Name: analyzing mixed-frequency linear circuits
Uploaded: 2025-05-22T09:06:14.000Z
Duration: 1 min 13 s
Channel: JoVE

Принцип суперпозиции находит применение при анализе линейных цепей переменного тока с несколькими независимыми источниками.
Рассмотрим схему, состоящую из двух резисторов, катушки индуктивности и двух независимых синусоидальных источников с разными частотами. Теорема о суперпозиции может быть использована для определения падения напряжения на резисторе.
Согласно теореме о суперпозиции, суммарный выход линейной схемы с несколькими входами равен сумме отдельных выходов с каждого входа, работающего независимо.
Таким образом, цепь можно разделить на несколько цепей, каждая из которых имеет один синусоидальный вход. 
Поскольку импеданс зависит от частоты, для каждой частоты существуют различные схемы в частотной области.
Правило деления напряжения и закон Ома применяются для определения выходного напряжения каждой цепи.
Полученные напряжения затем выражаются во временной области.
Общее выходное напряжение представляет собой сумму напряжений, производимых каждым источником в отдельности.
Когда это падение напряжения отображается графически, это приводит к несинусоидальному выходу, хотя все входы синусоидальные.

Теорема суперпозиции для цепей переменного тока

Представьте себе схему в устойчивом состоянии, где все ее входы синусоидальные, но не все они имеют одинаковую частоту. Такая цепь не классифицируется как цепь переменного тока (AC), и, следовательно, ее токи и напряжения не будут проявлять синусоидальное поведение. Однако, эту схему можно проанализировать, используя принцип суперпозиции.

Принцип суперпозиции предусматривает, что выход линейной схемы с несколькими конкурирующими входами эквивалентен совокупным выходам, когда каждый вход функционирует независимо. Входными данными для схемы являются напряжения от независимых источников напряжения и токи от независимых источников тока.

Когда все входы, кроме одного, обнулены, остальные входы становятся источниками напряжения в 0 Вольт и источниками тока в 0 Ампер. Учитывая, что источники напряжения в 0 В соответствуют коротким замыканиям, а источники тока, равного 0 А, соответствуют разомкнутым цепям, источники, связанные с другими входами, заменяются разомкнутыми цепями или короткими замыканиями. Остается схема в устойчивой фазе, с одним синусоидальным входом, которая квалифицируется как цепь переменного тока и анализируется с использованием фазоров и импедансов.

Следовательно, принцип суперпозиции используется для преобразования схемы с несколькими синусоидальными входами на разных частотах в несколько отдельных схем, каждая из которых имеет один синусоидальный вход. Затем каждая из этих цепей переменного тока затем анализируется с использованием фазоров и импедансов для определения ее синусоидального выходного сигнала. Совокупность этих синусоидальных выходов будет совпадать с выходом исходной схемы.

Принцип суперпозиции находит применение при анализе линейных цепей переменного тока с несколькими независимыми источниками.
Рассмотрим схему, состоящую из...

Education

JoVE Core

Electrical Engineering

Электротехника

Unit 1

Раздел 1

AC Circuit Analysis

Анализ цепей переменного тока

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

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 leaves a steady-state circuit with only one sinusoidal input.

That single-input circuit can then be treated as an AC circuit. It is analyzed with phasors, which represent sinusoidal signals, and impedances, which describe how circuit parts oppose current in AC. The same process is repeated for each source in the original circuit.

The final result is found by adding the separate sinusoidal outputs together. This sum matches the output of the full circuit with multiple sinusoidal inputs at different frequencies. In this way, superposition turns one mixed-input circuit into several simpler AC circuits.

Analyzing Mixed-Frequency Linear Circuits

analyzing-mixed-frequency-linear-circuits

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 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...

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 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 superposition breaks a linear circuit with multiple sinusoidal sources into simpler AC circuits, then combines the results into one output.

Analyzing Mixed-Frequency Linear Circuits | 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

Analyzing Mixed-Frequency Linear Circuits

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