How many

多少

using this karnaugh map

使用这个卡诺图，

you are produce output for each one.

你可以为每一个生成输出。

And the conductor were attacked towards each other. So the power in this conductor

而且这些导体是相互攻击的。所以这个导体中的功率

based on the if the currents are moving in the same direction, if the current is moving in the same direction, they will be attacked each other. while if the current is not flowing in the different direction, they will be

取决于电流是否沿同一方向流动。如果电流方向相同，它们就会相互吸引。而如果电流方向相反，它们就会相互排斥。

so if the current are moving in the same direction, it will have the opposite direction. because the current will produce a magnetic field. you can use the right hand rule.

所以，如果电流方向相同，就会产生相反的方向。因为电流会产生磁场。你可以使用右手定则。

to define how the magnetic field look like. if the current is in direction, the magnetic field will go inside.

来确定磁场的形状。如果电流方向是这样的，磁场就会指向内部。

and outwards in this direction. and if there's current here, it has a different direction.

而在这个方向上，则会指向外部。如果这里有电流，它的方向就不同了。

the current here and you have the magnetic field.

这里有电流，就会产生磁场。

i think this all come over in the high school, right?

我想这些在高中都学过，对吧？

and we know what is charge. and people used to represent the charge.

我们也知道什么是电荷。过去人们常用这种方式来表示电荷。

q is a symbol. q is a symbol of charge. and the unit is coulomb.

Q是一个符号。它代表电荷。它的单位是库仑。

and we can just use c for simplification.

为了简化，我们也可以直接用C来表示。

and one charge, for example, one charge actually has very small, very small than a coulomb.

比如，一个电荷实际上只有非常非常小的库仑量。

and we know the current is defined by the electric charge. by the time, right?

我们知道，电流是由电荷定义的。是单位时间内的电荷量，对吧？

so if you change the equation a little bit, you know the electric charge or the current.

所以，如果你稍微改变一下这个公式，你就能得到电荷或电流。

it's defined as the quantity of the electricity passing a given.

它的定义是：通过某一给定点的电量。

one coulomb of one ampere.

一安培就是一库仑。

it's maintained for one second.

而且这个电流持续一秒钟。

so q is i times t. so one coulomb is one ampere times one second.

所以，Q等于I乘以T。也就是说，一库仑等于一安培乘以一秒钟。

one ampere times one second, you will get one.

一安培乘以一秒钟，结果就是一。

one ampere times one second, you will get one.

一安培乘以一秒钟，结果就是一。

so coulomb is equivalent to an ampere times a second.

所以，库仑等同于安培乘以秒。

so that's the reason when you buy a battery, it always tells you ampere hours, right?

这就是为什么，当你购买电池时，它总是会标明安培小时数，对吧？

when you buy a battery for anything, there will be a symbol on the battery that tells you how big is the battery. how many electric charge. all stored in the battery. It's used the ampere hours. Ampere hours is exactly current times the time.

当你购买任何电池时，电池上都会有一个标识，告诉你电池的容量大小。也就是它有多少电荷量。全部都储存在电池里。它用安培小时来表示。安培小时其实就是电流乘以时间。

voltage is also quite simple. you know what is current. and voltage is we use a symbol v. to represent the voltage. the unit is volt. you can just simply use the v. for short of volt.

电压也很简单。你知道什么是电流。而电压就是，我们用符号V来表示。用来代表电压。它的单位是伏特。你可以直接用V。这是伏特的缩写。

and voltage is defined as the difference of potential energy. okay, two points.

电压被定义为电势能的差值。好的，两点。

the difference of potential energy between two points of a conductor. carrying one ampere of current when the power is dissipated. it means two points is equal to one volt.

即导体上两点之间的电势能差。当消耗一瓦功率时，通过一安培的电流。也就是说，这两点之间的电压等于一伏特。

So based on this definition, the voltage is the power is dissipated.

所以，根据这个定义，电压就是功率的耗散。

in one watt. one ampere of current.

等于一瓦。一安培的电流。

so voltage is the power divided by the current.

所以，电压等于功率除以电流。

or the power equals to the current times the voltage.

或者，功率等于电流乘以电压。

and the potential difference can also be written as

电势差也可以表示为

because we know the current.

因为我们知道电流。

You know the current is electric current divided by the time.

你知道电流等于电荷量除以时间。

so when you write the formula,

所以当你写出公式时，

power times the time is the current.

功率乘以时间就是电流。

so the voltage, the potential difference can also be written as the energy divided by the electric charge.

那么，电压或电势差也可以表示为能量除以电荷量。

Okay, resistance. so this is the famous Ohm's law. If you know the voltage and you know the current, you can use the Ohm's law. get the resistance.

好的，电阻。这就是著名的欧姆定律。如果你知道电压和电流，就可以使用欧姆定律。从而求出电阻。

Power, in the order to the power, we use the symbol P. It is the symbol P. P is the power, and the unit is watt. And power, the voltage is equivalent to the power divided by the current. So power is equal to the voltage times the current.

功率，我们用符号P来表示。这就是符号P。P代表功率，单位是瓦特。而功率，电压等于功率除以电流。所以功率等于电压乘以电流。

And the efficiency, we already have a derivation here. power is equal to current times voltage. And the current can also be written by electric current divided by time.

至于效率，我们这里已经有一个推导了。功率等于电流乘以电压。而电流也可以表示为电荷除以时间。

And voltage is the energy divided by the electric charge. So you are here.

而电压等于能量除以电荷。所以，你在这里得到了功率。

Power is the energy divided by the time. I think at the high school, you already know energy is the power times the time. So this is the general formula.

功率等于能量除以时间。我想在高中时，你们已经学过能量等于功率乘以时间。所以，这就是通用公式。

okay, now there is a new symbol.

好的，现在出现了一个新的符号。

you probably don't know it. I think

你们可能不认识它。我想，

this will be covered by your mathematic course.

这门课应该会在你们的数学课程中讲到。

but we know the energy, the energy is the power multiplied by the time, right? so this is the power. But sometimes the power is not constant one. power is changing. it's changing with the time. it's not a constant value, right?

但是我们知道，能量就是功率乘以时间，对吧？所以这就是功率。但有时候，功率并不是恒定的。功率是在变化的。它随着时间的推移而变化。它不是一个恒定值，对吧？

if you have a constant value, you know how to calculate the energy.

如果你有一个恒定值，你就知道如何计算能量。

we know how much energy is produced. it's just multiply power. and you can get the energy. but sometimes the power, it's changed. it's changing with the time.

我们知道产生了多少能量。只需要把功率乘以时间。这样就能得到能量。但有时候，功率是变化的。它随时间而变化。

so in this case, how can we get the energy?

那么在这种情况下，我们该如何计算能量呢？

a simple way is

一种简单的方法是，

if the power is discrete, you can divide

如果功率是离散的，你可以将其分解。

it's in regular. into very small pieces of rectangular.

它是规则的。把它分成非常小的部分，也就是矩形。

right? And then we age each rectangle here.

对吧？然后，我们对这里的每个矩形进行标记。

Then we will get the error.

这样我们就能得到卡诺图。

you can get the error. but the curve is continuous. that's the reason we use this symbol.

你可以得到这个错误。但是这个曲线是连续的。这就是为什么我们使用这个符号。

we use this symbol to represent the sum of the irregular rectangles.

我们用这个符号来表示不规则矩形的和。

if the curve is discrete, you can use the sum of each rectangle, right? i equals sum from one to i don't know how many rectangles you have.

如果曲线是离散的，你可以用每个矩形的和，对吧？i等于从1到求和，我不知道有多少个矩形。

these are the areas. become the curve. I mean the curve is continuous. we use this symbol. don't use this sum. anymore.

这些就是面积。就变成了曲线。我的意思是，这条曲线是连续的。我们使用这个符号。不要使用这个求和符号。不再使用了。

and i start from one.

我从1开始。

it's the area.

这就是面积。

it's on the curve.

它在曲线上。

any questions?

好的，有什么问题吗？

any questions?

还有问题吗？

and we know almost know, right? you know the resistance. the resistance equals to the voltage. divided by the current. it's almost know, almost know. actually there are three or actually there are four. three. three different types of resistors. a fixed resistor, fixed resistor, which means the value. which is the resistance is a constant. you can either use this symbol. to represent a fixed resistor. or you use this one. but this one is not fixed anymore. it's obsolete. we don't use this symbol for fixed resistor anymore. so this symbol represent the resistor. represent an resistor.

我们几乎都知道了，对吧？你知道电阻。电阻等于电压。除以电流。这几乎都知道了，几乎都知道了。实际上，有三个，或者说是四个。三个。三种不同类型的电阻。固定电阻，也就是说，它的值是固定的。也就是说，电阻是一个常数。你可以使用这个符号。来表示固定电阻。或者使用这个。但是这个就不再是固定的了。它已经过时了。我们不再用这个符号来表示固定电阻了。所以，这个符号代表电阻。代表一个电阻。

but sometimes the resistor can be changed. it's defined as variable. variable resistor, variable resistor. and we use this symbol to represent the variable resistor. which means it's adjustable. it's adjustable. You can switch the resistance.

但是有时候，电阻是可以改变的。这被称为可变电阻。可变电阻，可变电阻。我们用这个符号来表示可变电阻。这意味着它是可以调节的。它是可调的。你可以改变电阻值。

and the last type is a potentiometer.

最后一种类型是电位器。

which means, for example, the resistance can be adjusted, based on the heat, or based on the light. or based on humidity. of course, or chemical activity.

也就是说，例如，电阻可以调节，这取决于温度，或者取决于光线。或者取决于湿度。当然，还有化学活性。

so for example, for this one, potentiometer, which means if you apply different focus, if you press the resistor, it will give you a resistance. but if you press harder, it will give you a different resistance.

比如这个电位器，意思是如果你施加不同的电压，如果你按下电阻，它会产生一个电阻值。但如果你按得更用力，它产生的电阻值也会不同。

The resistance is depending on external conditions. It can be either heat or light. or focus.

电阻取决于外部条件。可能是热量，也可能是光线。或者是电压。

The resistance is always consuming. consuming power.

电阻总是会消耗能量。消耗功率。

which means the power will be dissipated.

这意味着功率会被耗散掉。

and you know that the resistor R here. and then you know the current. there will be a voltage drop.

大家知道，这里的电阻是R。然后你知道电流。就会产生电压降。

i times r. it's the voltage drop.

就是I乘以R。这就是电压降。

i times r. it's the voltage drop.

就是I乘以R。这就是电压降。

so it consumes the power. if the original voltage source is 100 volt. and this resistor consume 30 volt. and you will only get 70. so resistor were consuming power.

所以它会消耗功率。如果原来的电压源是100伏特。而这个电阻消耗了30伏特。那么你只能得到70伏特。所以，电阻消耗了功率。

so we see the lack of voltage power.

所以我们看到了电压功率的不足。

so we see the lack of voltage power.

所以我们看到了电压功率的不足。

and 1 kilowatt hour, we have a a new ignition policy. which is defined as conductance.

而1千瓦时，我们有一个新的点火策略。它被定义为电导。

1 kilowatt hour is defined as conductance. use the symbol g to represent conductance.

1千瓦时被定义为电导。用符号g来表示电导。

So the current can also be written by g. times v. Because g is the conductance. 1 kilowatt hour is defined as conductance.

因此，电流也可以表示为g。乘以v。因为g是电导。1千瓦时被定义为电导。

but sometimes the resistor is not just a constant value. So depending on how long the conductor is, how long we probably know if the conductor is quite long, it will consume a lot. so the resistance is dependent. it's dependent on the length of the conductor. and also dependent on the cross-section area. A.

但是，有时候电阻不仅仅是一个恒定值。所以，这取决于导体的长度。我们大概知道，如果导体很长，它会消耗很多能量。所以，电阻是依赖于这个的。它取决于导体的长度。同时也取决于横截面积。A。

and rho is the resistivity.

而ρ代表电阻率。

So the longer the conductor is, the higher

所以，导体越长，电阻就越大。

the resistance.

也就是电阻越大。

And also the cross section.

还有横截面积。

and based on the law. defining the resistance.

这也是基于定律的。它定义了电阻。

and the rho here is the resistivity, right? rho is the resistivity. Resistivity is a constant value. so it's it's already given. so you don't have to remember it. All the different conductors will have some fixed value of resistivity.

这里的ρ就是电阻率，对吧？ρ是电阻率。电阻率是一个常数。所以，它是一个恒定值。它已经给出了。所以你不需要记住它。不同的导体都有固定的电阻率值。

And you can look at the table and get the value of the resistivity.

你可以查表得到电阻率的数值。

And here you can see for the wood,

在这里你可以看到，对于木材来说，

it has a very large resistivity. And the power of eight.

它的电阻率非常大。是10的8次方。

and the ten to the power of eleven.

还有10的11次方。

So this is the resistivity of wood. That's the reason wood is not conductive.

这就是木材的电阻率。这就是为什么木材不导电的原因。

Wood is not conductive.

木材不导电。

so we know the resistance. It depends on resistivity. the length of the wire. and also the cross section area. But sometimes, the resistance also

所以我们知道电阻。它取决于电阻率。还有导线的长度。以及横截面积。但是，有时候电阻也

depends on the temperature. temperature of the conductor.

取决于温度。也就是导体的温度。

And the value of the resistivity, together on the previous slide.

而电阻率的数值，在之前的幻灯片上已经给出了。

So it's at a current at zero degree. so here is the resistivity, right? and this is the value. at zero degree. But if the temperature have changed,

所以这是在0度时的值。这里就是电阻率，对吧？这就是数值。在0度时。但是，如果温度发生了变化，

the resistance will be changed.

电阻也会改变。

for a change that way. and we have another variable which is defined as temperature coefficient. this alpha coefficient. It represents how big is the change of the resistance. because of the temperature change.

就是这样变化的。我们还有另一个变量，叫做温度系数。就是α系数。它表示电阻变化的幅度。这是由于温度变化引起的。

And if the resistance is at zero degree, and we use alpha zero.

如果电阻在零度时，我们使用α0。

alpha zero to refer the temperature coefficient.

用来表示温度系数。

Thank you.

谢谢。

Okay. The conversion is 10 minutes.

好的。转换时间是10分钟。

How can you get the current resistance? Just using the equation.

那么，你怎么得到当前的电阻呢？只要用这个公式就可以了。

R at 80 degree. equals to R0.

80度时的电阻R。等于R0。

times 1 plus alpha 0. multiplied by my delta theta. delta theta is 70 minus 0, right? So this is the delta theta.

乘以1加上阿尔法0。再乘以温度变化量Δθ。Δθ就是70减0，对吧？所以这就是Δθ。

This is the change of the temperature. So once we know the change of the temperature, and you know the temperature coefficient,

这是温度的变化。那么，一旦我们知道了温度的变化，也知道了温度系数，

and the resistance at the zero degree, you will get the corresponding resistance at the current temperature.

还有0度时的电阻值，就能算出当前温度下的电阻值。

And the temperature coefficient is also given. It is also given here. alpha zero is the temperature coefficient. at the temperature zero degree.

温度系数也已经给出了，就在这里。α0就是温度系数。即在0度时的温度系数。

so that's the reason we use alpha zero. so here the temperature is 20 degree, you will get a

所以这就是我们使用α0的原因。这里温度是20度，你会得到一个

okay, here is something different. and we have used zero. so that's the difference. if we use 20 degree as a reference, so we should change the equation.

不同的结果。我们这里用的是0。这就是区别所在。如果我们以20度作为参考，那么我们就需要改变公式。

because now we use 20 degree as a reference. instead of zero.

因为现在我们用的是20度作为参考。而不是0度。

so if you use a zero degree as a reference, you can get the

所以，如果你用0度作为参考，你就能得到

you can use this equation. to get the resistance and the ambient temperature. but sometimes the resistance at zero degree is unknown. but it only gives you a good example. the temperature coefficient alpha 20 degree

这个公式。从而得到电阻和环境温度。但有时候，0度时的电阻值是未知的。不过，这只是给你一个很好的例子。温度系数α在20度时，

that's the value.

就是这个值。

so if this kind of value is given,

所以，如果给出了这样的数值，

similarly, you can use the same equation to compute how big is the resistance at 30 degree.

同样，你也可以用同样的公式来计算30度时的电阻值。

similarly, you can use the same equation to compute how big is the resistance at 30 degree.

同样，你也可以用同样的公式来计算30度时的电阻值。

and now you can use that R 30 degree

现在，你可以用R30度

to

来

to

来

now the theta is 50. minus 20. it's the theta set up.

计算。这里的θ是50。减去20。这就是设定的θ值。

now the theta is 50. minus 20. it's the theta set up.

计算。这里的θ是50。减去20。这就是设定的θ值。

theta set up is 50 minus 20.

θ设定为50减20。

but the resistor is actually the resistor is actually quite small, right?

但是，这个电阻实际上非常小，对吧？

so we cannot directly write down. the resistance of each resistor. Instead, we use symbols. we use the counter function.

所以我们不能直接把它写出来。每个电阻的阻值。相反，我们使用符号来表示。我们使用计数函数。

to represent how big the resistance is.

用来表示阻值的大小。

there's something missing here. this resistor has four color bands. and there's some definition here. you can know. it's actually four bands yellow. kilo ohms.

这里好像少了点什么。这个电阻有四个色环。这里有一些定义。你可以知道。实际上，是四个黄色的色环。千欧姆。

and for this one, it has five color bands, right? so we can get 68 kilo ohms. and there's a small tolerance. which means that it can either be slightly higher or slightly lower. but the range is within 5 percent. and for this one, the range is within 10 percent.

而这个有五个色环，对吧？所以我们可以得到68千欧姆。还有一个小的容差。这意味着，它的阻值可能是稍微高一点或稍微低一点。但误差范围在5%以内。而对于这个，误差范围是10%。

so when you have a resistor, it will be shown like this. you can see different colors. and based on the definition or based on the table here,

所以，当你有一个电阻时，它会显示成这样。你可以看到不同的颜色。根据这里的定义或表格，

you can finally determine how big actually the resistance is.

你最终可以确定电阻的实际值是多少。

for example, the first digit, the first digit, we've written the value.

例如，第一位数字，我们已经写好了它的数值。

for example, the first digit, the first digit, we've written the value.

例如，第一位数字，我们已经写好了它的数值。

and for example, for the left one, we only have two digits. and for the first one is the yellow one. yellow is actually four, right? yellow color means it's four.

再比如，左边的这个，我们只有两位数字。第一个是黄色的。黄色实际上代表四，对吧？黄色表示四。

and okay, this is the purple one. Let's find the purple, right? and this is seven, right?

好的，这是紫色的。那我们找一下紫色的，对吧？这个是七，对吧？

and there's also a multiplier. Multiplier is the third part.

还有一个乘数。乘数是第三个部分。

and the color actually is the green.

而颜色实际上是绿色。

the first part of them is four, right? For the resistor at the left side. and the second part is seven. and the multiplier is the third one. it is ten. the power of ten.

左边的电阻，第一部分代表四，对吧？第二部分是七。乘数是第三个。它是十。也就是10的幂。

the first part of them is four, right? For the resistor at the left side. and the second part is seven. and the multiplier is the third one. it is ten. the power of ten.

左边的电阻，第一部分代表四，对吧？第二部分是七。乘数是第三个。它是十。也就是10的幂。

the first part of them is four, right? For the resistor at the left side. and the second part is seven. and the multiplier is the third one. it is ten. the power of ten.

左边的电阻，第一部分代表四，对吧？第二部分是七。乘数是第三个。它是十。也就是10的幂。

so that's the reason why it is seven times 0.7.

所以这就是为什么它是7乘以0.7的原因。

times 10 to the power of three ohm.

10的3次方欧姆。

and the last one is the tolerance.

最后一个部分是容差。

the last one is the tolerance. The silver is in 10% or 5%. the color is yellow, yeah? it's gray, right? Gray is exactly 10%.

也就是容差。银色的代表10%或5%。颜色是黄色的，对吧？是灰色的，对吗？灰色正好是10%。

the actual resistance is within 10% of 4.7 kilo ohm.

所以实际电阻值会在4.7千欧姆的上下10%范围内。

is it clear?

清楚了吗？

no.

没有。

so in a real resistor, it will not tell you how big or how large is the resistor. so you have to determine the resistance by yourself. based on the color. based on the color bands. for example, the left one. for the left one, we have four bands, right? Four color. we have yellow. then yellow, you have purple. then we have gray.

那么，在一个实际的电阻中，它不会告诉你电阻的具体阻值是多少。所以，你必须自己测量电阻值。根据颜色来判断。也就是根据色环。比如，左边这个。左边这个有四个色环，对吧？一个是黄色。然后是黄色，还有紫色。接着是灰色。

the first, this is the first digits, the second digits.

第一个，这是第一位数字，第二位数字。

this is the actual value, the actual value of the resistor.

这就是电阻的实际阻值。

and for the first digit, the color is yellow, right? yellow is exactly four.

对于第一位数字，颜色是黄色，对吧？黄色代表的数值就是4。

four has the yellow color. so this is the reason the first digit is four. and the second band is purple. when you look at the second table, the purple is seven, right? the second color is seven.

4对应的颜色就是黄色。所以，这就是为什么第一位数字是4的原因。第二位色环是紫色。大家看第二张表格，紫色代表7，对吧？第二位数字就是7。

so you will get seven here.

所以这里得到的是7。

and the third band, the third one is red. right?

第三位色环，是红色。对吗？

so red is the multiplier.

红色代表乘数。

it's exactly 10 to the power of two.

也就是10的2次方。

and the last band, the last color band, it represents the tolerance. it's exactly 10%.

最后一位色环代表误差范围。正好是10%。

so similar. i'm not quite sure whether you have seen the resistor in the real life. in our daily life. where do you have seen a real resistor?

差不多就是这样。我不太确定大家有没有在现实生活中见过电阻。在我们的日常生活中。你们在哪里见过真正的电阻呢？

it's always a powder like this. there are different powders on the resistor. different colors, different value, determine actually how to how much the resistance is.

它总是像这样的粉末状。电阻上有不同的粉末。不同的颜色代表不同的阻值，决定了如何确定电阻的大小。

so this is one kind of coding system for the resistor. but there's another

所以，这是一种电阻的编码系统。不过，还有另一种。

What is resistance? And here we have a very simple solution. You have water resource, right? We have water resource plus three resistors. And because the resistors are connected in series. so the current is the same for all resistors.

什么是阻力？这里我们有一个非常简单的解决方案。你们有水资源，对吧？我们有水资源，再加上三个电阻。而且这些电阻是串联连接的。所以，电流在所有电阻中都是相同的。

and the voltage, if you add the voltage for each resistor together, you will get the total voltage.

至于电压，如果把每个电阻上的电压加起来，就能得到总电压。

so the total voltage also equals the sum of the voltage across each element. in the circuit.

所以，总电压等于每个元件两端电压之和。在电路中。

across each element in the circuit. V1, sorry.

也就是电路中的每个元件。V1，抱歉。

and the voltage for each resistor. And if you add them together, you will get the total voltage.

就是每个电阻上的电压。如果你把它们加起来，就能得到总电压。

okay, now a different circuit. the resistors are connected in parallel. so in this case, they have the same voltage. every resistor will have the same voltage. so v1 equals to v2, equals to v3. and equals to the voltage source.

好的，现在来看另一个电路。这些电阻是并联连接的。所以在这种情况下，它们两端的电压是相同的。每个电阻上的电压都是一样的。所以V1等于V2，也等于V3。而且等于电源电压。

but the total current, the total current equals the sum of each

但是，总电流等于各个支路电流之和。

the total current equals the sum of the current in each branch. i will go through this quickly because i think you are familiar with this.

总电流等于每个支路电流的总和。我会快速过一遍，因为我想大家对此已经很熟悉了。

they are familiar with the parallel circuit and the series circuit.

大家应该对并联电路和串联电路都很熟悉。

and if he did a lot of researches here, like this one, we have r1, r2, r3. and you have so many resistors. and how can we calculate the total voltage? the total voltage is the sum of the voltage across each resistor. so you just add them. you add v1, v2, v3 until vn. you will get the total voltage.

如果他在这里做了大量的研究，比如这个，我们有R1、R2和R3。还有非常多的电阻。那么我们该如何计算总电压呢？总电压等于每个电阻两端的电压之和。所以你只需要把它们加起来。把V1、V2、V3一直加到Vn。这样就能得到总电压了。

so if the resistors are connected in series, series combination. the total resistance, you just add them. add each resistance together. because they are connected in series.

所以，如果电阻是串联连接的，就是串联组合。总电阻就是把它们加起来。把每个电阻的值相加。因为它们是以串联方式连接的。

okay, parallel. the current is the same. it's the sum of the currents in each branch, right? but they have the same voltage. each resistor have the same voltage.

好的，接下来是并联。电流是相同的。总电流等于各个支路电流之和，对吧？而且它们的电压是相同的。每个电阻两端的电压都一样。

so based on all this law, you are getting this equation. finally, you can compute. you can compute the total resistance. and this equation, I think, is also taught in high school.

所以，根据所有这些定律，你就能得到这个公式。最后，你就可以进行计算了。你可以计算出总电阻。我认为这个公式，也在高中里学过。

the total resistance. and the conductance, conductance is actually the inverse. and conductance is the inverse resistance.

总电阻。而电导，其实就是它的倒数。电导就是电阻的倒数。

so you can also use the resistance or use the conductance. to get the final total conductance.

所以，你也可以用电阻，或者用电导来计算。得出最终结果。总电导。

okay, that is part h e q e t.

好的，这就是关于H、E、Q、T的部分。

i think we will talk further tomorrow.

我想我们明天会再详细讨论。

The current in the orange color is flowing through the Karnaugh map. They are produced for each possible combination. At the Karnaugh map, we work on the possible combinations. For example, if there is a current in this Karnaugh map and also a current in this Karnaugh map, based on the current moving in the same direction, they are attacking each other.

橙色电流流经卡诺图。它们针对每种可能的组合产生。在卡诺图上，我们处理各种可能的组合。例如，如果这张卡诺图中有一股电流，而这张卡诺图中也有一股电流，那么基于电流沿同一方向移动，它们会相互攻击。

The current will produce a magnetic field. You can use the right-hand rule to determine how the magnetic field looks. The direction of the current determines whether the magnetic field goes inside or outside. This is the magnetic field. Given the current here, you can determine the force. The current is here, and you have the magnetic field.

电流会产生磁场。你可以使用右手定则来判断磁场的方向。电流的方向决定了磁场是向里还是向外。这就是磁场。已知此处电流方向，你就可以确定力的方向。电流在这里，同时你有了磁场。

We know the middle one is charge. People use a symbol to represent the charge. Here is a symbol of charge, and the unit is—

我们知道中间那个是电荷。人们用一个符号来表示电荷。这是电荷的符号，其单位是——

One charge actually has a much larger contribution. The current is defined by…

一个电荷实际上贡献大得多。电流的定义是……

Electric charge is defined in terms of time. If we consider the definition of current, it is defined as the quantity of electricity passing through a point per unit time. Specifically, one coulomb Q is the charge transported when a current of one ampere is maintained for one second.

电荷是以时间来定义的。如果我们考虑电流的定义，它被定义为通过某一点的电量与单位时间的比值。具体而言，一库仑 Q 就是当一安培的电流维持一秒时所传输的电荷量。

In SI base units, one coulomb equals one ampere times one second. So coulomb is equivalent to ampere-second. That is the reason why, when you buy a battery for an IC, it always tells you ampere-hours.

在国际单位制基本单位中，一库仑等于一安培乘以一秒。因此库仑等价于安培秒。这就是为什么当你购买用于集成电路的电池时，它总是以安培小时来标注。

There will be a symbol for a battery that tells you how many electric charges are stored in a battery. The unit used is ampere-hours. Ampere-hours is exactly current times time. Voltage is also quite simple. We use the symbol V to represent voltage, and the unit is volt. You can just use V for short of volt.

电池有一个符号，告诉你电池中储存了多少电荷。使用的单位是安培小时。安培小时就是电流乘以时间。电压也很简单。我们用符号V表示电压，单位是伏特。可以直接用V作为伏特的缩写。

It is defined as the difference of potential energy between two points of a conductor carrying one ampere of current when the power dissipated between these two points is equal to one watt.

它定义为：当载有一安培电流的导体上两点之间消耗的功率等于一瓦特时，这两点之间的电位能之差。

Power equals voltage times current. P = V × I. The potential difference can also be written as voltage.

功率等于电压乘以电流，即 P = V × I。电势差也可以写成电压的形式。

Power is electric current multiplied by voltage. When you multiply power by time, that gives you energy.

功率等于电流乘以电压。将功率乘以时间，就得到能量。

Power times time is energy. The voltage, or potential difference, can also be written as the energy divided by the electric charge. Now for resistance — this is the famous Ohm's law. If you know the voltage and you know the current, you can use Ohm's law to get the resistance.

功率乘以时间等于能量。电压，即电势差，也可以写成能量除以电荷量。接下来是电阻——这就是著名的欧姆定律。如果你知道电压，又知道电流，就可以用欧姆定律求出电阻。

For power, we use the symbol P. The unit of power is the watt — this is the standard unit, so to simplify, the unit is watt. Voltage is equivalent to power divided by current. Therefore, power is equal to voltage times current. The unit is watt.

对于功率，我们使用符号 P。功率的单位是瓦特——这是标准单位，因此简化来说，单位就是瓦特。电压等于功率除以电流。因此，功率等于电压乘以电流。单位是瓦特。

We already have the equations. How are you supposed to find voltage? And how could we use Karnaugh maps to find at which point?

我们已经有了方程。那电压应该怎么求？我们如何用卡诺图来确定在哪个点？

Power is energy divided by time. As a customer, you already know that energy equals power multiplied by time. So this is an example. Now there is a new symbol—probably you don't know it yet.

功率等于能量除以时间。作为用户，你们已经知道能量等于功率乘以时间。所以这是一个例子。现在有一个新符号——你们可能还不认识它。

This will be covered by your mathematical course. We know that energy is power multiplied by time. However, sometimes the power is not a constant value; it changes with time. If you have a constant value—for example, 100 watts—you know exactly how much energy is delivered.

这部分内容将在你们的数学课程中讲到。我们知道，能量等于功率乘以时间。然而，有时功率并不是一个恒定值，它会随时间变化。如果功率是恒定的——例如100瓦——你就能确切知道输出了多少能量。

You know how much energy you use for a given time. If you have a constant power at a time, you can simply multiply to get the energy. But sometimes the power is changing with time. So how can you get the energy?

你知道在给定时间内消耗了多少能量。如果功率在某时刻是恒定的，你可以直接相乘得到能量。但有时功率随时间变化。那么，如何求得能量呢？

A simple way is: if the power is discrete, we can divide this irregular shape into very small rectangular pieces.

一种简单的方法是：如果功率是离散的，我们可以将这个不规则形状分割成非常小的矩形块。

In a Karnaugh map, each valid grouping must form a rectangle. For instance, if the product term PA corresponds to a rectangle, the grouping is correct. However, if the shape is not a rectangle—for example, a curve—you will get an error. A curve is continuous rather than discrete, which is precisely why we use the rectangular symbol to denote valid groupings.

在卡诺图中，每个有效的分组必须构成一个矩形。例如，如果乘积项 PA 对应一个矩形，则该分组是正确的。然而，如果形状不是矩形——比如一条曲线——就会产生错误。曲线是连续的而非离散的，这正是我们使用矩形符号来表示有效分组的原因。

The sum of the rectangles gives you the area. If the curve is discrete, you use the sum of each rectangle: i equals the summation from 1 to n, where n is however many rectangles you have. This gives you the area under the curve. However, if the curve is continuous, you use a different symbol. You do not use the summation symbol; you use the integral symbol.

矩形面积之和即为所求面积。如果曲线是离散的，则对每个矩形求和：i 等于从 1 到 n 的求和，其中 n 为矩形的个数，由此得到曲线下方的面积。然而，如果曲线是连续的，则需要使用另一个符号——不是求和符号，而是积分符号。

This is an i-star for one. This area—you don't have a problem. Any questions?

这是一个 i 星对应 1。这个区域——没有问题。有什么问题吗？

Resistance. Resistance.

电阻。电阻。

Resistance. Resistance.

电阻。电阻。

Ohm's law, right? So Ohm's law involves the resistance. The resistance equals the voltage divided by the current. This is Ohm's law. And actually there are three types of resistors. A fixed resistor means the value, which is the resistance, is a constant. The resistance is a constant value. If I use this symbol, this symbol represents a fixed resistor. Or I use this one, but this one is not used anymore. It's obsolete. We don't use this symbol for fixed resistor anymore. We use the symbol above. So this symbol represents a resistor, represents a resistance. And sometimes the resistance can be changed.

欧姆定律，对吧？欧姆定律涉及电阻。电阻等于电压除以电流。这就是欧姆定律。实际上电阻器有三种类型。固定电阻器的含义是，其阻值是一个常数，即电阻值是恒定的。如果我使用这个符号，它代表一个固定电阻器。或者我用这个，但这个已经不再使用了，它已经过时了。我们不再用这个符号表示固定电阻器，而是用上面那个符号。所以这个符号代表电阻器，代表电阻。而有时候电阻是可以改变的。

This symbol is defined as a variable resistor. It represents a variable resistor, which means it is adjustable—you can change the resistance. The last type is a potentiometer.

这个符号定义为可变电阻器。它代表可变电阻器，意味着它是可调节的——你可以改变电阻值。最后一种类型是电位器。

This means, for example, the resistance can be adjusted based on heat, based on light, or based on humidity, or force, or chemical activity. For example, for this one, potentiometer—this means if you apply different force, if you press the resistor, it will give you a resistance. But if you press harder, it will give you a different resistance.

这意味着，例如，电阻可以根据热量、光照、湿度、力或化学反应进行调节。以电位器为例——如果你施加不同的力，按压电阻器，它会给出一个电阻值；但如果你按得更用力，它就会给出一个不同的电阻值。

It depends on external conditions. It can be either heat, light, or force, etc. A resistor is always consuming power, which means the power is dissipated by the resistor.

它取决于外部条件，可以是热、光或力等。电阻器始终在消耗功率，也就是说功率被电阻器耗散。

If you have a resistor R here and you know the current, then the voltage drop is I times R. This voltage drop means it consumes power. If the original voltage source is 100 volts and this resistor consumes 30 volts, then you only get 70 volts as output. So resistors consume power.

如果这里有一个电阻R，并且你知道电流，那么电压降就是I乘以R。这个电压降意味着它消耗了功率。如果原始电压源是100伏，而这个电阻消耗了30伏，那么你只能得到70伏的输出。所以电阻会消耗功率。

If you have a resistor R here and you know the current, then the voltage drop is I times R. This voltage drop means it consumes power. If the original voltage source is 100 volts and this resistor consumes 30 volts, then you only get 70 volts as output. So resistors consume power.

如果这里有一个电阻R，并且你知道电流，那么电压降就是I乘以R。这个电压降意味着它消耗了功率。如果原始电压源是100伏，而这个电阻消耗了30伏，那么你只能得到70伏的输出。所以电阻会消耗功率。

We see the result of one condition part. This part is written as one divided by R. For one divided by R, we have a new condition, which is defined as conductance.

我们可以看到其中一个条件项的结果。该项写为R分之一。对于R分之一，我们有一个新的条件，定义为电导。

One divided by R is defined as conductance. We use the symbol G to represent conductance. So the Karnaugh map can also be written by G. The Karnaugh map can also be written by G times V, because G is the conductance—G is one divided by R.

一除以R定义为电导。我们用符号G表示电导。因此，卡诺图也可以用G来书写。卡诺图还可以用G乘以V来书写，因为G就是电导——G等于一除以R。

This is C, common for all. The resistor is not a constant value; it depends on how long the conductor is. If the conductor is quite long, it will consume a lot of power. So the resistance depends on the length of the conductor, and also depends on the cross-section area A. And rho is...

这是C，对所有情况通用。电阻器的值不是常数，它取决于导体的长度。如果导体很长，它会消耗大量功率。因此，电阻取决于导体的长度，也取决于横截面积A。而ρ是……

Resistivity. The longer the conductor is, the higher the resistance. The longer the wire, the higher the resistance. And also the cross section.

电阻率。导体越长，电阻越大。导线越长，电阻越大。还有横截面积。

The rho here is the resistivity. Resistivity is a constant value. So resistivity, rho, is a constant value. It is already given, so you don't have to memorize it. All the different conductors will have some fixed value of resistivity.

这里的 ρ 是电阻率。电阻率是一个常数值。所以电阻率 ρ 是一个常数，它已经给出了，你不需要去记忆它。所有不同的导体都会有某个固定的电阻率值。

A Karnaugh map does not have a table. You get the value of the row, get the value of the region. Here you can see that the group of four covers a very large region. The power of eight...

卡诺图没有表格形式。你获取行的值，获取区域的值。在这里你可以看到，四个一组的覆盖区域非常大。八的幂次……

The reason water is not a conductor lies in its resistivity. Resistance depends on resistivity, the length of the conductor, and the cross-sectional area. But sometimes, resistance also...

水不是导体的原因在于其电阻率。电阻取决于电阻率、导体长度和横截面积。但有时，电阻还……

The temperature of the Karnaugh map. The value of the resistivity — we had that on the previous slide. It is correct at zero degrees. So here is the resistivity, and these are the values at zero degrees. But if the temperature has changed…

卡诺图的温度。电阻率的值——我们在上一张幻灯片中已经讲过。它在零度时是正确的。所以这里是电阻率，这些是零度时的值。但如果温度发生了变化……

Resistance changes due to temperature. We define another variable called the coefficient of temperature, denoted by alpha. Alpha represents how much the resistance changes because of a temperature change.

电阻会因温度而变化。我们定义另一个变量，称为温度系数，用α表示。α代表电阻因温度变化而改变的程度。

At 70 degrees, we have the temperature is 70 degrees. How can we get the current resistance using the equation? R at 70 degrees equals R zero times one plus alpha zero multiplied by delta theta. Delta theta is 70 minus zero.

在70度时，温度为70度。我们如何利用公式求得当前电阻？R在70度时等于R零乘以一加阿尔法零乘以德尔塔西塔。德尔塔西塔等于70减零。

This is the change of temperature, delta theta. Once you know the change of temperature and the temperature coefficient, and the resistance at zero degrees, R at zero degrees equals one kilo-ohm, you can determine the resistance at a different temperature.

这是温度的变化量，即 delta theta。一旦你知道了温度的变化量和温度系数，以及零度时的电阻——R 在零度时等于一千欧姆——你就可以求出不同温度下的电阻值。

Alpha zero is the concentration coefficient at temperature zero degrees — that is the reason they use the notation alpha zero. Here the temperature is twenty degrees. In this case, we use twenty degrees as the reference instead of zero degrees, so we need to change the equation correspondingly.

阿尔法零是温度为零度时的浓度系数——这就是使用符号阿尔法零的原因。这里温度为二十度。在这种情况下，我们以二十度作为参考而非零度，因此需要相应地修改方程。

If you use zero degree as reference, you can use this equation to obtain the resistance and component temperature. However, sometimes the resistance at zero degree is unknown. It only gives you, for example, the temperature coefficient at twenty degrees, that is, R at twenty degrees. So if you use this—

如果你以零度作为参考，你可以利用这个方程来求得电阻和元件温度。然而，有时零度下的电阻是未知的。它只会给你，例如，二十度时的温度系数，也就是 R 在二十度时的值。所以如果你使用这个——

If this value is given, similarly you can use the same equation to compute the output of the resistance at 30 degrees. Now you can use R at 50 degrees or 20 degrees. The delta theta is 50 minus 20.

如果已知该值，类似地，你可以使用同一方程计算电阻在30度时的输出。现在你可以使用50度或20度时的R值。delta theta等于50减去20。

But the resistor is actually...

但电阻实际上是……

You cannot directly write to the physical address. Instead, we use CPOS. We use the counter-fund to represent how big the memory system is. Here is an example.

你不能直接向物理地址写入数据。取而代之的是，我们使用CPOS。我们用计数器来表示内存系统的大小。下面是一个例子。

This resistor has four color bands. Based on the definition here, we can know it is actually 4.7 kilo-ohms. And for this one, it has five color bands, so we can get 68 kilo-ohms. And then there is a small tolerance. This means it can either be slightly higher or slightly lower. The range is within 5 percent for this one, and for this one, the range is within 10 percent. So when you have a resistor, it will show like this.

这个电阻有四个色环。根据这里的定义，我们可以知道它实际上是4.7千欧。而这个有五个色环，所以可以得到68千欧。然后还有一个小的容差。这意味着它可能略高或略低。这个的范围在5%以内，而这个的范围在10%以内。所以当你有一个电阻时，它会显示成这样。

You can see different colors on the resistor. Based on the definition of Ohm's table, you can finally determine how big the resistance is. For example, the first digit represents the value. For the second one, we only have two digits. The first line is yellow — yellow is actually 4, right? Yellow color means 4.

你可以在电阻上看到不同的颜色。根据欧姆表的定义，你最终可以确定电阻值有多大。例如，第一个数字代表数值。对于第二个例子，我们只有两个数字。第一行是黄色——黄色实际上代表4，对吧？黄色意味着4。

This is defined as proper. There is also a much higher value — the third power. The power is three.

这被定义为 proper。还有一个更高的值——三次幂。幂次为三。

The first band gives the first significant digit of the resistance. The second band gives the second significant digit. The third band is the multiplier, which is ten to the power of a number. So that is the reason why the value is 4.7 times ten to the power of three ohms. The last band indicates the tolerance. The tolerance is either ten percent or five percent. The color is yellow.

第一环给出电阻值的第一位有效数字。第二环给出第二位有效数字。第三环是倍率，即十的某次幂。这就是为什么该电阻值为4.7乘以十的三次方欧姆。最后一环表示容差。容差为百分之十或百分之五。颜色为黄色。

For an exact component, the tolerance is less than ten percent, but the actual resistance is within ten percent of 4.7 kΩ. Is it clear?

对于精确元件，其容差小于百分之十，但实际电阻值在4.7千欧的百分之十以内。清楚了吗？

Your resistor will not tell you how large the resistance is. You have to determine the resistance by yourself, based on the color bands. For example, for the left one, you have four bands, four colors: yellow, purple, gray, white. The first band gives you the first digit, the second band gives you the second digit. This is the actual value of the resistance.

电阻器不会告诉你阻值有多大。你必须根据色环自行确定电阻值。例如，左边这个有四条色环：黄色、紫色、灰色、白色。第一条色环给出第一位数字，第二条色环给出第二位数字。这就是电阻的实际值。

After reading success. For the first digit, the color is yellow. Yellow is exactly four, so the first digit is four. The second bound is purple. When you look at the second table, purple is seven, so you get seven here. The third bound is red. Red is a multiplier, exactly ten to the power of two. And the last bound is gray.

读取成功后，第一位数字对应的颜色是黄色，黄色恰好代表四，因此第一位数字是四。第二个边界是紫色，查看第二张表可知紫色对应七，所以这里得到七。第三个边界是红色，红色是一个乘数，恰好是十的二次方。最后一个边界是灰色。

Tolerance for race and exactly the opposite. I will talk about what we have seen in our daily life, what we have realized—it always covers...

关于种族宽容以及与之完全相反的态度。我将谈谈我们在日常生活中所见到的、所意识到的——它总是涵盖……

Like this, they are different colors on the resistor. Based on colors, you can determine actually how much the resistance is. This is one kind of coding system for the resistor. But there's another coding method.

像这样，电阻器上有不同的颜色。根据颜色，你可以确定电阻值是多少。这是电阻器的一种编码系统。但还有另一种编码方法。

This is called SMD—Surface Mount Device. SMD resistor and capacitor. Sometimes the chip is very small. The smallest chip is 3.5 nanometers. I am not quite sure what the smallest scale of our process is.

这叫做SMD——表面贴装器件。SMD电阻和电容。有时芯片非常小。最小的芯片是3.5纳米。我不太确定我们工艺的最小尺度是多少。

The chip is very small, the size of a small mark. But if we have a chip, we also have a resistor on the chip. In this case, we use a different coding system. For example, we just write down the code 01 on the resistor, and based on the table, we know how big the resistance is. Then we have a new frame.

芯片非常小，只有一个小标记那么大。但如果我们有芯片，芯片上也有电阻。在这种情况下，我们使用不同的编码系统。例如，我们只需在电阻上写下代码01，然后根据表格就能知道电阻值有多大。接着我们就有了一个新的框架。

Voltage is divided among various resistors. Here we have a very simple circuit: a 40-volt source plus three resistors. Because the resistors are connected in series, the current is the same for all the resistors. As for the voltage, if you add the voltage for—

电压分配在各个电阻之间。这里我们有一个非常简单的电路：一个40伏电源加上三个电阻。由于电阻是串联连接的，流过所有电阻的电流相同。至于电压，如果你把各电阻上的电压相加——

The total voltage equals the sum of voltages across each element in the circuit, V₁.

总电压等于电路中各元件两端电压之和，即 V₁。

Now, a different subject: resistors connected in parallel. In this case, every resistor has the same voltage. So V₁ equals V₂ equals V₃, and equals the voltage source.

现在换一个主题：电阻并联的情况。在这种情况下，每个电阻上的电压相同。因此，V₁等于V₂等于V₃，且等于电源电压。

The total current equals the sum of the current in each branch. Now consider a parallel circuit versus a series circuit. If you add a large number of resistors here—R1, R2, R3, and so on—how do you compute the total voltage? The total voltage is the sum of the voltage across each resistor. You simply add them together.

总电流等于各支路电流之和。现在考虑并联电路与串联电路的区别。如果你在此处加入大量电阻——R1、R2、R3，依此类推——如何计算总电压？总电压等于每个电阻两端电压之和，只需将它们相加即可。

If the resistors are connected in series, the total resistance is simply the sum of each individual resistance, because they are connected in series. In parallel, the total current is the sum of the currents in each branch, but each resistor has the same voltage across it.

如果电阻串联连接，总电阻等于各电阻之和，因为它们是以串联方式连接的。在并联电路中，总电流等于各支路电流之和，但每个电阻两端的电压相同。

Based on this equation, you can compute the total resistance. This equation is also called total resistance. We know that conductance is actually the inverse of resistance.

根据这个方程，你可以计算总电阻。这个方程也被称为总电阻。我们知道电导实际上是电阻的倒数。

The last slide.

最后一张幻灯片。

S is related to energy and mass. If we consider anti-mass—

S与能量和质量相关。如果我们考虑反质量——