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Download "Урок 253. Задачи на расчет электрических цепей - 3"

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00:00:10
today's two lessons we are built according to
00:00:13
this plan, now we are still solving problems on
00:00:16
calculating electrical circuits
00:00:18
in the next lesson I will tell you
00:00:21
something new and so the topic is
00:00:26
solving the problem,
00:00:37
homework, taking into account the
00:00:40
new material that will be in the second
00:00:42
lesson,
00:00:43
notes from Myakishev’s textbook, paragraph
00:00:55
106
00:01:00
Gelfgat problem book and one problem number 11 8 and
00:01:13
4 more problems according to Rymkevich with numbers 787 788
00:01:31
794 and 796
00:01:40
this is for tomorrow tomorrow we have 2
00:01:44
lessons in the second and fourth but
00:01:46
then we also have practice
00:01:48
for half the class and so we are working today
00:01:52
on the problem book gald farba let's start task
00:01:56
number 19 11 the task is accompanied by an
00:01:59
electrical circuit so let's
00:02:01
start with it and then write down a short
00:02:03
condition the circuit consists of three resistors
00:02:06
and this circuit is called a four-terminal circuit it
00:02:10
has two poles two inputs
00:02:17
and then there is some circuit and 2 outputs
00:02:29
here is the output inside the four-terminal circuit three
00:02:37
resistors their resistance r1 upper
00:02:51
r2 on the right side of the output
00:02:55
r3 and this is what is known about the properties of this
00:03:01
four-terminal network if a
00:03:04
voltage u1 of 100 volts is applied to the input of the electrical circuit, then the voltage
00:03:07
at the output of the op-amp is 340
00:03:16
volts if a voltage of 100 volts is applied to the input
00:03:23
u1 turn of resistor r1 then on the output
00:03:27
will be voltage u 3,
00:03:30
which is precisely the
00:03:31
voltage across resistor r3, such
00:03:34
designations we have, while
00:03:37
current i2 1 ampere flows through resistance r2,
00:03:45
this is the first piece of information, the
00:03:49
second piece of information, if
00:03:52
voltage u 3 is applied to the output of the circuit, stroke wipe
00:03:58
stroke 60 volts then the voltage at the input
00:04:05
turns out to be equal
00:04:07
to one stroke 15 volts, determine the
00:04:13
resistance values ​​r1 r2 and r3
00:04:21
r1 r2 and r3 this is the task, so if
00:04:31
100 volts are applied to input a, the output will be
00:04:34
40, let’s imagine this better, let
00:04:38
’s draw the current source, it’s clear that this
00:04:42
cannot be a galvanic cell
00:04:43
just an ordinary galvanic cell
00:04:45
gives one and a half lithium 3 volts here
00:04:48
we definitely have a battery connected to the
00:04:53
input and it gives voltage u1 it
00:04:58
comes out while voltage u 3
00:05:01
can be measured with a voltmeter, of course, you
00:05:05
can of course not draw all this,
00:05:08
but it’s easier to imagine the condition of the
00:05:12
problem therefore it’s easier to solve,
00:05:13
also note that in such a connection, a
00:05:17
current i2 of 1 ampere flows through resistor r2,
00:05:21
let’s show it like this and now
00:05:27
let’s try, in this mode of
00:05:30
operation of the four-terminal network, to extract the maximum
00:05:32
information about the resistors that we know, we
00:05:38
know the voltage on this resistor is 100
00:05:42
volts and the voltage on this resistors
00:05:45
please tell me if we can then
00:05:47
find out the voltage on resistor r2 so
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the blinker says that we need to subtract from u1 from
00:05:58
3 and we will get the voltage on the second
00:06:00
resistor let us designate it as u2 and
00:06:07
then really look at the voltage
00:06:10
here u2 the voltage here at 3
00:06:14
the voltmeter does not count this like a break in the
00:06:16
electrical color of an ideal voltmeter
00:06:18
while these two resistors are connected to each other
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in series, in this case the
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voltage between this point and this
00:06:26
voltage on the entire serial
00:06:28
chain is equal to the sum of y 2 + y 3 but this
00:06:32
voltage is exactly y 1 so we
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find out that y 2 equals 1 minus y,
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here’s the first step, but now it immediately
00:06:44
becomes clear how to find the
00:06:46
resistance r2 according to the conditions of the problem, the current
00:06:49
through this resistor is given, which means to
00:06:51
find the resistance r2, we’ll use
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Ohm’s law, write out all three
00:06:56
variants of writing the classic Ohm’s law on the side
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and divide y equally by r
00:07:01
further variations in the level and r and finally r is
00:07:07
equal to y divided at the bottom by the
00:07:11
voltage across the resistor r2 we know the strength of
00:07:14
the current we also know through it, which means we
00:07:17
can find the resistance r2
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r2 equals q 2
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ou-2 we have q1 minus 3 divided by
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barely is
00:07:34
good we continue to think further, what
00:07:39
other ideas do you have? You can find r3
00:07:43
well, how do you divide the morning
00:07:49
into barely, and why do you divide the morning into barely a
00:07:56
series connection? with a
00:07:58
series connection, the currents through
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both resistors will be the same, which means
00:08:04
here, too, the current dances,
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turns here, yes, here, too, the current
00:08:08
will be barely,
00:08:10
therefore we know the voltage on this resistor,
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by the condition a it is equal to three times the current
00:08:19
through this resistor, we also know by this
00:08:21
way we can say that p 3
00:08:25
is equal to the voltage on this resistor and
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3 divided by barely
00:08:32
wounded 3 is ready to find the
00:08:39
resistance of the resistor r1 here it’s
00:08:42
not difficult guess that this circuit
00:08:44
will not be able to tell us anything about this
00:08:49
resistor because no matter what the voltage is
00:08:53
here, no matter what currents flow here,
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this resistor is connected directly
00:08:58
to the current source, which means the voltage on
00:09:02
it is known about the strength of the current, we
00:09:05
cannot get information from this circuit,
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we have already got everything we could, now all that remains is
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to use the second part of the problem condition
00:09:13
when the four-terminal network is turned on
00:09:16
backwards, let's draw this circuit again, here is the
00:09:22
resistance r2, here are
00:09:26
the conclusions,
00:09:34
now the source is connected to the output,
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here is the
00:09:46
source voltage, 3 of them have q3
00:09:51
stroke, let's finish drawing this resistor
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though We won’t need it for work anymore,
00:10:00
this is resistor r2,
00:10:03
this is resistor r1, but this resistor r is
00:10:10
now you to the output, no, now this will be
00:10:13
the input, connect the voltmeter,
00:10:19
the voltmeter shows the voltage u1 stroke,
00:10:23
one stroke, let’s remember that
00:10:31
we already know r2, we want to find r1 what
00:10:35
ideas will we have,
00:10:41
let's climb along the same path
00:10:44
that we followed, what
00:10:45
we did with you, we found this
00:10:48
voltage as the difference, this is minus this,
00:10:51
let's denote this voltage with two
00:10:54
strokes and then this voltage u 3
00:11:05
stroke represents the sum of r2 y q2
00:11:09
stroke and one touch because, again,
00:11:12
this voltmeter does not show itself in any way,
00:11:15
that is, it does not branch off the current to itself,
00:11:17
so here
00:11:19
the same current flows through the resistor r1 and r2, and
00:11:22
since, among other things, they are
00:11:24
connected in series, the voltage
00:11:27
here plus the voltage here equals
00:11:31
voltage 3 stroke here you can do what you can do,
00:11:38
you can find the current strength through this
00:11:43
resistor marked with
00:11:45
two strokes and 2 strokes, the current strength through
00:11:54
this resistor
00:11:55
is equal to the voltage u2 stroke OU-2 stroke
00:11:59
this will be the difference of 3 pieces of them minus one
00:12:04
stroke u3 stroke minus y1 stroke divided
00:12:09
by resistance r2 like this and
00:12:15
resistance r2 we know, let's
00:12:18
take the value of r2 from here
00:12:20
then we will get it and two strokes
00:12:25
equals utrish 3 minus y1 stroke divided
00:12:32
by r 2 divided by this one let's write it like this y
00:12:34
one minus 3 will go here in
00:12:37
the denominator
00:12:41
but it will barely go into the numerator so that it turns out to be a
00:12:45
dimensionless fraction multiplied
00:12:47
by some value so and what does this
00:12:50
give us so we know the current through this
00:12:53
resistor
00:12:56
where it flows next, it flows all through
00:13:00
resistor r1 because the voltmeter does not
00:13:02
branch itself into the cake, which means there
00:13:05
will also be a current in force and 2 stroke voltage we
00:13:10
know the current strength we have already found it too, here it is the
00:13:15
current strength the voltage is given according to the conditions of
00:13:20
the problem, which means we can find
00:13:22
the resistance r1 we get from this
00:13:34
version of
00:13:35
Ohm’s law by dividing the voltage
00:13:37
voltage we have one stroke
00:13:41
one stroke per current strength through this
00:13:45
resistor then there are on and 2 pieces of them now all that
00:13:51
remains is just a stroke to take from here and we
00:13:55
get r1 equals the fraction and 2 stroke is
00:14:03
in our denominator so we turn the
00:14:06
fraction q1 minus u 3 divided by y 3 stroke minus
00:14:10
one stroke now this will barely go into
00:14:16
the denominator
00:14:20
and y is one stroke in the numerator here is 3, the formula is a
00:14:30
dimensionless fraction and here the ratio of
00:14:32
voltage to current has the dimension of
00:14:35
resistance, it remains to be calculated, well, let's
00:14:46
probably start with the card r3
00:14:50
equals the morning 40 volts divided by and 2 by
00:14:59
1 ampere equals 40 it is further r2r 2
00:15:10
equals the
00:15:12
fraction y 1 out of 100 minus 40 100 minus 40
00:15:23
divided by and 2 by 1 volt per ampere it
00:15:30
turns out 60 and finally r3
00:15:37
r3 equals 1 minus u 3 100 minus 40
00:15:45
100 minus 40 divided by y 3 prime minus
00:15:50
y1 prime 60 minus 15 units and I don’t write
00:15:57
they will already be reduced volts and with volts
00:16:00
one stroke 15 volts
00:16:05
divided by 1 ampere the answer will be in ohms
00:16:10
numerator 60 here 60 minutes 15-45 and here
00:16:15
we reduce 15 by 15 it turns out 60 divided
00:16:18
by 320
00:16:24
so resistance r1 here you need to
00:16:32
write r1 before
00:16:34
resistance r1 I use this formula
00:16:36
he considered this is 20 he resistance r2
00:16:41
this is 60 ohm resistance r 3 this is
00:16:45
40 and here is such a problem here we had the
00:16:49
opportunity to solve everything in a general form
00:16:51
because the lyrical tapeworm is especially
00:16:53
difficult if the circuit is complex it is possible to solve the
00:16:57
problem in parts it is not necessary to obtain
00:16:59
all the formulas in general,
00:17:01
now we will solve a similar
00:17:04
problem,
00:17:05
but on the contrary, we will not write a single formula,
00:17:08
that is, I will show you another style for
00:17:11
solving similar problems,
00:17:13
although on a test it is difficult to
00:17:15
convey your thoughts to the person who will
00:17:19
check it worked, but you can
00:17:22
quickly solve problems problem number 19 20
00:17:31
1920 farm
00:17:38
this electrical circuit is like this
00:17:44
and the values ​​of the resistors are written directly on it
00:17:50
this resistor is 10 kohms
00:17:58
I won’t write units, we will all
00:18:02
mean in ohms then here’s another 10 kohms here
00:18:10
the resistor is also 10 m further the current source
00:18:24
is drawn here galvanic element, but
00:18:27
since it says seven and a half volts,
00:18:29
I’ll still write the batteries, I’ll draw a
00:18:32
battery of 7 and 5 volts, here the ammeter is turned on, here are the
00:18:46
connections, here are the connections, and finally, here’s
00:18:49
another resistor,
00:18:58
its resistance is 15 and now in
00:19:04
order for us to communicate we need
00:19:07
designations this resistor we will have r1
00:19:13
what could we call these elements
00:19:15
this r2
00:19:18
this r3hab what is below r4
00:19:26
well the question is the following: what
00:19:30
current flows through an ammeter
00:19:32
with a negligible internal
00:19:35
resistance
00:19:36
to the circuit shown shown in this
00:19:39
figure what is the current through the ammeter an
00:19:43
ideal ammeter with negligible
00:19:45
internal resistance
00:19:54
if you are dealing with an ammeter, you
00:19:57
can imagine that there is nothing here,
00:19:59
that is, there is just a wire going on and let's
00:20:03
try to find the distribution of currents in
00:20:06
this circuit, just if we replace
00:20:08
the ammeter with a conductor, it is easier to see
00:20:10
what type of connection is here and we will
00:20:15
look for currents not definitely right here
00:20:17
and in different parts of the circuit, look,
00:20:20
if it’s a vampire, there’s no meter, they forgot, and
00:20:24
resistor r1 is connected, its left
00:20:31
terminal is connected to the negative pole of
00:20:34
the battery, and the right terminal is just a
00:20:37
wire, and the right terminal is connected to
00:20:39
the positive, then this resistor is
00:20:41
simply connected to the current source
00:20:44
we know its resistance 5015 them we
00:20:48
know the voltage of the current source we can
00:20:50
find the current through this resistor it
00:20:55
does not depend on any other parts of
00:20:58
the circuit because the resistor is directly
00:21:00
connected to the current source
00:21:01
how to find the current here everything else
00:21:05
throw out the
00:21:08
voltage divide by the resistance
00:21:10
as you get half an ampere, let's
00:21:14
write the current flows from plus to minus,
00:21:17
which means here is the negative terminal, here
00:21:20
we have a current of 0.5 amperes, the
00:21:26
whole current is the same 0.5 amperes,
00:21:35
let's try to move forward with the project,
00:21:39
there is no ammeter, please tell me
00:21:42
how these two resistors are connected to each other,
00:21:45
what is it? -is it a standard connection or an
00:21:47
exotic one?
00:21:49
These two beginnings are connected to one
00:21:58
point in the electrical circuit and the two ends are
00:22:03
connected to another point in the electrical
00:22:06
circuit. How are they connected in parallel then
00:22:10
if each of them is 10 ohms then what is the
00:22:12
total resistance
00:22:14
5 means these two resistors together have a
00:22:18
resistance of 5th and now look
00:22:21
this pair of resistors is connected to
00:22:24
these resistors in what way
00:22:27
in series means these two
00:22:30
resistors look they are connected to the
00:22:32
positive pole then the current flows
00:22:35
through resistor r2 which is connected
00:22:38
in series
00:22:39
here 5 here 10 total resistance
00:22:43
15 what current flows through this resistor
00:22:48
also
00:22:49
that and a half divided by 15 half an ampere
00:22:55
05 amperes are
00:23:03
these half amperes
00:23:06
from which two identical
00:23:10
resistors are
00:23:11
connected in parallel, since they are connected in
00:23:14
parallel the voltage on them is the same, which
00:23:17
means the current strengths in them are also the same,
00:23:20
these two currents adding up give us half an
00:23:24
ampere each of these currents
00:23:26
what a fight for 025
00:23:30
so here a current of 025 amperes flows and here a
00:23:37
current of 025 amperes flows and they merge
00:23:43
to form a current of half an ampere,
00:23:45
well now we have come close to
00:23:49
solving the problem, look at this
00:23:52
node, a
00:23:56
current of half an ampere flows out of it along this branch and along this one
00:24:00
branches current is a quarter ampere then what
00:24:02
current flows through this conductor 075 the problem is
00:24:06
solved and equals 0 75 amperes here is a
00:24:12
completely different approach another style of
00:24:14
solving problems by the way you can
00:24:16
check yourself look let's come from the other
00:24:19
end we used this node
00:24:22
let's come from this end here
00:24:25
2 currents are combined half an ampere means through the
00:24:28
current source what kind of current flows
00:24:29
1 ampere 1 ampere now a quarter ampere leaves from this one
00:24:36
ampere to the left 025
00:24:43
amperes which means once from one ampere an
00:24:46
outflow of a quarter ampere branched off here
00:24:49
we have three quarters left of the
00:24:51
same 075 amperes 075
00:24:55
amperes it turns out and so you can
00:24:58
solve similar problems,
00:25:01
let's move on to another popular type of
00:25:12
problem for calculating the resistance of all sorts of
00:25:15
meshes problem number 19 16 number 1916
00:25:23
Hyde Park you need to find the resistance
00:25:32
between points a and b of a
00:25:34
wire mesh like this, draw a
00:25:40
rhombus like this, it is advisable that you
00:25:44
leave some space on the right because that now we
00:25:46
will redraw this rhombus and then
00:26:01
this whole connection is like this,
00:26:12
here the conductor comes off and here
00:26:17
the conductor comes off. and this is point b, the
00:26:22
resistance of each link of this grid
00:26:25
is small, you need to find the resistance of the entire
00:26:29
grid as a whole, how these elements of each of them are connected to each other,
00:26:37
these elements of each of them can
00:26:40
be considered a resistor, points must be
00:26:42
placed everywhere, that is, here is a resistor that is
00:26:46
simply not drawn here, these are all
00:26:48
resistors, each has the resistance r is
00:26:50
small
00:26:51
as they are connected to each other in
00:26:53
parallel in series in different ways and
00:26:57
not in parallel and inconsistently,
00:26:59
so if we want to solve this problem
00:27:02
we need to use some kind of trick, we
00:27:04
need to somehow redraw
00:27:08
this circuit, somehow modify it, but in
00:27:10
such a way that the resistance and at the same time, it did
00:27:13
n’t change, you can do the following,
00:27:17
imagine that you cut
00:27:21
here and here
00:27:23
this diagram was torn apart like this, let’s
00:27:28
see what happens, redraw
00:27:31
the diamond again
00:27:38
and now here we have a gap, so
00:27:50
and so,
00:27:54
and here we have a gap,
00:28:08
everything else remains the same -as before. b.
00:28:17
and here is the connection and now
00:28:31
please tell me is there a potential difference
00:28:35
between these two points here on
00:28:38
these streets oh no the circuit is symmetrical
00:28:41
so the voltage between this point and
00:28:44
this this this is the
00:28:47
same imagine that we
00:28:49
connected with a wire here in which
00:28:52
direction the current will flow to the left or to the right
00:28:55
considerations of symmetry will not flow anywhere,
00:28:58
therefore, if we connect with
00:29:01
a wire, that is, if we restore the
00:29:03
previous configuration, then the electrical
00:29:06
properties will remain the same, these
00:29:08
points we will call these
00:29:10
potential these potential points
00:29:13
can be connected
00:29:14
or can be broken, so this circuit will
00:29:19
have exactly the same resistance is like
00:29:21
this, but this is a simpler diagram,
00:29:27
now let’s use this same idea further
00:29:29
where else we can make a break here
00:29:38
and draw this diamond again with
00:29:44
these additional breaks we’ll
00:29:46
get
00:29:55
. b. and then this section
00:30:06
now looks like this, this section looks
00:30:12
now like this, here we have a gap, which means this is
00:30:19
what this section will look like, so and
00:30:23
so, and this section here,
00:30:30
we draw the gap like this, so and so,
00:30:43
we make electrical connections,
00:30:55
the resistance of this circuit will be the same
00:30:58
because here, too, there is no
00:31:01
potential difference for reasons of
00:31:03
symmetry,
00:31:04
that is, if we connect these two points, it
00:31:06
won’t flow like that, so it doesn’t matter whether there are
00:31:08
connections here or not, it was here,
00:31:12
and so the resistance of this chain and this
00:31:15
chain is the same, each link has a
00:31:17
resistance r small in addition, pay
00:31:21
attention that this whole chain consists of two
00:31:24
halves on the left and exactly the same on the right, which means that
00:31:27
if we find the resistance of the left
00:31:29
half, then the resistance of the right half will be
00:31:31
exactly the same, and so let’s deal with the left
00:31:33
half, what does it consist of, the
00:31:36
resistance of this piece r
00:31:39
the resistance of this piece era how
00:31:42
they are connected to each other
00:31:44
in series what is the total
00:31:46
resistance 2r to these two r
00:31:51
but exactly the same
00:31:53
resistance 2r how are they connected to each other
00:31:56
in parallel 2r connected
00:32:00
in parallel to 2r how much will be r and so this
00:32:04
whole little diamond between
00:32:07
these points
00:32:08
has a resistance r now look at
00:32:11
this diamond, this link is
00:32:14
connected in parallel or in
00:32:15
series, and this
00:32:18
link, this diamond has a resistance r,
00:32:21
each of these links also has a
00:32:24
resistance r,
00:32:25
they are connected in series, which means
00:32:28
together, this entire section of the circuit, as I see it in
00:32:31
dotted lines,
00:32:35
has a resistance of
00:32:38
3p here this is r plus this is r and this is r3 r3
00:32:49
we take the next step here is this piece of
00:32:54
wire what resistance does it have 4 1
00:32:59
2 3 4 sections power means here we have 4
00:33:03
r and this section is connected to 3p
00:33:07
as in parallel means total
00:33:10
resistance let's take a different
00:33:12
color chalk total resistance this
00:33:14
section
00:33:23
is the resistance of two resistors, one
00:33:26
of which is 4r, this second one is 3r, which means
00:33:29
we get 4 r multiplied by 3 r
00:33:35
divided by 4 r plus r numerator 12 in the
00:33:43
denominator 7 equals 12 sevenths r this is the
00:33:49
resistance between these two
00:33:51
points 12 sevenths we go further from this
00:33:59
point to point b there is a section with
00:34:02
resistance r and here a section with
00:34:05
resistance r
00:34:06
means 12 sevenths we still need to add 2r
00:34:10
and we get 12 sevenths r + 2 r
00:34:17
equals how much 2r is 14 7 and 12
00:34:24
sevenths twenty-six sevenths
00:34:26
twenty-six sevenths is what this is the
00:34:32
resistance of this left half of the
00:34:35
electrical circuit,
00:34:36
but exactly the same resistance is on the
00:34:39
right half and they are connected to
00:34:41
each other in parallel then what will be the
00:34:44
total resistance to p and twenty-
00:34:53
six-sevenths in half, that is, 13 sevenths
00:34:57
equals 1 2 multiplied by I’m just
00:35:01
writing this for clarity twenty-six
00:35:04
sevenths of r or r & b equals 13 sevenths of
00:35:14
r little
00:35:17
so what idea did we use now and
00:35:20
cry potential points can be connected
00:35:23
or breaks can be made cuts in an
00:35:27
electrical circuit to form
00:35:29
these potential points no current flows through them
00:35:32
therefore their
00:35:35
presence of connections there or the lack of
00:35:37
connections does not in any way affect the overall
00:35:39
resistance of the chain, well, let's
00:35:41
take a break and in the next lesson
00:35:43
we will deal with new material

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Урок физики в Ришельевском лицее

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