Name: impedance matching with mesh currents
Uploaded: 2024-04-11T12:48:49.000Z
Duration: 1 min 12 s
Channel: JoVE

在无线电通信中，阻抗匹配电路用于匹配无线电发射机和接收机的阻抗。
考虑一个基本的阻抗匹配电路，该电路由一个电阻器组成，后接一个电容器和一个电感器，二者并联连接。
如果已知输入和输出信号，则可以计算流经电容器的电流。
从输入电压的时域表达式中提取的角频率，用于确定电感器和电容器的阻抗。
变换后的频域电路以极坐标形式表示阻抗以及输入和输出信号。
进行分析时，需为每个网孔分配网孔电流，并对每个网孔应用基尔霍夫电压定律，从而得到一组线性联立方程。
这些方程可以用矩阵形式表示，然后利用克莱姆法则求解网孔电流。
公共支路中的电流是网孔电流之差。
通过代入计算得到的网孔电流，可求得电容器上的电流的极坐标形式，然后将其转换到时域。

交流电路的网格分析

在无线电通信领域，必须能够考虑到阻抗匹配的重要性，并以此来确保无线电发射器和接收器之间的信号能够进行有效传输。实现这种平衡需要使用阻抗来匹配电路，其中的基本配置包括电阻器、电容器和电感器。 协调这些阻抗的过程起始于对输入信号和输出信号的清晰理解。一旦明确这些信号，下一步就是需要计算出流过该电路中电容器的电流。 从输入电压的时域表达式中所得到的角频率起着至关重要的作用。它是用来确定电感器和电容器中阻抗值的主要因素。 <img alt="Equation1" src="/files/ftp_upload/15111/15111_PT_Equation_1.svg" style="text-align: center; height: 16px;"> <img alt="Equation2" src="/files/ftp_upload/15111/15111_PT_Equation_2.svg" style="text-align: center; height: 40px;"> 然后将电路变换到频域。这种表示形式包括阻抗、输入信号和输出信号，并且全部以极坐标的形式来进行表示，从而以此来简化分析过程。为了能够更深入地研究电路的运行原理，便为其分配了网格电流，同时并应用了网格分析中的基本原理——基尔霍夫电压定律 (KVL)。重要的是，网格分析特别适用于平面电路。 这种细致的分析结果能够得出一组联立的线性方程，同时还可以用矩阵的形式来进行表示。克莱默规则的作用是揭示网格电流，从而以此来确定电容器中的共享电流。 提供替换计算出的网格电流能够得出流经电容器的电流，并将其以极坐标形式来进行表示。因此，这些数据都会被巧妙地转换到时域中，并以此来理解和优化阻抗中的匹配电路。

在无线电通信中，阻抗匹配电路用于使无线电发射机和接收机的阻抗相匹配。  
考虑一个基本的阻抗匹配电路……

Education

教育

JoVE Core

Electrical Engineering

电气工程

Unit 1

第一单元

AC Circuit Analysis

交流电路分析

KEY TERMS AND CONCEPTS

关键术语与概念

Mesh Analysis for AC Circuits

impedance matching with mesh currents

Impedance matching in radio communication uses resistor, capacitor, and inductor circuits to help signals move efficiently between a transmitter and a receiver. The circuit must balance the input and output signals before the current in the capacitor can be found. This makes the problem a good example of AC circuit analysis for high school students.

The first step is to identify the input voltage and the output signal. The angular frequency is then taken from the time-domain form of the input voltage. That frequency is used to find the impedance of the inductor and the capacitor, since both depend on frequency in AC circuits.

Next, the circuit is moved into the frequency domain. In this form, the signals and impedances are written in polar form, which makes the math easier to handle. Mesh currents are then assigned to the planar circuit, and Kirchhoff's voltage law, or KVL, is applied around each loop.

The result is a set of linear equations. These equations can be written in matrix form. Cramer's rule is then used to solve for the mesh currents and find the current shared by the capacitor. After substituting the mesh currents, the capacitor current is converted from polar form back into the time domain to complete the impedance-matching analysis.

Impedance Matching with Mesh Currents

sinusoidal sources

AC Waveform Basics and Phase

Alternating current, or AC, changes direction and magnitude over time, unlike direct current, or DC, which flows in one direction with constant polarity. AC is the main form of electricity used for modern power transmission and distribution because it can travel long distances efficiently. In homes, power supplies use sinusoidal sources to deliver electricity. A sinusoidal source creates a voltage that rises and falls in a smooth wave-like pattern. This waveform is called a sinusoid, which is...

graphical and analytic representation of sinusoids

graphical-and-analytic-representation-of-sinusoids

Measuring Sinusoid Phase on an Oscilloscope

Measuring sinusoid phase on an oscilloscope helps compare two sinusoidal voltages with the same amplitude and period. An oscilloscope is an instrument used to display and analyze waveforms. The process starts by reading the peak-to-peak value, which is twice the amplitude of the sinusoid. This shows the maximum voltage swing of the waveform. Next, the period and angular frequency are measured. The period is the time for one complete cycle. Angular frequency, often written as omega, is the rate...

phasors

Phasors for AC Circuit Analysis

Phasors are a mathematical tool for analyzing alternating current, or AC, circuits. They represent sinusoids with complex numbers. The phasor magnitude matches the sinusoid amplitude, and the phasor angle shows the phase measured from the positive x-axis. Phasors make AC circuit analysis simpler by removing time from the current and voltage expressions. That lets an AC circuit be treated like its equivalent direct current, or DC, form for calculations. Phasors can also be written in...

phasor arithmetics

Phasor Rules for AC Circuit Analysis

Phasor rules make AC circuit analysis easier by linking sinusoids to their matching phasors. A phasor is a complex-number form of a sinusoidal signal. This connection helps explain how alternating current changes with time. Differentiation and integration work in both the time domain and the phasor domain. If you take the derivative of a sinusoid in time, its phasor is multiplied by jω, where j is the imaginary unit and ω is the angular frequency. If you integrate a sinusoid in time, its...

phasor relationships for circuit elements

phasor-relationships-for-circuit-elements

AC Circuit Phasors: Resistor, Inductor, Capacitor

AC circuit phasors help show how voltage and current behave for resistors, inductors, and capacitors. Phasor representation moves these relationships from the time domain to the frequency domain, which makes AC circuit analysis easier. For a resistor, Ohm's law still applies. The voltage across the resistor and the current through it stay in phase, so their phasors point in the same direction. An inductor works differently because its voltage depends on the rate of change of current. The...

kirchoff's laws using phasors

AC Circuit Rules with Phasor Diagrams

AC circuit analysis uses Kirchhoff's voltage law and Kirchhoff's current law with phasors. In household wiring, AC power from a distribution panel is sent to appliances connected in parallel. These rules help describe how voltage and current behave in sinusoidal steady state. Kirchhoff's voltage law, or KVL, says that the sum of phasor voltages around a closed loop in an AC circuit is zero. The voltages can first be written in the time domain and then changed into phasor form. In phasor...

impedances and admittance

AC Circuit Response: Impedance and Admittance

AC circuit response changes with frequency, and impedance describes how resistors, inductors, and capacitors affect sinusoidal current flow. In phasor form, voltage and current are used to show this behavior in a clear way. Impedance is a key idea in AC circuit analysis because it measures opposition to current in an alternating circuit. These components behave differently in AC than in DC. In DC circuits, inductors act like short circuits and capacitors act like open circuits. In AC circuits,...

impedance combination

Series and Parallel Impedance Rules

Impedance in series and parallel circuits follows the same rules used for resistors. A string of Christmas lights is a simple example. Each bulb represents one impedance element. In a series connection, the current stays the same through every component. This series behavior matches Kirchhoff’s Voltage Law, or KVL. The total impedance in the circuit is the sum of the individual impedances. The source voltage is then divided among the components based on their values. This is the voltage...

node analysis for ac circuits

AC Circuit Node Method in an Angioplasty Model

AC circuit node analysis is used to study an angioplasty system with a catheter turbine. The system helps remove plaque deposits from coronary arteries. Its electrical model is similar to a dual-node RLC circuit driven by a current-controlled voltage source. The analysis begins by collecting the input source voltage, inductance, and capacitance values. These measurements are used to find the driving voltage for the catheter shaft. The circuit is then viewed in the frequency domain, where...

mesh analysis for ac circuits

Impedance matching in radio communication uses resistor, capacitor, and inductor circuits to help signals move efficiently between a transmitter and a receiver. The circuit must balance the input and output signals before the current in the capacitor can be found. This makes the problem a good example of AC circuit analysis for high school students. The first step is to identify the input voltage and the output signal. The angular frequency is then taken from the time-domain form of the input...

source transformation for ac circuits

AC Source Conversion in Circuit Analysis

AC source conversion is a circuit analysis method used in the frequency domain. It changes a voltage source in series with an impedance into a current source in parallel with the same impedance, or it converts a current source back into a voltage source. The source values must keep the correct relationship during each change. This method helps solve for an unknown voltage in a circuit that includes a current source and components such as resistors, capacitors, and inductors. Each of these...

th&#233venin equivalent circuits

Finding Voltage and Impedance in Circuit A

The household power distribution system can be simplified with Thevenin's theorem. In this system, distribution lines and transformers form the main network. Electrical appliances can be treated as load impedance, which makes the circuit easier to analyze. An AC circuit can be split into two parts, circuit A and circuit B, with a single pair of terminals between them. Circuit A can then be replaced by its Thevenin equivalent circuit. That equivalent has a voltage source in series with an...

norton equivalent circuits

Norton Current in AC Circuit Analysis

Norton’s theorem helps simplify complex AC circuits by replacing a two-terminal linear network with a smaller equivalent circuit. That equivalent circuit uses an impedance in parallel with a constant current source. The method lets students break one complicated circuit into two parts and analyze them more easily. To build the Norton equivalent, the source is replaced by its internal impedance. The result is the Norton impedance, which is the same as the Thévenin impedance. This parallel...

superposition theorem for ac circuits

Analyzing Mixed-Frequency Linear Circuits

Analyzing mixed-frequency linear circuits starts with the principle of superposition. In a linear circuit, the total output from several inputs is the sum of the outputs from each input acting alone. Here, the inputs are the voltages from independent voltage sources and the currents from independent current sources. When one source is kept active, every other independent source is set to zero. A 0-V voltage source becomes a short circuit, and a 0-A current source becomes an open circuit. This...

op amp ac circuits

Analyzing an Op Amp Audio Filter

An op amp audio filter shapes an amplified sound signal by strongly reducing the lower-frequency parts. In this circuit, the focus is on a basic filter setup used after an amplifier in an audio system. The filter uses an operational amplifier in an inverting arrangement with resistors R1 and R2 and a capacitor C1. The analysis starts with a known input signal and asks for the output signal. To solve it, the capacitor impedance is found first. That requires the angular frequency from the...

design example: capacitance multiplier circuit

design-example-capacitance-multiplier-circuit

Capacitance Multiplier Circuit for Larger Capacitance

A capacitance multiplier circuit creates a larger effective capacitance from a small physical capacitor. In integrated circuit technology, this approach is useful when a small capacitor is not enough. The circuit can multiply capacitance by as much as 1000, so a 10-pF capacitor can act like a 100-nF capacitor. The design uses two op-amps. The first op-amp works as a voltage follower, which helps isolate the generated capacitance from the loading caused by the second stage. The second op-amp...

key terms and concepts

Learn how mesh currents, KVL, and Cramer's rule are used to analyze an AC impedance-matching circuit and find capacitor current in radio communication.

Impedance Matching with Mesh Currents | JoVE K12

JoVE Core Electrical Engineering offers easy-to-understand animated videos that cover fundamental concepts of electric circuits along with circuit analysis techniques. It incorporates first-order and second-order circuits using operational amplifiers and circuit transformation concepts. The inclusion of practical design examples and real-world applications through scientist-in-action videos adds significant educational value, making complex concepts accessible.

electrical engineering

Impedance Matching with Mesh Currents

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