Oh, thanks. How can we get the maximum y?

哦，谢谢。我们怎样才能获得最大的y值？

y is the power.

y是幂。

a and b of x1. so the process is to compute the first order. first order derivation. so the maximum power is achieved. when the first order derivation is zero. and get the maximum.

x1的a和b。所以，这个过程就是计算一阶导数。也就是求一阶导数。这样就能获得最大功率。当一阶导数等于零时。就能得到最大值。

and get the maximum p.

得到最大的P。

and then this can be the process. and how can you compute the first order derivation?

这就是整个过程。那么，如何计算一阶导数呢？

and by we use this equation, it's the first order derivation. and just use this equation you can find this expression. and we input zero. so we can get maximum. and get maximum p.

我们使用这个公式，它就是的一阶导数。只要使用这个公式，你就可以得到这个表达式。然后我们输入零。这样就能得到最大值。得到最大的P。

okay, if you want to this equation equal zero, that means

好的，如果你想让这个方程等于零，那就意味着

the numerator should be zero. the numerator is the upper part. this is the numerator. and it equals zero.

分子应该等于零。分子就是上面的部分。这就是分子。它等于零。

so based on this equation, you can find the sampling resistance should be equal to the load resistance. and the first order derivation will be zero. that means when the first order derivation is zero, you can get the load resistance.

所以，根据这个方程，你可以得出，采样电阻应该等于负载电阻。而且一阶导数等于零。这意味着，当一阶导数为零时，你可以得到负载电阻。

that's the reason you can get the conclusion. when the load resistance equals the sampling resistance, you can get maximum p. this is the process. this is the process that we can get the conclusion.

这就是你能得出这个结论的原因。当负载电阻等于采样电阻时，你就能获得最大的P值。这就是整个过程。这就是我们得出这个结论的过程。

and it's the second way of proof.

这是第二种证明方法。

can also be matched by formula.

也可以用公式来推导。

and we know the theory what is the best argument, the maximum power is consumed. and the denominator is minimum.

我们知道，从理论上讲，最佳情况是最大功率被消耗。而分母是最小的。

and similarly, if you want to achieve the minimum value, this here can be the first all equation and set zero. and the other equation is same calculation. and a lot of research has been put to preserving the system. you want to get the maximum power from it.

同样地，如果你想达到最小值，这里就可以是第一个等式，并将其设为零。另一个等式也是同样的计算过程。而且，有很多研究都致力于保护这个系统。你的目标是从中获取最大功率。

and here is the conclusion. it's the relation of the p, power and the resistance.

这就是结论。这是功率和电阻之间的关系。

it's the power. The y-axis is the power and the x-axis is the load resistance.

这是功率图。y轴代表功率，x轴代表负载电阻。

so when the load resistance equals to seven resistance. we can achieve the peak point.

所以，当负载电阻等于7欧姆时。我们就能达到峰值点。

this happens when the load resistance is the same resistance.

这种情况发生在负载电阻与内阻相等的时候。

then we can calculate how much power is actually delivered.

这样，我们就可以计算出实际输送了多少功率。

the maximum power transfer happens when the load resistance equals the same resistance.

最大功率传输发生在负载电阻等于内阻的时候。

okay, let's just replace the load resistance with the same resistance. it can simplify the expression. it's like x squared minus i squared r squared is our r squared. i squared resistance is the power. if you replace rl is rth, you can get this equation. this is maximum power. this is the value of maximum power.

好的，让我们把负载电阻换成相同的阻值。这样可以简化表达式。就像x的平方减去i的平方乘以r的平方，等于我们的r的平方。电流的平方乘以电阻，就是功率。如果你把RL换成Rth，就能得到这个公式。这就是最大功率。这就是最大功率的数值。

because the resistance and the load resistance is equal, the same resistance, so

因为电源内阻和负载电阻相等，所以阻值相同，

half of the power is is wasted. and the maximum power is transferred to the load.

那么有一半的功率被浪费了。而最大功率则传输给了负载。

because of the internal resistance.

这是因为存在内阻。

only half of the power is transferred to the load. So half is wasted.

只有一半的功率传输到了负载上，所以另一半就被浪费了。

and only the half is transferred to the load. then you can calculate the efficiency. but the definition of the efficiency is the output power divided by the input. output power divided by the input power times 100, then you get the efficiency. our transfer to load is current square times resistance.

只有另一半传输给了负载。然后，你就可以计算效率了。效率的定义是输出功率除以输入功率。再乘以100，就能得到效率。传输到负载的功率等于电流的平方乘以电阻。

and this is the total power.

这就是总功率。

the output power divided by the input power.

输出功率除以输入功率。

you will get the efficiency. it's 70%.

这样就能得到效率。是70%。

that's the reason half of the power is wasted. on the internal resistance.

这就是为什么有一半的功率被浪费了。是在内部电阻上。

and only half of power is transferred to load. that means the efficiency is only 50%. how can you increase the efficiency?

只有一半的功率传输到了负载上。这意味着效率只有50%。那么，如何提高效率呢？

it's not very efficient. but it can get maximum power. unfortunately.

效率并不高。但它能获得最大功率。但不幸的是，

so 50% of the power is wasted.

有一半的功率都被浪费了。

that's a great efficiency, right?

这效率确实很高，对吧？

because of the resistance. so the terminal voltage is also half.

这是因为存在电阻。所以端电压也减半了。

the e is the voltage source. and the terminal only gets a half of the voltage source.

这里的E代表电压源。而负载只获得了电压源的一半电压。

so here is the calculation. a maximum power transfer does not mean maximum efficiency.

那么，这里有一个计算。最大功率传输并不意味着最高效率。

maximum efficiency is only 50%.

效率只有50%。

so there are some different cases for the utility companies. utility companies is the company provide electricity. electricity for the users.

所以对于电力公司来说，情况有所不同。电力公司就是提供电力的公司。为用户提供电力。

so they also want to achieve maximum power transfer. but instead they want to seek high efficiency.

所以他们也希望实现最大功率传输。但与此同时，他们又追求高效率。

so they should reduce the internal resistance.

因此，他们应该降低内阻。

in order to increase the efficiency. and they want to minimize the power losses. in order to get a more efficient transmission.

以提高效率。他们还希望将功率损耗降到最低。从而实现更高效的传输。

because that can save a lot of money. a lot of power is wasted.

因为这样可以节省大量资金。因为有很多电力都被浪费了。

so the maximum power transfer is mainly used for signal and communication.

所以，最大功率传输主要应用于信号和通信领域。

because we don't want the signal becoming weaker and weaker. We don't want the signal to become weaker and weaker. We want the signal to become deeper and deeper. in a different city, but still receive it. even though the people is far away from me. That's the reason you want to achieve maximum power transfer.

因为我们不希望信号变得越来越弱。我们不想让信号越来越弱。我们希望信号越来越强。即使在不同的城市，也能接收到。即使那个人离我很远。这就是为什么你要实现最大功率传输的原因。

But for the utility companies,

但是对于电力公司来说，

they don't want the maximum power transfer. instead, they want to minimize the power losses. during the transmission.

他们并不希望实现最大功率传输。相反，他们希望将功率损耗降到最低。在传输过程中。

So they want to increase the efficiency. they don't want to waste the money, right? And for the signals and the communications,

所以，他们想要提高效率。他们不想浪费钱，对吧？而对于信号和通信来说，

they don't mind how big the losses during the transmission.

他们并不在乎传输过程中的损耗有多大。

but what matters is the signal can be received. can be received clearly.

重要的是，信号能够被接收。而且能够清晰地接收。

if the people is far away. Any questions?

即使那个人离得很远。有什么问题吗？

Okay, I've finished the maximum power transfer.

好的，我已经讲完了最大功率传输的内容。

in order to solve this problem, firstly you have to, we know the conclusion, right? when the load resistance equals the series resistance,

为了解决这个问题，首先，我们必须知道结论，对吧？我们知道，当负载电阻等于内阻时，才能实现最大功率传输。

that means we have to find

这意味着，我们必须找到

the series resistance.

那个内阻。

Okay, five steps.

好的，一共五个步骤。

let's repeat the five steps. first step is remove the

让我们再重复一下这五个步骤。第一步是，移除

load, right?

负载，对吧？

and then the second step is open the terminals.

然后第二步是，打开端子。

and it's already found. A and B.

现在已经找到了，就是A和B。

the third step is

第三步是，

we set the sources to zero.

将电源置零。

that means for voltage source,

这意味着，对于电压源来说，

we set it into a short.

要把它设为短路。

so there's no voltage drop.

这样就没有电压降了。

that means we can draw the circuit like this. the voltage is short circuit.

也就是说，我们可以把电路画成这样。电压源是短路的。

a and b, 150 ohms.

A和B之间是150欧姆。

so how these two resistors are connected each other?

那么，这两个电阻是如何连接的呢？

they're connected in parallel.

它们是并联的。

so the resistance,

所以，电阻是

it should look like this, because they're connected in parallel.

应该看起来像这样，因为它们是并联的。

and the fourth step is, we put the source back.

第四步是把电源放回去。

and then you can put the whole circuit power.

然后，你可以把整个电路通电。

that means here is open. so the same voltage.

这意味着这里处于开路状态。所以电压是相同的。

a and b should be the voltage of this resistor. it's voltage 120 ohms, this resistor.

A和B应该是这个电阻的电压。这个电阻的阻值是120欧姆。

and we know the total voltage, right? so the voltage for this one.

我们知道总电压，对吧？那么，这个电阻上的电压是多少？

would be 360 times 120.

应该是360乘以120。

so this is the voltage.

这就是电压。

and we know the same voltage and the same resistance. so you can build the equivalent circuit. the equivalent circuit is a voltage source, Vth. and then in series with the Rth. and then here is the terminal a and b.

我们知道电压相同，电阻也相同。所以，你可以构建一个等效电路。这个等效电路是一个电压源，即Vth。然后与Rth串联。这里是端子A和B。

maximum power transfer happens when the load here.

当负载在这里时，会发生最大功率传输。

and the same resistance.

而且电阻相同。

so, uh, firstly,

所以，首先，

the series and parallel.

是串联和并联。

voltage source, uh, for the voltage, you have the open circuit voltage.

对于电压源来说，你有开路电压。

here is the total voltage.

这里是总电压。

and then you have to compute the series resistance. it's actually it's a parallel.

然后，你需要计算串联电阻。实际上，它是并联的。

so you can get the total.

所以你可以得到总电阻。

and the maximum power of the magnet under the normal resistance is also 25.

在正常电阻下，磁铁的最大功率也是25。

because the internal resistance is

因为内部电阻是串联的。

and then we look at how much power it is transferred.

然后我们来看一下传输了多少功率。

here you can compute the total p. so you can get it.

在这里，你可以计算出总功率P。这样你就能得到结果了。

once you know the current, you can calculate the time t before you get the power.

一旦你知道了电流，就可以计算出得到功率所需的时间t。

and here is more small examples.

这里还有几个小例子。

he only chose the series circuit, right? and internal resistance is 25. and load resistance is 5 ohm. it can be changed from 0 to 100.

他只选择了串联电路，对吧？内阻是25。负载电阻是5欧姆。它可以从0到100之间变化。

and the voltage is 100 volt.

电压是100伏特。

and by ohm's law, you can get the current. which is the voltage divided by the total resistance. and power is the current square times the load resistance. this is the power transfer of the loads.

根据欧姆定律，你可以算出电流。也就是电压除以总电阻。而功率等于电流的平方乘以负载电阻。这就是负载的功率传输。

and once you know these values, these three values, you can change the resistance from 0 to 100.

一旦你知道了这三个值，你就可以将电阻从0调整到100。

and you can compute, for example,

然后你可以计算，比如，

for example, current. and for example, power.

电流是多少。还有功率。

and when you plug the numbers together, you will get this figure. the y-axis is the power, x-axis is the resistance.

当你把这些数值代入计算，就会得到这个结果。Y轴是功率，X轴是电阻。

so this figure is just a plot of this table.

所以这张图只是这张表格的可视化呈现。

and here you can see when the load resistance is 20, you will get the

在这里你可以看到，当负载电阻为20时，你会得到

maximum.

最大功率。

and this is also defined a match condition. or resistance matching. resistance matching means the load resistance equals to the circuit resistance. so it's the resistance matching or match condition.

这也被称为匹配条件。或者说是电阻匹配。电阻匹配的意思是，负载电阻等于电路电阻。所以这就是电阻匹配或匹配条件。

so here is a general rule. The maximum power is transferred. from an active device. Active device could be a battery or power supply.

这里有一个通用规则：最大功率会被传输。这是从有源器件传输的。有源器件可以是电池或电源。

to an external device.

传输到外部设备。

what is the independence of the source? it has the same meaning as the resistance.

那么，电源的内阻是多少呢？它的含义与电阻相同。

impedance is at the same meaning of a resistance.

阻抗的含义与电阻是一样的。

so when the resistance of the external device exactly matches the resistance of the source, you will get maximum power.

所以，当外部设备的电阻与电源的内阻完全匹配时，你就能获得最大功率。

okay, and there's another example.

好的，还有一个例子。

solve the problem in five minutes.

请在五分钟内解决这个问题。

also maximum power turning on.

同样也是关于最大功率传输的。

that's more easier, right? voltage is given. This is the same voltage. also the same resistance.

这个更容易理解，对吧？电压已经给出了。这是相同的电压。电阻也是一样的。

so you can work something out.

所以你可以算出来。

because i think it's much simpler and it's easy. so let's have a solution.

因为我觉得这要简单得多，也更容易。那我们来看一下解法。

if you want power to happen, you want a lot of resistance.

如果你想获得最大功率，就需要很大的电阻。

so when the v on, v the r is also 0.535. so the maximum power happens

所以，当Von为0时，Vr也是0.535。这样，最大功率就出现了。

and then you can calculate the power. power is the current square times the resistance. this is the voltage. voltage is 4. 4 divided by the total resistance. and then you can get the power.

然后你就可以计算出功率了。功率等于电流的平方乘以电阻。这是电压。电压是4。4除以总电阻。这样就能得到功率了。

but if the losses within the battery

但是，如果电池内部的损耗

because of this small r. is the internal resistance. the internal loss should be smaller. should not exist.

是因为这个小的内阻r。这就是内阻。所以内部损耗应该更小。不应该存在。

empty box.

空盒子。

so here we have to use the small r. the current power is i squared r. it's the internal losses. the internal loss is smaller, smaller than 20.

所以这里我们必须使用小的r。功率就是I的平方乘以R。这就是内部损耗。内部损耗应该更小，小于20。

you know the resistance, right? You know the internal resistance. so the current is smaller than 3.5.

你知道电阻吧？你知道内阻。所以电流应该小于3.5。

r is smaller than 0.56.

R应该小于0.56。

that means the resistance, the big r, should be larger.

这意味着，电阻，也就是那个大R，应该是更大的。

larger than 125. the larger of the load resistance, the smaller of the current.

大于125。负载电阻越大，电流就越小。

and the voltage divided by the current is the total resistance.

电压除以电流就是总电阻。

i don't want to teach any new knowledge because i think your time is saved.

我不想讲新的知识了，因为我觉得大家的时间很宝贵。

How can we get the maximum of y? y is equal to a times e to the power of negative x. So the process is to compute the first-order derivative. The maximum power is achieved when the first-order derivative is zero. When the first-order derivative is zero, we can get the maximum P.

如何求 y 的最大值？y 等于 a 乘以 e 的负 x 次方。求解过程是计算一阶导数。当一阶导数为零时，功率达到最大值。即一阶导数等于零时，可以得到最大 P。

How can you compute the first-order deviation? You use this expression, derive this expression, and set it equal to zero. Then you get back from power, get back from P.

如何计算一阶偏差？使用该表达式，对其求导，并令其等于零。然后从功率反推，从P反推。

The numerator is the upper part. Setting the numerator equal to zero, based on this equation, we can find that the series resistance should be equal to the load resistance. The first-order deviation must be zero. When the first-order deviation is zero, we obtain load resistance equals series resistance.

分子是上半部分。令分子等于零，根据该方程可以得出串联电阻应等于负载电阻。一阶偏差必须为零。当一阶偏差为零时，可得负载电阻等于串联电阻。

And that's the reason you can get the conduction. But the lower resistance equals the same resistance because I get maximum power from the bottom. This is the process that you can get the induction, you get the conduction. And it's the second way of proof, that also divided by...

这就是你能获得传导的原因。但较低的电阻等于相同的电阻，因为我从底部获得了最大功率。这就是你能获得感应、获得传导的过程。这是第二种证明方法，同样除以……

The first eigenvalue of the Laplacian operator corresponds to the minimum value of the denominator. Similarly, to achieve the minimum value, you take the first-order equation, set it to zero, and then obtain the second-order equation. When the system equals zero, you get the maximum or minimum problem.

拉普拉斯算子的第一特征值对应于分母的最小值。类似地，为了达到最小值，你取一阶方程，令其为零，然后得到二阶方程。当系统等于零时，你就得到了极大值或极小值问题。

Regarding X-ray emission, there is characteristic radiation. The power is denoted by X₀, and X_max represents the lower resistance threshold. Now, what is this lower resistance? If we want to change the tube voltage—that is, the kV—the tube voltage determines the outcome.

关于X射线发射，存在特征辐射。功率用X₀表示，X_max代表下电阻阈值。那么这个下电阻是什么？如果我们想改变管电压——即千伏值——管电压决定了结果。

When the load resistance equals the Thevenin resistance, the maximum power is transferred to the load. To find how much power is actually delivered, recall that power equals current squared times resistance. If we replace the load resistance R with the Thevenin resistance R_th in the expression, we can simplify and obtain this equation. This gives us the value of the maximum power.

当负载电阻等于戴维南电阻时，最大功率传输到负载。为了求出实际传输的功率，回忆功率等于电流的平方乘以电阻。如果我们将表达式中的负载电阻R替换为戴维南电阻R_th，就可以简化并得到这个方程。这就给出了最大功率的值。

Because the load resistance is equal to the same resistance, half of the power is dissipated because of the internal resistance, and half of the power is transferred to the load. Half of the power is wasted because of the internal resistance. Only half of the power is transferred.

因为负载电阻等于内阻，所以一半的功率因内阻而耗散，另一半功率传输给负载。一半的功率因内阻而被浪费掉了。只有一半的功率被传输出去。

Half is wasted because it goes to the internal resistance. The other half is the output power. Then you can calculate the efficiency. The definition of efficiency is the output power divided by the input power times 100%. The output power is I squared times R.

一半被浪费了，因为它消耗在内阻上。另一半是输出功率。然后你可以计算效率。效率的定义是输出功率除以输入功率再乘以100%。输出功率等于I的平方乘以R。

This is the total power. Efficiency is the output power divided by the input power. That is the reason half of the power is wasted on the internal resistance; only half is useful.

这是总功率。效率等于输出功率除以输入功率。这就是为什么一半的功率浪费在内阻上，只有一半是有用的。

The power is unconverted. That means the efficiency is only 50%. It is not very efficient. We get maximum power unconverted. So 50% of the power is wasted.

功率未被转换，这意味着效率仅为50%。这并不高效。我们得到的最大功率是未转换的，因此50%的功率被浪费了。

The Carnot efficiency is also half of the total efficiency. E is the work resource, and the Carnot engine gets half of the work resource. So here is the condition: maximum power transfer occurs at not the maximum efficiency.

卡诺效率也是总效率的一半。E是做功资源，卡诺热机获得做功资源的一半。因此条件如下：最大功率传输并不发生在最大效率处。

There are some different cases for utility companies. A utility company is the company that provides electricity for users. They do not want to achieve maximum power transfer; instead, they want to seek high efficiency.

公用事业公司有一些不同的情况。公用事业公司是为终端用户提供电力的公司。它们并不希望实现最大功率传输，而是追求高效率。

Internal combustion engines have long been used in propulsion systems. Electric motors, on the other hand, increase efficiency. We want to minimize power losses in transmission and distribution. The same principle applies to electric motors: if power is wasted, we care about it. Therefore, the maximum power transfer theorem is mainly used for signal and communications applications, because we need to optimize power delivery in those contexts.

内燃机长期以来被用于推进系统。而电动机则提高了效率。我们希望最小化传输和配电过程中的功率损耗。同样的原则也适用于电动机：如果功率被浪费，我们就会关注它。因此，最大功率传输定理主要用于信号与通信应用，因为在这些场景中我们需要优化功率的传递。

As you go further from the source, the signal becomes weaker and weaker. If you want to cover a long distance—for example, between different cities—you need to receive a very strong signal at the far end. Even though the signal is weak at that distant point, you still need to achieve a strong signal there. That is why you must place amplifiers along the route. For utility companies, they go...

随着你离信号源越来越远，信号会变得越来越弱。如果你要覆盖很远的距离——例如在不同城市之间——你就需要在远端接收到非常强的信号。尽管在那个远处信号已经很弱了，你仍然需要在那里实现强信号。这就是为什么你必须在沿途放置放大器。对于公用事业公司来说，他们会……

In power systems, instead of applying the maximum power transfer theorem, the goal is to minimize power losses during transmission, thereby increasing efficiency. For signals and communications, however, the magnitude of the loss during transmission is not the primary concern; what matters is whether the signal can be successfully received.

在电力系统中，目标不是应用最大功率传输定理，而是在传输过程中最小化功率损耗，从而提高效率。然而，对于信号与通信而言，传输过程中的损耗大小并非主要关注点；关键在于信号能否被成功接收。

W is zero, V of E. The T point is far away. Any questions? Now we move to the maximum power transfer problem.

W为零，V等于E。T点距离较远。有问题吗？现在我们进入最大功率传输问题。

We know the conclusion: the load resistance equals the Thevenin resistance. That means we have to find the Thevenin resistance. What is the Thevenin resistance? Five steps. Let me repeat the five steps. First step: remove the load.

我们已经知道结论：负载电阻等于戴维南电阻。这意味着我们需要求戴维南电阻。什么是戴维南电阻？五个步骤。我再重复一下这五个步骤。第一步：移除负载。

The second state is open at the terminals. We have already found A and B with marks. The third state is when the source is set to zero. Setting the source to zero means the voltage source becomes a short circuit, so there is no voltage drop across it. Therefore, we can draw the equivalent circuit as follows: the voltage source is replaced by a short circuit between A and B. The resistors are 130 ohms and 30 ohms. Now, how are these two resistors connected?

第二种状态是端子开路。我们已经用标记求出了A和B。第三种状态是将电源置零。电源置零意味着电压源变为短路，因此其两端没有电压降。所以我们可以画出如下等效电路：电压源被A和B之间的短路替代。电阻分别为130欧姆和30欧姆。那么这两个电阻是如何连接的？

Connected in parallel, right? Connected in parallel. So the series resistance does not look like this; it looks like connected in parallel.

是并联连接，对吧？并联连接。所以串联电阻不是这样的；它是并联连接的。

The first step is to put the source back. Then we compute the open-circuit voltage. Since here it is open circuit, the 70 volts at A and B should be the voltage across this resistor — this resistor with 170 ohms. And we know the total voltage.

第一步是将电源还原。然后我们计算开路电压。由于此处是开路状态，A和B两端的70伏即为该电阻两端的电压——这个170欧姆的电阻。而总电压我们是已知的。

So the voltage for this one would be 360 times 1 amp. This is the voltage. We now have the Thevenin voltage and the Thevenin resistance, so we can compute the equivalent circuit.

所以这个电压为360乘以1安培。这就是电压值。现在我们已经得到了戴维南电压和戴维南电阻，因此可以计算等效电路。

The circuit has a voltage source E_th, and here is R_th, and here is the load R_L. Maximum power transfer happens when the load equals the Thevenin resistance, that is, R_L equals R_th.

该电路包含一个电压源E_th，这里是R_th，这里是负载R_L。当负载等于戴维南电阻时，即R_L等于R_th，发生最大功率传输。

The voting is also for the carbon. This is a problem of voting. And then you have to come up with the second system. The second system is actually the voting system.

投票也是针对碳的。这是一个投票问题。然后你必须想出第二个系统。第二个系统实际上就是投票系统。

Banknotes are equivalent tokens. In normal cases, the results are fine because of self-consistency. At the same time, how much power is transferred? If you transfer 5000 dollars, you know the total resistance, and you know the voltage, so you can get the current.

纸币是等价代币。在正常情况下，由于自洽性，结果是没问题的。与此同时，传递了多少功率？如果你转移了5000美元，你知道总电阻，也知道电压，所以你可以求出电流。

Once we have problems, this kind of functional power is based more on...

一旦我们遇到问题，这种功能性权力更多地基于……

Here, for the circuit to be working, the internal resistance is 25 ohms, and the load resistance is variable—it can be changed from zero to one hundred ohms. The source voltage is also 100 volts. By Ohm's law, you can get the current, which is the voltage divided by the total resistance. And power is the current squared times the load resistance—this is the power transferred to the load.

这里，为了使电路正常工作，内阻为25欧姆，负载电阻是可变的——可以从零变到一百欧姆。电源电压也是100伏。根据欧姆定律，可以求出电流，即电压除以总电阻。而功率等于电流的平方乘以负载电阻——这就是传输到负载上的功率。

You can change the resistance from zero to one hundred, a range of load resistances, and you can compute the corresponding current and the corresponding power. When you plot the numbers together, you get this figure. The x-axis is the resistance. So this figure is just a plot of this table. And here you can see when the load resistance is seventy-five...

你可以将电阻从零变到一百，即一系列负载电阻值，然后计算对应的电流和对应的功率。将这些数值一起绘制出来，就得到这张图。横轴是电阻。所以这张图就是这张表的图形化表示。在这里你可以看到，当负载电阻为七十五时……

If you are given the heat flux, you are given the heat boundary condition. This is also defined by mass convection or resistance. Mass resistance is the velocity resistance. Velocity resistance equals the surface resistance. So it is the resistance by mass convection. Here is a general curve. The maximum power is one curve.

如果给定热流密度，则给定的是热边界条件。这也可以由质量对流或阻力来定义。质量阻力即速度阻力。速度阻力等于表面阻力。因此它是由质量对流产生的阻力。这里是一条一般曲线，最大功率对应一条曲线。

An active device can be powered by a battery or a power supply. An external device has an impedance. The impedance of the external device exactly matches the impedance of the source. Impedance has the same meaning as resistance. The resistance of the external device exactly matches...

有源器件可以由电池或电源供电。外部器件具有阻抗。外部器件的阻抗与源阻抗完全匹配。阻抗与电阻含义相同。外部器件的电阻完全匹配……

This is the existence of a source for maximum power transfer. Here is another example. Solve the problem in five minutes. Also maximum power transfer.

这就是最大功率传输源的存在性。这里是另一个例子。请在五分钟内解决这个问题。同样是最大功率传输。

This approach is much easier because the voltage is given. The voltage is the same, and the resistance is also given. Therefore, you do not need to work hard to calculate the voltage and the current in the circuit.

这种方法要容易得多，因为电压是已知的。电压相同，电阻也是已知的。因此，你不需要费力去计算电路中的电压和电流。

I'm unable to process audio files directly. Please provide the raw transcript text for the 1774.5–1815.0 second segment, and I will clean and reformat it into a formal lecture transcript as requested.

我无法直接处理音频文件。请提供1774.5–1815.0秒时间段的原始转录文本，我将按要求将其整理为正式的课堂讲稿。

Because as we discussed in the first case, maximum power transfer happens when the load resistance equals the internal resistance. When V on, V on is also there by 75, but V on...

因为正如我们在第一个案例中讨论过的，最大功率传输发生在负载电阻等于内阻时。当V on时，V on也是75，但V on……

Power is the current squared times the resistance. This is the voltage: voltage equals current times resistance. You can get the current squared times the resistance, and you get the power. But if the losses are in the battery because of the small r—the internal resistance—then the internal losses should be smaller than 20 watts. So at the end, you have to use the small r: the current squared times r gives the internal losses. The internal losses should be smaller than 20 watts.

功率等于电流的平方乘以电阻。电压等于电流乘以电阻，由此可得电流的平方乘以电阻即为功率。但如果损耗在电池内部，是由小 r——即内阻——引起的，那么内部损耗应小于 20 瓦。所以最终你需要用小 r：电流的平方乘以 r 得到内部损耗，该内部损耗应小于 20 瓦。

If you know the total resistance and the internal resistance, the current should be smaller than 0.56. That means the resistance, the big R, should be larger than 100. The larger the load resistance, the smaller the current. And the voltage divided by the current equals the total resistance. Total resistance minus the internal resistance gives the load resistance.

如果你知道总电阻和内阻，电流应该小于0.56。这意味着电阻，即大R，应该大于100。负载电阻越大，电流越小。电压除以电流等于总电阻。总电阻减去内阻得到负载电阻。