1. Logarithmic coordinate diagram
(1) Concept derivation
To study the frequency, we must use the logarithmic coordinate diagram. The logarithmic diagram can show the impact of the relative change of frequency on the circuit performance in a wide range, and this effect can not be achieved by the ordinary linear coordinate diagram.
This may be a bit abstract. You may understand it by taking a real example of consumption. For example, if you want to buy a steamed bread, if the price of steamed bread rises from 1 yuan to 5 yuan, you may feel expensive; Next, you plan to buy a mobile phone. If the price of the mobile phone changes by 5 yuan, you may not care at all. When you buy a mobile phone, you care about the price changes of hundreds to thousands of yuan; Next, if you plan to buy a car, if the price of the car changes by hundreds of yuan, you may not care at all. What you care about is the price change of 10000 yuan.
The objective units of several hundred yuan when buying a car, a few hundred yuan when buying a mobile phone and a few yuan when buying steamed bread are the same, which are “Yuan”, but for your psychology, the evaluation criteria for them are different in different price ranges. This is where the logarithmic coordinate map can play a great role. It can reflect the impact of “relative change” in different intervals.
(2) Characteristics of logarithmic coordinate graph
Logarithmic coordinates take the logarithm of the original value, and then allocate the logarithmic result to the coordinate axis in proportion. The commonly used logarithmic coordinates usually take 10 as the bottom, sometimes take 2 as the bottom and E as the bottom. This situation will be specially explained next to the figure.
Logarithmic coordinates include “semi logarithmic coordinates” and “full logarithmic coordinates”. The abscissa of semi logarithmic coordinate diagram is logarithmic coordinate, and the ordinate is still ordinary coordinate; The ordinate and abscissa of the full logarithmic coordinate graph are logarithms. There is a variant of this graph, that is, the ordinate is expressed in decibels (decibels are also a kind of logarithm). We will look at the decibel graph in the next section. In this section, we first focus on some characteristics of ordinary logarithmic coordinate graph. The following figure shows the legend of abscissa of a semi logarithmic coordinate graph from 1 to 10:
As can be seen from the figure, when the abscissa changes from 1 to 2, it occupies about 30% of the span, from 1 to 3 occupies about half of the span, from 1 to 5 occupies about 70% of the span, and from 9 to 10 only occupies less than% 5 of the span. This is very consistent with our previous intuition. We will pay more attention to the price from 1 to 2, but we may not pay so much attention to it from 9 to 10.
Then we enlarge the range of abscissa to 1 ~ 1000, as shown in the following figure:
It can be seen from the figure that the abscissa length of 1 ~ 10, 10 ~ 100 and 100 ~ 1K is the same, that is, the span occupied by them on the figure is the same for every 10 times of the increase of the abscissa value. This is an important feature of logarithmic coordinate graph!
Abscissa values can increase not only to a large extent, but also to a small extent. If you want to study small-scale problems, you can also expand the logarithmic coordinates to the left infinitely. For every 10 times reduction, their spans on the graph are still the same. The following figure is a logarithmic coordinate graph in the range of 0.01 ~ 10:
Generally, when using a logarithmic coordinate graph, we only need to take the abscissa range we are concerned about (such as 0.1 ~ 10, or 1 ~ 10000). It can expand infinitely to the left (never reaching 0) and to the right (never reaching infinity).
Finally, when the logarithmic coordinate diagram is used for graphical evaluation, the size measured on the diagram needs to be changed back to the original value by anti number calculation. The method is as follows:
When you need to calculate the actual value of the abscissa of point a, first measure the distance from point a to 10x as Da, and then measure the dimension from 10x to 10x + 1 as D (review: each 10 times span is equal), then the calculation formula of the actual value of point a is:
(3) Tenfold range
Finally, let’s talk about a unit that often appears when using logarithmic graph: decade.
In the ordinary linear coordinate system, the unit measurement of the general abscissa is 1. For example, for the linear coordinates and straight line y = KX + C in the following figure:
For every additional unit of X (from X1 to x2), y increases K（ Δ Y = k * 1 = k), written as an expression:
So the question comes: what is the unit of abscissa on a logarithmic graph? Since the scale of the abscissa has become LG (x), let the abscissa of the two points on the logarithmic graph be X1 and X2 respectively, then one unit of the abscissa is:
According to the nature of logarithm:
It can be seen from the results of the above formula that the distance of one unit of the abscissa on the logarithmic graph is 10 times, such as from 1 to 10, from 2 to 20, from 30 to 300, etc. as long as X2 is ten times of x1, it is one unit on the abscissa of the logarithmic graph.
In circuit analysis, the frequency is often used as the abscissa of the logarithmic graph, and the saying “n / ten octave frequency range” often appears, which means that every 10 times of the frequency as the abscissa increases, the ordinate increases by n.
According to the formula of LG (x2 / x1) above, we can also see that as long as the ratio of x2 / X1 is equal, their distance on the logarithmic abscissa is equal. For example, from 1 → 2, from 3 → 6 and from 100 → 200, the ratio between them is 2, so the distance on the log abscissa is also equal. If it is used to describe the frequency, it can be said as: M / double frequency range, which means that every time the frequency as the abscissa is doubled, the ordinate is increased by M.
2. DB and gain
(1) Power gain
Decibel unit is a common unit in an electronic circuit or signal processing, but its definition may confuse beginners, especially how the coefficient 20 comes from, so we need to talk about the causes and consequences here. After understanding the process, you will find that the decibel unit is still very useful.
Bell is named in memory of bell, the inventor of the telephone. At first, this unit was used to measure the power gain in the communication system. It is defined as:
Recall that there was no modern information theory at the end of the 19th century and the beginning of the 20th century. At that time, people felt that communication was long-distance transmission power. As long as the power was large, everything would be easy to do. The greater the power, the stronger its anti-interference ability and better able to withstand all kinds of attenuation in the process of transmission. Therefore, the ratio of power is a very important unit:
Later, it was found that the ratio is often too large or too small. Large numbers such as 10000 and 100000 often appear when measuring power amplification, and small numbers such as 0.01 and 0.001 often appear when measuring power attenuation. It is inconvenient to use, so a logarithm LG was added in front, which is Bell’s definition formula. This makes it refreshing to use. For example, the power amplification of 1000 times is 3bel, and the power attenuation to 0.001 times is – 3bel.
Later, people found that the unit was still a little uncomfortable. For example, it was 3 bell at 1000 times, 3.7 bell at 5000 times, and 4 bell at 10000 times. Although it is a big change scale from 1000 times to 5000 times, it only increases by 0.7 from 3 to 3.7 on the bell scale, which is a little understatement. So people defined a more subdivided unit: decibel (DB). One bell is equal to 10 decibels, and the decibel value is equal to the original bell value multiplied by 10. In this way, the magnification of 1000 times is 30dB, and the magnification of 5000 times is 37db, an increase of 7 decibels, so people feel much better subjectively (sweat…). Decibel is defined as:
It should be noted that the above is only the DB expression of “power gain”, and the DB expression of voltage gain is different. See below.
(2) Voltage gain
Gradually, it is found that in many applications, power amplification is not necessary, as long as the voltage signal or current signal is amplified, so a logarithmic unit used to express the voltage gain is needed. Since the decibel unit has been widely used, the original decibel unit can be directly used instead of inventing a new unit by using the conversion relationship between power and voltage. See the figure below:
When the voltage changes from v1Increase to V2At load RLIf the power increases from P1 to P2 without change, the relationship between power and voltage should be:
Substitute it into the previous DB expression:
Therefore, when the gain of “voltage” is expressed in decibels, the previous coefficient becomes 20. Later, when people got used to the decibel unit of voltage gain, connect the R of the circuit before and after the gainLThe condition of “equality” is ignored, but the above formula is purely used to represent the voltage gain, for example, in the figure below,
Voltage signal V1It is amplified to V by the amplification circuit2, although from v1Input resistance RI and V seen at the terminal2Load resistance RLCompletely different, but people still use the above DB voltage gain expression to represent the voltage gain.
Similarly, the current gain expressed in decibels is the same, and its expression is:
Gradually, in electronic circuit systems, people use decibels more to represent the gain of voltage or current, but less to represent the power gain.
Later, because the unit of decibel was so easy to use, people began to abuse it and expressed all ratios (gain or attenuation) in decibel units indiscriminately. For example, for the physical quantity “frequency”, which is completely independent of power, if a circuit is improved and its bandwidth frequency is increased from 1kHz to 10kHz, people will also say that its bandwidth is increased by 20dB (meaning that the bandwidth is increased by 10 times), and so on
The following table shows the corresponding relationship between several common DB values and gain multiples:
There are several common decibel values that we need to remember (circled with red lines in the figure). For example: – 6dB means attenuation to half of the original, – 3dB means attenuation to 0.707 times of the original; 6dB means twice the original amplification, 20dB means 10x amplification and 40dB means 100x amplification.
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