 

So the final time-domain result plots are as follows:

(2) Multi-variable system, eigenvalue and Jacobian matrix
Now let's consider there are multiple input and output system,

We need to find eqiliriums first, let  

that is f1(x1,x2)=0, f2(x1,x2)=0, (those can be linear or non-linear) then we obtain the equilibriums.  
We need to find whether these equiliriums are stable.


Our previous assertion then is to find the Jacobian Matrix at the specified equilibrium. If all the eigenvalues have negative real part, then the equilibrium is stable!

Questions:
(1) What is Jacobian Matrix?
Answer: Jacobian matrix is a first-order derivative of functions with respect to variables, which is corresponding to a slope in a single-variable function.
In this case, Jacobian matrix is as follows (a denotes the equilibrium point)
J|a=[df1/dx1, df1/dx2;]
      [df2/dx1, df2/dx2]  |a


(2) What is eigenvalue?
If J is multiplied with vector v, J can be viewed as a linear transformer of v. J basically stretches and rotate v. If v is only stretched or shrunk, without changing direction, then v is a eigenvector of J, meaning it is owned by J. The scale of stretching and shrinking then is called eigenvalue! Those can be expressed as



Eigenvalues can be found by solving



OK. Now that we have cleared the above two important concept, we now ask why eigenvalue especially its real part matters?

If we are at an arbitrary point (system state) that is described by the differential equation , how do we find the solution for each state. To put it another way, what are xs at time t?

To solve differential equations, we use Laplace Transform. 
But what is this about?

Laplace Transform is extended from Fourier Transform. Fourier says any mathematical function can be decomposed to linear combination of sinusoidal functions with different magnitude and frequencies. He then applied Euler's formula and established this famous transform:

Euler's formula: e^{ix}=cosx + isinx

Fourier's Transform

Fourier's transform possesses a clear physical meaning, but also has some weakness when applied to non-integral functions. Therefore, Laplace extend this transform by add a factor so that the function will always be damping or decaying with time. This has become famous Laplace's Transform.

Note s=sigma t+ jwt, a complex number.

If we solve the system differential equations, we usually get solutions somewhat like

where lambda is the eigenvalue of the Jacobian matrix. If it has negative real part, it means xt will decay to some equilibrium point, stable! Conversely, if it has positive real part, xt will explode as time goes to infinity!!!!