The main content of this chapter is to study the influence of the frequency change of the input signal on the amplifier gain. Before starting the formal research, this section also introduces two tools, one is the normalized unit and the other is the Bode diagram.

## 1. Normalized unit

Generally, the influence of the frequency of the input signal on the gain of an amplification circuit is typically shown in the figure below:

Figure 10-2.01

As can be seen from the figure, this is a semi logarithmic coordinate diagram, with abscissa (frequency) as logarithmic coordinate and ordinate (voltage gain) as linear coordinate. In the figure, we can observe that the voltage gain can reach the maximum in the intermediate frequency range (about 100Hz ~ 100kHz) (150 in the figure). When the input signal is lower than 100Hz or higher than 100kHz, the voltage gain will decrease.

Generally, we will pay more attention to the left and right frequency points (F in the figure) when the voltage gain decreases to 0.707 (i.e. 1 / √ 2) of its maximum gain_{L}And F_{H}）, because from the original circuit definition, when the voltage drops to 0.707 times, the output power will be reduced to 1 / 2 of the original according to the conversion relationship between voltage and power. When the output power drops to less than half of the original, we usually think that this circuit is no longer working normally.

The frequency range between these two frequency points is called:**bandwidth**（bandwidth）。 Corresponding f_{L}And F_{H}be called:**cut-off frequency**(cutoff frequency) or**Inflection frequency**（corner frequency）。

In many cases, when comparing the performance of different amplification circuits, we only pay attention to their frequency bandwidth rather than the real amplification factor (because the amplification factor can usually be adjusted by configuring different resistors, and the bandwidth is usually determined by the design structure of the circuit or the properties of the device, which is generally unable to be adjusted). Therefore, people use the vertical coordinate**normalization**(normalized) method, the value of the ordinate is expressed as the ratio of the actual gain and the maximum gain value of each frequency point, so as to complete the normalization. The maximum value of the ordinate is 1 and dimensionless, as shown in the following figure:

Figure 10-2.02

After normalization, we can compare the bandwidth performance between different amplification circuits. For example, by comparing the normalized gain frequency diagrams of the following two amplification circuits, we can clearly see that the frequency performance (bandwidth) of the amplifier in the left figure is better than that in the right figure:

Figure 10-2.03

## 2. Bode diagram

If the ordinate is further changed and the ordinate is also expressed in decibels, the so-called**Bode diagram**（Bode plot）。 After the normalization diagram in figure 10-2.02 above becomes Bode diagram, it is shown in the following figure:

Figure 10-2.04

In the figure, the value of the original ordinate of 0.707 has now changed to – 3dB:

Now, the ordinate has also become a logarithmic coordinate, a “total logarithmic coordinate map”. According to the broad definition of DB in the previous section, for the ratio of attenuation to 0.707 times, people generally directly say that the attenuation is 3dB (or the gain is – 3dB), and no longer pay attention to the nature of the physical quantity itself- 3dB is an important indicator, which will often be encountered later.

You may feel a little confused. The normalized unit diagram is also very good. Why do you add trouble to yourself by changing the ordinate into DB? In fact, the advantage of Bode diagram is that it can turn the nonlinear “gain frequency” curve into a linear graph. We will actually use Bode diagram in the next section to analyze the simplest RC circuit, and you will understand the benefits of Bode diagram.

### (1) Basic parameters of RC circuit

The figure below shows a simple RC high pass filter circuit. Although it is not an amplification circuit, we can convert the output voltage v_{o}And input voltage V_{i}The ratio of is regarded as voltage gain to analyze the relationship between voltage gain and frequency. (the following symbols of AC voltage and current are represented by their phasor symbols)

Figure 10-2.05

When the circuit works at medium and high frequency, capacitor C can be regarded as short circuit and output v_{o}Equal to input V_{i}, its gain reaches the maximum value of 1, that is:

When the circuit works at low frequency, capacitor C will hinder the AC, making the output v_{o}Less than input V_{i}, according to the basic knowledge of capacitor in circuit principle, the expression of capacitive reactance is:

(Note: Although capacitive reactance x_{C}The unit of is also ohm, but it is an imaginary number in phasor value.)

Output voltage V_{o}The expression is:

Therefore, the amplitude expression of the output voltage is:

Its actual gain a_{v}Is the ratio of output voltage to input voltage amplitude, i.e.:

As can be seen from the above formula, when the frequency f tends to 0, X_{C}Tends to infinity, and the actual gain tends to 0; When the frequency f tends to infinity, X_{C}Tends to 0 and the actual gain tends to 1.

Now let’s calculate the cut-off frequency f_{L}: we want the frequency to be f_{L}Actual gain a_{v}And theoretical maximum gain a_{vmax}The ratio of is 0.707:

The above formula can be solved when f is:

The normalized gain of voltage amplitude is 0.707. The voltage amplitude gain frequency diagram is roughly shown in the following figure:

Figure 10-2.06

### (2) Use Bode diagram

OK, now let’s see how to linearize the above frequency diagram using Bode diagram. We have obtained the expression of normalized gain of voltage amplitude as follows:

Take the f calculated above_{L}A slight change can be obtained:

Substituting this (1 / 2 π C) into the normalized gain of the above formula, we can get:

Now, the normalized gain of this voltage amplitude is expressed in decibels. The calculation process is as follows:

When f is much greater than the cut-off frequency f_{L}(f) in the above formula_{L}/f) The value of item 2 will be much less than 1, so it can be approximated; And when f is much less than f_{L}When, 1 in the above formula can be approximated; Therefore, the gain of voltage amplitude can be written as a piecewise function:

If the above piecewise function is drawn on the Bode diagram, the 0dB of the first formula is a horizontal line coincident with the x-axis, which is relatively simple. The key is the second formula, which needs to be simplified again to see its linearity:

In the above formula, the preceding term – 20lg (f)_{L}）Is a constant. In the case of log abscissa, LG (f) can be regarded as a whole, that is, the x-axis factor. Therefore, the pure expression of the above formula on the log coordinate graph can be regarded as:

This is a linear expression, but it is only linear on the “DB logarithm diagram”. If the above piecewise function is drawn on the bird diagram, it is shown in the figure below:

Figure 10-2.07

Its frequency f is less than F_{L}The voltage amplitude normalized gain anorm is approximately a slope k equal to 20 and passes through F on the abscissa_{L}Straight line of point; When frequency f is greater than f_{L}When,. The voltage amplitude normalized gain anorm is approximately a horizontal straight line coincident with the x-axis.

According to our explanation of the use of “ten octave” in the previous section, this straight line with a slope of 20 can be called: 20dB / ten octave. In practical use, sometimes people think that the coefficient of 20 is too large and inconvenient to write; For the same slope, sometimes a smaller unit is used: / octave (that is, the number of decibel changes in the ordinate when the frequency is doubled). “20dB / octave” is approximately equivalent to “6dB / octave”. The conversion method is as follows:

When F2 / F1 = 2:

The final problem is, after this linearization, in F ≫ F_{L}And f ≪ f_{L}The approximation is better in the case of F, but it is close to F_{L}The position is not accurate and needs to be corrected manually. So let’s calculate a few more f, which is closer to F_{L}Value at:

Then fit them to the Bode diagram, so the real voltage amplitude gain curve is shown as the green curve in the following figure:

Figure 10-2.08

After understanding how the Bode diagram linearizes the curve, we can use the Bode diagram not only in the normalized unit, as long as the ordinate is any ratio (such as V)_{o}/V_{i}）, can be represented on the bird diagram.

Finally, it is worth mentioning that this linear approximation method is very useful in the era when there was no computer before (byrd first published this Byrd diagram method in 1945). It can help people greatly reduce the amount of calculation and facilitate the analysis of circuits or systems. This method is not outdated even today, and needs to be well mastered, because it can quickly grasp the main characteristics of the system.

However, in today’s era when computers have become popular, it is not difficult to draw accurate gain diagrams or response diagrams. For example, using Python’s SciPy and Matplotlib libraries can quickly draw accurate bode diagrams and other curves, and such good software is actually free. I recommend learning to live up to the good times of this era.

### (3) Phase analysis

Bode diagram can only linearize the gain amplitude, but can not linearize the phase response. We can only honestly calculate several typical values, and then fit these points into a curve on the diagram. Here we will demonstrate how to draw the logarithmic diagram of the phase response of the RC circuit above.

Rewrite the original expression of the voltage gain above as follows:

Its phase response θ The expression is:

When f ≪ f_{L}When:

When f ≫ f_{L}When:

Calculate the values at several typical frequencies:

According to the phase values of these points, we can fit an approximate phase response curve, as shown in the figure below:

Figure 10-2.09

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（ end of 10-2）