AC Circuits: Alternating Current, Waveforms, and Formulas

AC circuits (alternating current circuits) are electrical circuits where the current and voltage periodically change their direction and magnitude over time. This property is essential in transmitting electrical power and household appliances, usually represented by a sine wave. The main components include resistors, capacitors, and inductors, which can be combined to create circuits with various functionalities. Understanding AC circuits requires analyzing waveforms and characteristics and calculating and designing skills. Predicting circuit behavior, identifying issues, and executing optimal designs through precise calculations and analyses are possible.

This article explains the operational principles and essential elements of AC circuits.

AC Waveform

An alternating current (AC) waveform refers to a pattern of current or voltage that periodically changes in amplitude and direction over time. This waveform is fundamental to the electricity used in homes and industries, allowing for the efficient transfer and utilization of electrical energy. Among AC waveforms, the sine wave, with its smooth fluctuations, is optimal for efficiently supplying power while minimizing energy loss. Understanding AC waveforms is essential for the stable and high-quality use of electricity.

What is AC（Alternating Current）?

Alternating Current (AC) refers to the flow of electricity in which the direction and intensity of the current change periodically over time. AC is the standard form of power used in homes and industries, and it can be efficiently transmitted from power stations to distant locations. The periodic nature of this current makes it suitable for operating various electrical devices.

The starting point for AC circuits is the AC power source, which typically has a sine wave waveform. A sine wave changes periodically, and its cycle and amplitude are crucial parameters. These periodic changes enable power supply and signal transmission.

• Period: The period is the time it takes for the waveform to complete one cycle, typically denoted by \(T\) and measured in seconds (s).
• Frequency: Frequency indicates how often the waveform oscillates per unit time, usually denoted by \(f\) and measured in hertz (Hz).
• Amplitude: Amplitude indicates the magnitude of the swing to the maximum or minimum values of the waveform, usually denoted by \(A\). It shows the maximum voltage or current values, indicating the waveform’s strength and energy.

Frequency is the inverse of period, representing the number of cycles per second, denoted in Hz. A higher-frequency waveform has a shorter period. For example, a 50 Hz power source periodically changes 50 times per second.

AC Waveform Types

AC circuits refer to circuits where current and voltage change over time. A characteristic feature is the AC waveform, typically a periodic waveform like a sine wave. AC waveforms come in various types, each with different characteristics.

Regular Periodic Waveforms

Sine Wave: A continuous and smooth periodic waveform with important parameters like amplitude, frequency, and phase angle.
Square Wave: Alternates between high and low voltage levels, showing abrupt changes.
Triangular Wave: Similar in shape to a sine wave but changes more abruptly.
Sawtooth Wave: Continuously rising waveform, primarily with a single increasing gradient, showing abrupt changes.

Period vs. Frequency

Frequency and period of time are fundamental characteristics of AC waveforms. Frequency indicates how often the waveform repeats per unit time, and period time represents the time it takes to complete one cycle.

• Frequency: Frequency indicates how many times a waveform repeats per unit of time. Generally, frequency is expressed in hertz (Hz), where 1 hertz indicates one cycle per second. The frequency (\(f\)) can be calculated using the following:

\(\text{Frequency (f) } = \displaystyle \frac{1}{\text{Period (T)}}\)

Here, \(f\) represents the frequency, and \(T\) represents the period. Frequency indicates the speed of oscillation or repetition rate of a waveform, where a high frequency suggests a short period and a low frequency indicates an extended period.

Period: The period is the time it takes for a waveform to complete one cycle. The period is usually expressed in seconds (\(s\)). The relationship between period (\(T\)) and frequency (\(f\)) is shown in the following:

\(\text{Period (T)} = \displaystyle \frac{1}{\text{Frequency (f)}}\)

Period time defines the periodicity of the waveform, with shorter period times indicating faster oscillations and longer period times indicating slower oscillations.

Average and RMS Value of AC and Voltage

Average Value of AC

The average value is an indicator that shows the mean amplitude of a waveform. Typically, the average value is denoted as “AVE” or “μ” and can be calculated using the following formula:

\(V_{\text{AVE}} = \displaystyle \frac{V_1 + V_2 + V_3 + \ldots + V_N}{N} = \displaystyle \frac{1}{T} \int_0^T V(t) \, dt\)

Here, Ave represents the average value, \(T\) represents the period, and \(V(t)\) represents the value of the waveform at time \(t\). The average value indicates the mean amplitude of the entire waveform, considering both the positive and negative regions. The average value identifies the center of the waveform; in symmetrical waveforms, the average value is zero.

The average value is crucial in various applications such as power supply, signal processing, and current control.

Power Supply: The average power value indicates power efficiency in AC power supply.

Signal Processing: This is for tasks like reducing the level of DC component noise.

Current Control: Control is essential in motor drives and control systems applications.

RMS Value of AC

The root mean square (RMS) value is an indicator of the average energy of a waveform：

\(V_{\text{RMS}} = \sqrt{\displaystyle \frac{V_1^2 + V_2^2 + V_3^2 + \ldots + V_N^2}{N}} = \sqrt{\displaystyle \frac{1}{T} \int_0^T [V(t)]^2 \, dt}\)

Here, RMS refers to the RMS value, \(T\) is the period, and \(V(t)\) represents the value of the waveform at time \(t\). The RMS value shows the average energy of the amplitude of the waveform. It is calculated as the square root of the average squared values of the waveform’s amplitude.

The RMS value is essential for power supply, signal processing, and efficiency.

Power supply: Indicates the actual voltage and current values, i.e., the actual power.
Signal processing: Assesses the strength of signal energy.
Power efficiency: Evaluate energy losses in the performance assessment of power equipment.

Essential Elements of AC Circuit

The essential elements of an AC circuit include the average value, RMS value, phase difference, complex numbers, impedance, and series and parallel circuits. The average value represents the time average of the waveform. In contrast, the RMS value indicates the value of a direct current voltage equivalent to the power effect of the AC waveform. The phase difference indicates the time shift between waveforms and plays a vital role in waveform harmonization. Complex numbers and impedance concepts are indispensable elements in AC circuit analysis. The impedance is calculated in series and parallel circuits using the components’ connection method.

Average Value and RMS Value

The waveform’s Average and RMS values are fundamental indicators in AC circuits. The Average value is the time average of the waveform, dividing the total area of the waveform by time. In contrast, the RMS value relates to the waveform’s amplitude and represents the value of a DC voltage with the same power effect as the AC waveform.

Phase Difference

In AC circuits, a phase difference might occur between multiple waveforms, indicating a temporal shift within the waveform cycle, which can affect characteristics like waveform harmony. Moreover, waveform delays represent temporal delays occurring within the circuit, which is crucial for signal processing and control systems and essential in waveform synthesis and circuit design.

Complex Numbers

Using complex numbers and polar coordinates is highly beneficial in analyzing AC circuits. Complex numbers consist of a real and an imaginary part, compactly representing waveform amplitude and phase information. Polar coordinates help convert complex number representations and intuitively understand waveform amplitude and phase.

Impedance

Impedance is equivalent to resistance in AC circuits but also considers reactance from components like coils and capacitors. It is represented as a complex number, and its magnitude and phase characterize the waveform’s properties.

Series and Parallel Circuits

Components are connected in series circuits, and their characteristics are calculated using Ohm’s law and phasor notation to determine total impedance and current. Components are connected in parallel circuits, and their characteristics are defined using the reciprocal rule and phasor notation to calculate total impedance and current.

As shown in the diagrams, circuits with resistors (\(R\)), coils (\(L\)), and capacitors (\(C\)) connected are known as RLC series circuits and RLC parallel circuits, respectively. They share the common characteristic of variable impedance with frequency. In series circuits, the impedance is affected by the sum of elements, whereas in parallel circuits, the impedance of each component coexists and influences the overall impedance.

Calculations of AC circuits

Calculations of AC circuits are crucial for designing and verifying AC systems where voltages and currents vary with time. In AC circuits, it is essential to understand the behaviors of basic components such as resistors, capacitors, and inductors, and based on this understanding, apply Ohm’s Law and Kirchhoff’s Law. Additionally, calculating impedance using complex numbers is indispensable, enabling accurate evaluation of the overall performance of AC circuits. This understanding and application are vital for achieving efficient and stable electrical systems.

Complex Numbers

Introduction to Complex Numbers

Complex numbers consist of real and imaginary parts and are usually expressed as \(a+bj\). Here, ” \(a\) ” represents the real part, and ” \(b\) ” represents the imaginary part. The imaginary unit ” \(j\) ” of complex numbers has the property \(j^2=-1\)：

\(z = A \cdot e^{j\theta}\)

where ” \(A\) ” is the magnitude, ” \(θ\) ” is the phase angle, and ” \(j\) ” denotes the imaginary unit.

Alternating current waveforms often utilize complex numbers. By combining amplitude and phase information, complex numbers help accurately capture changes in waveform amplitude and phase differences. Combining complex numbers to add and subtract makes it possible to synthesize multiple waveforms. Complex numbers are also used in impedance calculations. The impedances of coils and capacitors are represented as complex numbers, which aids in calculating combined impedance.

Electrical Impedance in AC Circuits

In alternating current circuits, impedance is a critical element that defines the relationship between current and voltage. Impedance can be calculated as a combination of resistance, inductance, and capacitance. It can be considered a complex resistance that specifies the relationship between current and voltage within an AC circuit.

Impedance formula

Impedance is generally expressed as a complex number consisting of a real part (resistance) and an imaginary part (reactance) and is represented as follows:

\(Z = R + jX\)

Here, \(Z\) is the impedance, \(R\) is the resistance, \(X\) is the reactance, and \(j\) is the imaginary unit.

Electrical resistance

Resistance is an element that directly controls the relationship between current and voltage. The resistance value (\(R\)) is expressed as the ratio of voltage (\(V\)) to current (\(I\)), following Ohm’s Law.

\(V = I \times R\)

Resistance affects power loss and signal attenuation, representing energy loss in AC circuits.

Resistance impedance depends solely on the resistance value and is expressed in the same units as resistance.

\(Z_R = R\)

Inductance

Inductance depends on the shape of the coil or coil winding and is an element in AC circuits that causes a delay in current. The inductance value indicates that voltage is induced when the current changes. This causes the voltage to rise when the current changes abruptly and to fall when the current changes slowly.

The impedance of a coil is represented by the coil’s inductance value (\(L\)) and angular frequency (\(ω\)) as follows:

\(Z_L = j\omega L\)

Capacitance

Capacitance is known as a capacitor, and it can store electrical charges. The capacitance value (\(C\)) indicates the accumulation of charge when there is a change in voltage. This allows for the discharge of charge when the voltage changes abruptly and for the supply of charge when the voltage changes gradually.

Capacitors are used in both direct current (DC) and alternating current (AC) circuits. In DC circuits, they store charge to supply energy, contributing to the stability of the circuit and signal rectification. In AC circuits, they exhibit frequency-dependent reactance and alter the phase difference of signals.

A capacitor comprises two conductors (usually metal plates) separated by a dielectric (insulator). The dielectric stores the charge, which is preserved as electric field energy. The capacitance of a capacitor (capacitance value, \(C\)) indicates the amount of charge it can store and is expressed in farads (\(F\)).

The impedance of a capacitor is expressed as follows, based on the capacitor’s capacitance (\(C\)) and angular frequency (\(ω\)).

\(Z_C = \displaystyle \frac{1}{j\omega C}\)

Phase Difference and Phase Shift in AC Circuits

Phase difference and reactance are essential in analyzing alternating current (AC) circuits. The phase difference indicates the temporal shift of the waveforms, and reactance represents the impedance of components such as coils and capacitors.

What causes a phase difference?

Reasons and Impacts of Phase Difference

Phase differences occur due to the different responses of coils (inductors) and capacitors to AC signals, which depend on frequency-sensitive reactance. Managing and adjusting phase differences are critical in the design and analysis of AC circuits because of the following impacts on circuit operation:

Phase difference formula

The phase difference in an AC circuit indicates the degree to which the current and voltage waveforms are shifted in time. The method of calculating the phase difference using radians is widely used.

1. Calculation of Angular Frequency:
   Using the normal frequency \(f\), the angular frequency \(ω\) is calculated as follows:
   \(\omega = 2\pi f\)
2. Calculation of Phase Difference:
   The phase difference \(ϕ\) is calculated as the product of time \(t\) and angular frequency \(ω\):
   \(\phi = \omega t\)
   For example, if the frequency \(f\) is 50 Hz, the corresponding angular frequency is:
   \(ω =2π×50\)
   If \(t\)=0.01 seconds, the phase difference is calculated as follows:
   \(ϕ＝(2π×50)×0.01\)

Calculations of reactance

Reactance represents the alternating current (AC) impedance of components like coils and capacitors and is denoted by “\(X.\)” Specifically, a capacitor’s reactance varies with the angular frequency \(ω\). Capacitors store electric fields and charge. When exposed to an AC signal, a capacitor first allows current to flow and accumulate charge; then, the current reverses and discharges the stored charge. This current delay, caused by the capacitor’s reactance, leads the current to voltage, a phenomenon observed as a phase difference.

The calculation of a capacitor’s reactance is as follows：

\(X_C = \displaystyle \frac{1}{2\pi fC}\)

Here, \(X_C\) represents the capacitor’s reactance, \(f\) is the frequency, and \(C\) is the capacitance of the capacitor. The reactance of a coil also varies proportionally with the angular frequency \(ω\). Therefore, as the frequency changes, the reactance of the coil also changes.

When an AC signal is applied to the coil, a magnetic field is generated within it, which affects its inductance. The generation and collapse of this magnetic field take time, resulting in a phenomenon where the current lags behind the voltage. This delay is observed as a phase difference.

Similarly, the reactance of the coil is calculated as follows:

\(X_L = 2\pi fL\)

Here, \(X_L\) represents the coil’s reactance, \(f\) is its frequency, and \(L\) is its inductance.

Application of Phase Difference in AC Circuits

The angular frequency affects the reactance of a coil, \(X_L=jωL\), and the reactance of a capacitor, resulting in a phase difference. When these components are incorporated into a circuit, the phase difference affects the entire circuit. For example, in an RC circuit (a combination of resistor and capacitor), the phase difference occurs due to the capacitor’s reactance and the resistor’s influence. Changing the frequency alters the reactance, changing the phase difference. This can be used to control the characteristics of the RC circuit based on frequency.

Design and Practical Application of AC Circuits

Steps in AC Circuit Design

Clarification of Objectives
The first step in circuit design is to clarify the design objectives. Identify what functions need to be achieved and what the constraints are.

Critical Points in Power Supply Selection

Choosing the proper power supply is critical to enhancing circuits’ performance, reliability, and efficiency and achieving design objectives. Plan thoroughly and select a power supply that meets the required specifications.

1. Voltage and Current Requirements
   Clearly define the voltage and current requirements for the circuit’s regular operation. These specifications ensure that the circuit functions stably and achieves its design goals. The voltage requirement refers to the voltage provided by the power supply, while the current requirement indicates the current consumed by the circuit. These specifications significantly impact the circuit’s performance and characteristics.
2. Stability and Reliability
   The selected power supply must ensure the stability of the circuit and provide reliable operation. Consider how stable the power supply is against voltage and current fluctuations and whether mechanisms to protect the circuit from overvoltage and overcurrent are necessary. A reliable power supply contributes to longer life and improved safety.
3. Efficiency and Power Consumption
   An efficient power supply contributes to energy savings and minimizes power consumption. This is especially important for battery-operated devices and applications where energy efficiency is crucial. An efficient power supply reduces heat generation and lowers energy costs.
4. Power Supply Noise and Ripple
   Noise and ripple from the power supply can negatively affect the regular operation of the circuit. When selecting a power supply, consider minimizing noise and ripple. A stable power supply improves signal quality and helps avoid issues caused by noise.
5. Protective Features
   Ensure that the selected power supply is equipped with circuit-protecting features. Features like overvoltage, overcurrent, and short-circuit protection enhance the circuit’s safety and ensure the equipment’s durability.
6. Supply Chain and Cost
   The availability and cost of the power supply should also be considered. Check whether a specific power supply will be available over the long term and whether the price fits within the budget. The reliability of the supply chain must also be considered.

How to Read a Schematic

Schematics are an essential communication tool in electronics and electrical engineering. They visually represent various components and connections to help understand the structure and operation of electronic circuits.

Schematic Symbols

In schematics, various electronic components are represented by symbols. Common component symbols include resistors, capacitors, coils, transistors, and diodes. Each element has a unique symbol that indicates its function.

Connection Lines
Connection lines represent the electrical connections between components. These lines are usually drawn as straight or curved lines, connecting component symbols to their junction points. When connection lines intersect, they are sometimes represented with a dot.

Power Supply
The power supply represents the elements that provide power to the electronic circuit. Power supplies are typically shown as symbols for voltage sources or current sources, indicating from which part of the circuit power is received.

Power Ground
The power supply and ground (ground) are essential elements in a schematic. Typically, the power supply is positioned at the top or left side of the circuit, and the ground is placed at the bottom or right side. This arrangement clarifies the path of current flow within the circuit.

LEDs and Their Applications

LEDs (Light-Emitting Diodes) are also widely used in AC circuits. This section explains how LEDs operate in AC circuits and their main applications.

Integration of LEDs into AC Circuits

LEDs typically operate on DC power, but a circuit is needed to integrate them into AC circuits. The following are the main methods for using LEDs in AC circuits.

Full-Wave Rectification

Full-wave rectification is a standard method to convert AC power to DC. Rectifier diodes are used to rectify both the positive and negative half-cycles, obtaining a DC supply. This DC supply powers the LED, causing it to light up. Bridge rectification is a variation of full-wave rectification, which uses four rectifier diodes to rectify AC power. This method provides high efficiency and stability and is widely used for powering LEDs.

Half-Wave Rectification

Half-wave rectification rectifies only the half-cycle of the AC power and lights up the LED. This method is less efficient than full-wave rectification and is limited to specific applications.

LEDs’ applications in AC circuits are varied and utilized in various fields due to their high efficiency and durability. LEDs are essential for inefficient lighting, information displays, communication, and control systems.

Advanced Calculations (Laplace Transform, Resonance, and others)

The article above explored the basic concepts of AC circuits, including impedance, capacitors, Ohm’s law, and the fundamentals of current and voltage. This basic knowledge prepares you to advance to more sophisticated calculations and circuit design.

Laplace Transform

The Laplace transform is a mathematical technique that converts signals or responses from the time domain to the complex frequency domain. This transformation turns differential equations into algebraic equations, making analysis easier. After performing the Laplace transform, the original signal or response is restored using the inverse transform.

Resonance

Resonance is a phenomenon in AC circuits that amplifies oscillations at a specific frequency. Vibrations at the resonance frequency maximize current and voltage, producing desirable effects for particular applications.
Calculating the resonance frequency and adjusting the bandwidth are crucial in designing resonance circuits. The selection of elements and design methods for resonance circuits is focused on achieving the desired resonance characteristics.

Using advanced calculation techniques like the Laplace transform and resonance to analyze and design complex AC circuits proves beneficial in real projects and problem-solving. Acquiring advanced calculation skills allows for more sophisticated design and analysis of AC circuits, identification and resolution of issues, and efficient design.

Power Factor and Power Consumption

Understanding power factor and power consumption is essential in AC circuit design. Power factor assesses the efficiency of power use, and power consumption impacts energy efficiency and costs.

Definition of Power Factor

Power Factor (\(PF\)) in AC circuits represents the ratio between the actual useful power (active power) and the apparent power (power transmitted from the power source to the consumer). The power factor typically ranges from 0 to 1, where a high power factor indicates efficient electricity use and a low power factor signifies wasteful consumption.

Importance of Power Factor

Power factor is a crucial metric for power supply systems; a low power factor places a load on power sources and transmission lines, leading to energy waste. Maintaining a high power factor improves the power supply’s efficiency and reduces power supply costs.

Calculation of Power Factor

The power factor is calculated using the following formula with active power (W) and apparent power ( \(VA\) ):

\(\text{Power Factor (PF)} = \displaystyle \frac{\text{Active Power (W)}}{\text{Apparent Power (VA)}}\)

For instance, if a device has an active power of 100W and an apparent power of 120VA, the power factor is calculated as follows:

\(\text{Power Factor(PF)} = 100/120=0.833\)

Importance of Improving Power Factor

Improving a low power factor enhances power supply efficiency and cost reduction. Here are some methods to improve the power factor:

Addition of Series Capacitors

Adding series capacitors to a circuit can improve the power factor. Capacitors carry a lagging phase and correct improper power factors.

Active Power Factor Correction Devices

Active power factor correction devices are inserted between the power source and the consumer to improve the power factor. These devices monitor the power factor and perform appropriate corrections.