How can a BLDC motor respond to a specific dynamic load demand without changing the supply voltage? Coming at BLDC motors from the theory side, my understanding has always been that they have a linear curve based on Kv, no-load current, and resistance, for a given applied voltage. Now the thing is, say they have a given &quot;system load&quot;, like a propeller, that varies during the operation. So the load demands a certain RPM/torque combination at time A, and then a different one at time B. My understanding was that in order to achieve both combination A and B, it has to vary the supply voltage so that it's linear torque-speed curve intersects this operating point (provided this point lies below it's highest torque-speed curve). What I can't seem to find is how this works without changing the supply voltage? If I use a PWM controller at 50% (but the supply voltage is the same), is the torque-speed curve really going to have a characteristic curve that responds as if the supply voltage was half? If not, how are these BLDC motors controller for dynamic applications where a specific torque and speed are needed at different times?
Does increasing voltage increase the torque of a brushless motor? 1.) I have a brushless motor with testing data at 12V. I want to know if I increase the voltage supply, will this increase the torque?   In my research I keep seeing "current controls torque, and voltage controls speed." But that doesn't seem quite right. Let's say I have a theoretical electrically powered unicycle that is weighted. With a 5V source and 2A, the motor can't move the unicycle. But now I increase the voltage to 10V, and I believe current will also increase by x2 if resistance is constant (V = IR). So now the torque increases and the unicycle can move itself. It that a reasonable sequence of events? Can I say that increasing voltage increases torque?   2.) A similar question: if P = VI = tau*omega, then P/omega = tau, so:  2*(P/omega) = 2*tau, which means increasing power will increase torque, correct? And since P = VI, increasing V will increase P, which will in turn increase torque. Is this accurate at all?
Expected value in an urn problem In an urn there are $n$ red balls and $m$ blue balls. I extract them without replacement. Let $X$=time of first blue. What is $E(X)$?  I found PMF of $X$ and it is, if $k &gt; n+1$  $$P(X=k) =\frac{n(n-1) \dots (n-k+1)m}{(m+n)(m+n-1) \dots (n+m-k+1)}$$  How could I evaluate $E(X)$?  Edit: I'm looking for an explicit form of $E(X)$ Suppose $\{f_n\} \subset L^+$ , $f_n \rightarrow f$ pointwise, and $\int f = \lim \int f_n &lt; \infty$. Then $\int_{E} f = \lim \int_{E} f_n$ for all $E \in \mathcal{M}$. However, this needn't be true if $\int f = \lim \int f_n = \infty$.   This is the exercise 13 on page 52, Real Analysis, by Gerald B. Folland. $L^+$ is the space of all measurable functions from $(X,\mathcal{M}, \mu)$ to $[0, \infty]$.  I can find a counterexample satisfying $\int f = \lim \int f_n = \infty$. Let $X = \Bbb N$, $\mathcal{M} = \mathcal{P}(\Bbb N)$ and $\mu$ be the counting measure. $f_n = \chi_{\{2i+1:i \leq n\} \cup \{2j:n \leq j \leq 2n \}}$. Then $\int f = \lim \int f_n = \infty$. But when $E = \{2k: k \in \Bbb N\}$,  Then $\int_{E} f =  0$ and $\lim \int_{E} f_n = \infty$.  But how to show the part "$f_n \rightarrow f$ pointwise, and $\int f = \lim \int f_n &lt; \infty$. Then $\int_{E} f = \lim \int_{E} f_n$ for all $E \in \mathcal{M}$"? Best way to extract a subvector from a vector? How do I time a method's execution in Java? Reading order for Dragonlance? Difference between standard error and standard deviation I'm struggling to understand the difference between the standard error and the standard deviation. How are they different and why do you need to measure the standard error? I dont have a clue about the proof in the reverse implication on the iff statement.  I will show what I have by now:  1) A cofinite topology implies that any infinite subset in $X$ is dense. Let $U\in\tau$ and is cofinite topology so $U=X-\bigcup_{k=1}^{n}\{x_k\}$.  Let $V\in X$ an infinite subset then $$V\cap U=V\cap(X-\bigcup_{k=1}^{n}\{x_k\})=(V\cap X)-(V\cap \bigcup_{k=1}^{n}\{x_k\})=V-(V\cap \bigcup_{k=1}^{n}\{x_k\})$$  and cause $|V|\ge\aleph_0$ and $|\bigcup_{k=1}^{n}\{x_k\}|=n\in\Bbb N\,$ we have that $|V\cap \bigcup_{k=1}^{n}\{x_k\}|\le n$. And finally is obvious that $V-(V\cap \bigcup_{k=1}^{n}\{x_k\})\ne\emptyset$ so any infinite subset is dense on $X$.  2) A cofinite topology implies a $T_1$-space. Let $x, y\in X$ and $U\in\tau: U=X-\{y\}$ so $x\in U\land y\notin U$. We can define too $V\in\tau:V=X-\{x\}$ so $x\notin V\land y\in V$ then a cofinite topology is a $T_1$-space.  But my problem is that I dont have a clue to prove the reverse implication i.e. if we have a $T_1$-space where every infinite subset is dense then it is a cofinite topology. Some hint? Some advice or correction in the proof of above? Thank you in advance.    I will complete the proof with the advice of @Cameron Buie.  1) A $T_1$-space implies that any finite subset is closed. Let the singleton $\{x\}\subset X$. Then $\overline{\{x\}}=\{x\}\cup\{x\}'$ where $\{x\}'=\{y:(\forall U\in\tau : y\in U)\land x\in U\}$, but because we are in a $T_1$-space exist some $U$ for any points $x,y\in X: x\in U\land y\notin U$ so $\{x\}'=\emptyset$ then $\overline{\{x\}}=\{x\}$ and is closed.  This means that for every $T_1$-space it cofinite sets are open.  2) If any infinite subset is dense on $X$ it means that the closure of any infinite subset is $X$ so the only closed sets are finite (and $X$ itself), so the topology is cofinite. I am trying to show that how the binary expansion of a given positive integer is unique.  According to this link, , All I see is that I can recopy theorem 3-1's proof?  Is this polished enough of an argument. Thanks How to succeed with making omelette Can "whatever" be split into two words? I tend to write, "say whatever they want", but I'm always tempted to write "say what ever they want". Is it acceptable to split the word in this context? How do you allocate more RAM to Minecraft? There is a lot of lag when I play Minecraft. I think this is due to the low amount of RAM I've allocated to Minecraft. (495MB)  How do you allocate more RAM to Minecraft?  I checked my other computer (Intel i5 Core) and it had the same amount of RAM allocated to Minecraft. However, Minecraft did not lag on that computer.  After I installed Optifine on this computer, the lag stopped (I don't know why cause I read that Optifine splits the load onto 2 cores, but I have only one core (At least I think so, a sticker on the laptop says Intel CORE i7)). Is tipping mandatory at Restaurants and Bars in Germany? In Bavaria, I paid for my drink, I asked her for "Ruckgeld", but only got the integer part of the money back, the rest of it, she kept it for tip! I looked her in the eye and she understood that I wasn't happy with that "initiative", but wouldn't blink an eye!  My company advised me to let it go, mentioning that if I was going to ask for the change, she would swear.  Notice how she didn't return me the correct amount back and then stare at me, demanding for tip, but she would decide for me!  Since, I will be around for a week or so, I was thinking of paying the exact amount of money next time. Will I get in an uncomfortable situation of having the waiter stare at me, demanding a tip? In other words, is tipping mandatory in Germany?    PS: Having experienced the tipping-cruelty in USA, and having tipped generously waiters across the Mediterranean for their supreme behavior, this Saturday night came as a shock, since I had the impression that this awful habit was only a USA-thing (you know, just like the fake audience laughing they have in USA TV series :) ). How can I add a "C" wire to my thermostat? After writing code to populate textboxes from an object, such as:  txtFirstName.Text = customer.FirstName; txtLastName.Text = customer.LastName; txtAddress.Text = customer.Address; txtCity.Text = customer.City;   is there way in Visual Studio (or even something like Resharper) to copy and paste this code into a save function and reverse the code around the equal sign, so that it will look like:  customer.FirstName = txtFirstName.Text; customer.LastName = txtLastName.Text; customer.Address = txtAddress.Text; customer.City = txtCity.Text; While I can post external texts into question and answer bodies, it preserves the old font style, spacing and size (although the markdown display won’t be affected by that.)   App Version: 1.6.6.2 Device: iPhone 6 OS Version: Version 11.0.3 (Build 15A432) A coin is tossed three times. The probability of zero heads is 1/8 and the probability of zero tails is 1/8. What is the probability that there is at least one head and at least one tail?  So, if P(zero heads)= 1/8 , then that should be the same of p(all tails)?  We would use the complement rule and Multiplication Rule?   P(at least one head) = 1 - P(no heads) = 1 - 1/8= 7/8 P(at least one tail) = 1 - P(no tails) = 1 - 1/8= 7/8  would I multiply the two values? How to use an ESC I'm new to RC related modules. Today I got my 's Skywalker 20A brushless esc. Manual is .    I also have a brushless outrunner motor (HL3512)    I don't have any device to program the esc's (even the seller hasn't mentioned about such requirement).   The  talks about programming by listening to the beep patterns. But for that it seems that the esc should be connected to a flight controller (it talks about throttle positions etc...). But I don't have any flight controller.  Can some one tell me how to use the esc with what I have? (I have 2200mAh LiPo and  too) What's the difference between these two sentences? And which one's grammatically correct? How come gas molecules don't settle down? If the earth's gravity exerts a net downward gravitational force on all air molecules, how come the molecules don't eventually lose their momentum and all settle down? How is the atmosphere is still miles thick after billions of years? One-dimensional [Noetherian] UFD is a PID