Introduction: This paper proposes a general method for drawing specific graphs on an oscilloscope using analog arithmetic circuits. By designing polar coordinate equations and analog circuits, the image of a specific function is converted into the polar coordinate form of XY signals and displayed in the Cartesian coordinate system. And the feasibility of this method is verified by the circuit example of Multisim simulation software, and graphics such as Cartesian heart line, improved rose line and heart-shaped curve are drawn. Among them, the heart-shaped line drawing circuit is comprehensive and has great significance.
关键词: Oscilloscope , analog operation circuit , polar coordinates , specific function image , XY channel , circuit design , Multisim simulation
01 Introduction
An oscilloscope is a commonly used electronic testing instrument that can be used to observe and analyze signal waveforms. Usually, the oscilloscope can display the waveform of the signal in the Cartesian coordinate system, but by using the X-Ychannels of the oscilloscope, we can observe the graph formed by the two signal waveforms in the Cartesian coordinate system.
In order to design the image of a specific function, this paper proposes a method using an analog computing circuit, which can conveniently draw a specific graph on an oscilloscope. In this method, the function equation is firstly designed based on the polar coordinate form, and then the conversion from the polar angle to the polar radius is realized by direct operation or the inverse function method by using the basic units such as the analog multiplier and the summation amplifier circuit of the analog operational amplifier circuit.
After the conversion is complete, an analog multiplier is used to convert the polar angle and polar radius into Xthe Ypolar form of the sum signal. Finally, by inputting these signals in polar coordinates X-Yinto the channel of the oscilloscope, we can draw the image formed Xwith Ythe signal in the Cartesian coordinate system, showing the specific graphics designed.
In order to verify the practicability of this method, Multisimseveral circuit examples are designed by using simulation software, and graphics such as Cartesian heart line, improved rose line and heart-shaped curve are drawn. These examples demonstrate the ability of the method in plotting different specific functions and demonstrate the effectiveness of the method. The drawing of the heart-shaped curve uses many links of analog computing circuits, which is the core design of this article. Through this example, we can see the robustness and practicability of this method.
To sum up, the general method of drawing specific graphics on the oscilloscope by using the analog operation circuit proposed in this paper provides us with a convenient tool. Through this method, we can directly display the image of a specific function graph on the oscilloscope by designing a suitable polar coordinate equation and an analog circuit, which provides a new practical means for signal processing, circuit design and teaching.
02 Design method
2.1 Overview
Signal graph drawing is to use the oscilloscope to display the signal on the display screen, and use the XY channel of the oscilloscope to draw a plane graph composed of two signals. Therefore, a circuit can be designed to draw the target function on the oscilloscope. Generally speaking, the closed graph composed of two signals is suitable for displaying through the XY channel, otherwise the corresponding relationship between the Y signal and the X signal may be infinitely extended, so this paper studies the drawing circuit of the closed graph composed of the two signals. If the equation f(x, y)=0 is given, the graph of the equation can be made under general conditions, so the two signals can be constructed through the equations satisfied by the x and y signals, so as to draw the curve represented by the equation. In order to achieve this goal, this paper proposes a general method to process and obtain target signal graphics by using analog computing circuits.
The method proposed in this paper is based on the polar coordinate decomposition of the equations satisfied by the target plane graph, which can efficiently design the circuit and draw the signal graph.
▲ 图2.1.1 信号图形绘制电路的基本框架结构
As shown in the figure above, it is a general framework for the design of circuits for drawing closed signal graphics. We have, the polar coordinate expression of a closed figure is as follows:
First, use the input sine signal to pass the phase advance link or directly use the cosine signal to obtain the important link sinθ, cosθ signal under the polar coordinate decomposition, where θ = ωt is the phase of the sine signal cosine signal. Then, through a certain operation relationship, that is, the function operation unit, the input signal can be transformed to obtain the polar diameter signal ρ. Finally, the x signal and the y signal can be obtained by performing ρcosθ and ρsinθ operations respectively through two multipliers. At this point, through the XY display mode of the oscilloscope, you can see the target signal graph appear on the display.
2.2 Design of the function operation unit
By analyzing the process of drawing the signal graph, it can be seen that the most complicated link is the design of the function operation unit, which is the core link of whether the circuit can correctly draw the target signal. In fact, functional computing units can be divided into two categories according to the functional form of the closed graph, that is, simple models with explicit polar coordinate equations and complex models without explicit polar coordinate equations. The following will introduce the design methods of the functional operation units for these two different models one by one.
2.2.1 Simple model-explicit equation method
When the polar coordinate expression of the objective function graph can be explicitly written as the following formula,
Then we can use the direct construction method of explicit equations to directly construct ρ by using basic arithmetic circuits that can realize the four arithmetic operations. For example, for the drawing of a circular curve, the polar coordinate equation is ρ = R, that is, ρ is a constant, and a DC signal can be directly used as ρ, and then an analog multiplier is used to construct x, y signals. This method is simple, intuitive, and easy to implement. Fortunately, a considerable part of function graphs can be explicitly expressed in the form of polar coordinate equations, and there are even books dedicated to recording these beautiful polar coordinate curves.
2.2.2 The method to solve complex closed graphs - inverse function circuit method
For some complex closed figures, their polar coordinate expressions may not have an explicit solution, so it is necessary to use the inverse function to obtain ρ by adding an inverse function link circuit. For example, the rectangular coordinate expression of the cardioid curve is ( x 2 + y 2 − 1 ) 3 − x 2 y 3 = 0 \left( {x^2 + y^2 - 1} \right)^3 - x^2 y^3 = 0(x2+y2−1)3−x2 y3=0 , it can be transformed into polar coordinate formρ 5 sin 2 θ cos 3 θ = ( ρ 2 − 1 ) 3 \rho ^5 \sin ^2 \theta \cos ^3 \theta = \left( {\rho ^ 2 - 1} \right)^3r5sin2icos3i=( r2−1)3 . This is a complex quintic equation with no explicit solution. By moving the two unknowns to both sidessin 2 θ cos 3 θ = ( ρ 2 − 1 ) 3 / ρ 5 \sin ^2 \theta \cos ^3 \theta = \left( {\rho ^2 - 1} \right)^3 /\rho ^5sin2icos3i=( r2−1)3/ p5 , also givef 1 ( ρ ) = f 2 ( sin θ , cos θ ) f_1 \left( \rho \right) = f_2 \left( {\sin \theta ,\cos \theta } \right )f1( p )=f2(sini ,cosθ ) equation. Through the inverse function circuit method, the sine and cosine signals are passed through the operation circuit to formf 1 ( sin θ , cos θ ) f_1 \left( {\sin \theta ,\cos \theta } \right)f1(sini ,cosθ ) signal potential; then through the inverse function link, its output is regarded as ρ, and ρ is constructed through a reasonable polarity operation to construct f1(ρ) signal and lead to the input terminal of the corresponding follower, and f2(sinθ, cosθ) signal, thus constructing a method of obtaining the polar radius in an inverse function link.
However, this method has major disadvantages. First, it is an equation of the form f1(ρ) = f2(sinθ, cosθ). If there is no explicit solution, there may be multiple solutions. The polarity in the process of constructing f1(ρ) You need to be very careful, and such circuits are often very complicated, and may even use electronic switches depending on the polarity of f1(ρ). Therefore, only a few circuits with explicit polar coordinate expressions are constructed in this paper.
▲ 图2.2.1 心形曲线
03Design Example
3.1 Cartesian heart line circuit
The Cartesian heart line is a curve that was first studied in depth by the French mathematician Descartes. The study of polar coordinate curves is of great significance in mathematics. The general formula in polar coordinate form is:
ρ = a ( b + sin θ ) ( a > 0 , b ≥ 1 ) \rho = a\left( {b + \sin \theta } \right)\,\,\,\left( {a > 0,b \ge 1} \right) r=a(b+sini )(a>0,b≥1)
This part uses analog circuit design and uses an oscilloscope to draw this magic curve.
3.1.1 Circuit Design Analysis
▲ 图3.1.1 笛卡尔心脏线电路
▲ 图3.1.2 仿真的结果
The circuit shown in the figure is a simple Cartesian heart line drawing circuit. Run this circuit and adjust the oscilloscope to XY mode to observe a typical Cartesian heart line.
The analog multiplier in the circuit satisfies uo = kux ⋅ uy u_o = ku_x \cdot u_yuo=k ux⋅uy. The operational relationship of the operational amplifier is
u O = R f ( u I 1 + u I 2 R ) u_O = R_f \left( { { {u_{I1} + u_{I2} } \over R}} \right) uO=Rf(RuI 1+uI2)
The resistance value of the resistor and the amplification ratio of the multiplier are both adjustable parameters.
First, the circuit uses AC signal sources V3 and V4 as sinusoidal signal sources. The actual circuit can use a function generator to generate a sine wave, and lead the sine wave through an integral or differential circuit with an appropriate proportional coefficient to obtain a phase ahead of the sine signal by π / 2 \pi /2The cosine signal of π /2 .
Then, the circuit leads the sinusoidal signal to a balanced resistor non-inverting adder composed of an integrated operational amplifier, and obtains the transfer function U 1 ( ω t ) U_1 \left( {\omega t} \right)U1( ω t ) by the parameters of the operational amplifier:
U 1 ( ω t ) = R f [ V 4 ( ω t ) R + V 5 R ] U_1 \left( {\omega t} \right) = R_f \left[ { { {V_4 \left( {\omega t} \right)} \over R} + { {V_5 } \over R}} \right] U1( t ) _=Rf[RV4( t ) _+RV5]
▲ 图3.1.3 典型的玫瑰线
▲ 图3.1.4 带有萼片的玫瑰线
▲ 图3.1.5 实现的电路
▲ 图3.1.6 频率之比非整数的图片
▲ 图3.1.7 工作在线性区域
▲ 图3.1.8 工作在非线性区域
▲ 图3.1.9 三叶草电路
▲ 图3.1.10 绘制的三叶草
▲ 图3.1.11 爱心电路设计
▲ 图3.1.12 爱心电路图形
04 Insufficiency and reflection
This paper introduces a method of drawing a specific function image on an oscilloscope by using an analog operation circuit, and demonstrates the feasibility and practicality of the method. However, there is still room for improvement and further research in this method.
First, although we have shown a few instances of specific function graphs in this paper, there are many other interesting functions and graphs that can be explored. Future research can further expand the scope of application, cover more different types of function images, and explore more innovative design methods and circuit structures. The accuracy and variety of drawn images can be further improved by exploring the properties of different functions and using more complex circuit structures.
Moreover, the research in this paper is limited to circuit verification using Multisim simulation software. Future work can consider implementing and testing these circuits on actual hardware platforms to verify their feasibility and effects in practical applications. With actual hardware implementation, the performance of the circuit can be more accurately evaluated and discrepancies between simulated and real environments can be resolved.
Finally, this article does not discuss in depth the drawing of the inverse function method. In fact, there are many polar coordinate expressions of function graphs that can only be expressed by implicit equations. This needs to be solved through further research in the future.
And in the process of designing and building the heart-shaped line drawing circuit in this paper, the square root operation circuit has a serious locking phenomenon, and this paper only adopts a more complicated solution. In fact, the method of eliminating this phenomenon is worthy of further research, and there is a more convenient solution to the locking phenomenon.
05 Epilogue
This paper introduces a general method to draw a specific function image on an oscilloscope by using an analog operation circuit, and verifies the feasibility and practicability of the method. By designing polar coordinate equations and analog circuits, we can convert the image of a specific function into the polar coordinate form of the XY signal and display it in the Cartesian coordinate system.
Through the research of this paper, we found that the signal graph drawing method based on analog computing circuit has important application value in the field of circuit design and signal processing. This method can not only realize the drawing of specific function images, but also can be used in the development and verification of signal processing algorithms, visual display of circuit design, and teaching and learning. Moreover, the signal drawing process of the analog operation circuit involves a lot of rich knowledge of the analog circuit, and the graphics drawing of the analog operation circuit can integrate more analog electronic technologies, which is of great significance for learning and teaching.
However, we also recognize that the method still has room for improvement and further research. Future work can consider adopting digital signal processing technology and more complex circuit structures to improve the accuracy and diversity of image rendering. In addition, the actual hardware implementation and the expansion of application fields are also important directions for future research.
We hope that the research results of this paper can provide valuable references for researchers in the field of circuit design and signal processing, and inspire more innovative ideas about circuit design and signal graph drawing. Through continuous exploration and improvement, we are expected to further improve the method and make greater contributions to the development of related fields.
references
[1] Tong Shibai, Hua Chengying. Basics of Analog Electronics Technology (Fifth Edition) [M]. Beijing: Higher Education Press, 2015. [2]
Basic courseware of Analog Electronics Technology
[3] GGB Drawing: Graceful Polar coordinate curve https://zhuanlan.zhihu.com/p/530461328
[4] Wolfram Mathworld Heart Curve : https://mathworld.wolfram.com/HeartCurve.html
● Links to related diagrams:
- Figure 2.1.1 The basic frame structure of the signal graph drawing circuit
- Figure 2.2.1 Cardioid curve
- Figure 3.1.1 Cartesian heart line circuit
- Figure 3.1.2 Simulation results
- Figure 3.1.3 Typical Rose Line
- Figure 3.1.4 Rose line with sepals
- Figure 3.1.5 Realized circuit
- Figure 3.1.6 A picture with a non-integer frequency ratio
- Figure 3.1.7 Working in the linear region
- Figure 3.1.8 Working in the nonlinear region
- Figure 3.1.9 Clover circuit
- Figure 3.1.10 Drawn clover
- Figure 3.1.11 Heart circuit design
- Figure 3.1.12 Heart circuit graphics