# Where to Find Alternating Current Circuits by Kerchner and Corcoran PDF Online

## Alternating Current Circuits by Kerchner and Corcoran PDF 103l

Alternating current (AC) is a type of electric current that changes its direction and magnitude periodically. Unlike direct current (DC), which flows in one direction only, AC can be used to transmit power over long distances efficiently and safely. AC is also widely used in many devices and applications that require varying voltages, frequencies, or phases.

## Alternating Current Circuits By Kerchner And Corcoran Pdf 103l

One of the classic books that covers the theory and practice of AC circuits is "Alternating Current Circuits" by Russell M. Kerchner and George F. Corcoran. First published in 1951, this book has been revised and updated several times to reflect the latest developments and standards in the field of electrical engineering. The book provides a comprehensive and rigorous treatment of AC circuit analysis, design, and testing.

The purpose of this article is to give an overview of the main topics covered in the book "Alternating Current Circuits by Kerchner and Corcoran" and explain some of the key concepts and principles involved. The article will also provide some examples and exercises to help you understand and apply what you learn. By reading this article, you will gain a solid foundation for further study or practice in AC circuits.

## Chapter 1: Fundamentals of Alternating Current

The first chapter introduces the basic concepts and definitions of alternating current (AC), such as sinusoidal waveforms, frequency, amplitude, phase, power, impedance, etc. It also explains how to use phasor diagrams and complex numbers to analyze AC circuits. Finally, it describes the common AC sources and loads, such as generators, transformers, resistors, capacitors, inductors, etc.

### Sinusoidal Waveforms

A sinusoidal waveform is a type of periodic waveform that has the shape of a sine or cosine function. It is characterized by three parameters: amplitude, frequency, and phase.

The amplitude (A) is the maximum value of the waveform in either direction. It is measured in volts (V) for voltage or amperes (A) for current.

The frequency (f) is the number of cycles or complete oscillations that the waveform makes in one second. It is measured in hertz (Hz) or cycles per second (cps).

The phase (φ) is the angle or position of the waveform at a given instant of time. It is measured in degrees () or radians (rad). The phase can be positive or negative, depending on whether the waveform is ahead or behind a reference waveform.

The general equation of a sinusoidal waveform is given by:

$$y(t) = A \sin(2\pi ft + \phi)$$

where y(t) is the instantaneous value of the waveform at time t, A is the amplitude, f is the frequency, and φ is the phase.

### Phasor Diagrams and Complex Numbers

A phasor diagram is a graphical representation of sinusoidal waveforms using vectors or arrows. A phasor is a vector that has a magnitude equal to the amplitude of the waveform and an angle equal to the phase of the waveform. Phasors can be used to simplify the analysis of AC circuits by converting time-domain equations into frequency-domain equations.

A complex number is a number that has both a real part and an imaginary part. A complex number can be written in rectangular form as:

$$z = x + jy$$

where x is the real part, y is the imaginary part, and j is the imaginary unit that satisfies $$j^2 = -1$$. A complex number can also be written in polar form as:

$$z = r \angle \theta$$

where r is the magnitude or modulus of the complex number, and θ is the angle or argument of the complex number. The relationship between rectangular and polar forms is given by:

$$x = r \cos \theta$$

$$y = r \sin \theta$$

$$r = \sqrtx^2 + y^2$$

$$\theta = \tan^-1(\fracyx)$$

A phasor can be represented by a complex number in polar form, where the magnitude is equal to the amplitude of the waveform and the angle is equal to the phase of the waveform. For example, a voltage phasor can be written as:

$$V = V_m \angle \phi_v$$

where Vm is the peak voltage and φv is the voltage phase. Similarly, a current phasor can be written as:

$$I = I_m \angle \phi_i$$

where Im is the peak current and φi is the current phase.

### Common AC Sources and Loads

An AC source is a device that produces an alternating voltage or current. An AC load is a device that consumes power from an AC source. Some of the common AC sources and loads are:

A generator is a device that converts mechanical energy into electrical energy by rotating a coil of wire in a magnetic field. A generator can produce AC voltage by varying the direction or speed of rotation.

A transformer is a device that transfers electrical energy from one circuit to another by using electromagnetic induction. A transformer can change AC voltage by varying the number of turns of wire in its primary and secondary coils.

A resistor is a device that opposes the flow of electric current by converting electrical energy into heat. A resistor has a constant resistance (R) that does not depend on frequency or phase.

A capacitor is a device that stores electric charge by separating two conductors with an insulator. A capacitor has a capacitance (C) that depends on its geometry and material. A capacitor opposes changes in voltage by charging and discharging.

An inductor is a device that stores magnetic energy by coiling a wire around a core. An inductor has an inductance (L) that depends on its geometry and material. An inductor opposes changes in current by inducing a voltage that opposes the change.

The behavior of resistors, capacitors, and inductors in AC circuits can be summarized as follows:

ComponentVoltage-Current RelationshipImpedancePhase Angle

ResistorV = IRZ = Rφ = 0

CapacitorV = \frac1C \int I dtZ = \frac1j\omega C = -jX_Cφ = -90

InductorV = L \fracdIdtZ = j\omega L = jX_Lφ = 90

where Z is the impedance, ω is the angular frequency (ω = 2πf), XC is the capacitive reactance, and XL is the inductive reactance. The impedance is the effective resistance of the component to AC, and it depends on both the frequency and the phase angle. The phase angle is the difference between the voltage and current phases for each component.

## Chapter 2: Series and Parallel AC Circuits

The second chapter explains how to apply Kirchhoff's laws and Ohm's law to AC circuits with series and parallel connections. It also shows how to calculate the equivalent impedance, current, voltage, power, and power factor of AC circuits. Finally, it introduces the concepts of resonance, filters, and quality factor to improve the performance of AC circuits.

### Kirchhoff's Laws and Ohm's Law for AC Circuits

Kirchhoff's laws and Ohm's law are fundamental principles that govern the behavior of electric circuits. They can be applied to both DC and AC circuits, with some modifications for AC circuits.

Kirchhoff's current law (KCL) states that the algebraic sum of currents entering a node (or junction) is zero. For AC circuits, this means that the phasor sum of currents entering a node is zero.

Kirchhoff's voltage law (KVL) states that the algebraic sum of voltages around a closed loop is zero. For AC circuits, this means that the phasor sum of voltages around a closed loop is zero.

Ohm's law states that the voltage across a resistor is proportional to the current through it. For AC circuits, this means that the phasor voltage across a resistor is proportional to the phasor current through it.

To apply these laws to AC circuits, we need to use complex numbers and phasor diagrams to represent voltages and currents. We also need to use impedances instead of resistances to account for the phase differences between voltages and currents.

### Equivalent Impedance, Current, Voltage, Power, and Power Factor of AC Circuits

The equivalent impedance of an AC circuit is the total impedance that an AC source sees when it is connected to the circuit. The equivalent impedance can be calculated by using series and parallel rules for impedances, similar to those for resistances in DC circuits.

The equivalent impedance of a series connection of impedances is the sum of their individual impedances: $$Z_eq = Z_1 + Z_2 + ... + Z_n$$

The equivalent impedance of a parallel connection of impedances is the reciprocal of the sum of their reciprocals: $$\frac1Z_eq = \frac1Z_1 + \frac1Z_2 + ... + \frac1Z_n$$

The current in an AC circuit is determined by the applied voltage and the equivalent impedance: $$I = \fracVZ_eq$$ where V is the phasor voltage of the source and I is the phasor current in the circuit.

The voltage across each component in an AC circuit is determined by the current through it and its impedance: $$V = IZ$$ where V is the phasor voltage across the component, I is the phasor current through it, and Z is its impedance.

The power in an AC circuit is the rate at which energy is transferred from the source to the load. The power can be calculated by using the following formulas:

The average power (P) is the product of the rms voltage and the rms current times the power factor (cos φ): $$P = V_rms I_rms \cos \phi$$ where Vrms is the rms voltage of the source, Irms is the rms current in the circuit, and φ is the phase angle between the voltage and the current.

The apparent power (S) is the product of the rms voltage and the rms current: $$S = V_rms I_rms$$ where Vrms and Irms are as defined above. The apparent power is measured in volt-amperes (VA).

The reactive power (Q) is the product of the rms voltage and the rms current times the sine of the phase angle: $$Q = V_rms I_rms \sin \phi$$ where Vrms, Irms, and φ are as defined above. The reactive power is measured in volt-amperes reactive (VAR).

The power factor (PF) of an AC circuit is a measure of how efficiently the circuit converts electrical energy into useful work. The power factor is defined as the ratio of the average power to the apparent power: $$PF = \fracPS = \cos \phi$$ where P, S, and φ are as defined above. The power factor can range from 0 to 1, with 1 being ideal and 0 being worst. A low power factor indicates that the circuit has a large reactive component that consumes energy but does not perform any useful work.

### Resonance, Filters, and Quality Factor of AC Circuits

Resonance is a phenomenon that occurs in AC circuits when the frequency of the source matches the natural frequency of the circuit. At resonance, the impedance of the circuit becomes purely resistive, and the current reaches a maximum value. Resonance can be used to enhance or suppress certain frequencies in AC circuits.

A filter is a device that allows certain frequencies to pass through while blocking others. Filters can be classified into four types based on their frequency response:

A low-pass filter allows low frequencies to pass through while blocking high frequencies.

A high-pass filter allows high frequencies to pass through while blocking low frequencies.

A band-pass filter allows a range of frequencies to pass through while blocking others.

A band-stop filter blocks a range of frequencies while allowing others to pass through.

A quality factor (Q) is a measure of how selective a filter is. The quality factor is defined as the ratio of the resonant frequency to the bandwidth of the filter: $$Q = \fracf_0B$$ where f0 is the resonant frequency and B is the bandwidth. The bandwidth is the difference between the upper and lower cutoff frequencies of the filter: $$B = f_H - f_L$$ where fH is the upper cutoff frequency and fL is the lower cutoff frequency.

A resonant circuit is a special case of a band-pass filter that has a very high quality factor and a very narrow bandwidth. A resonant circuit can be formed by connecting a capacitor and an inductor in series or parallel. The resonant frequency of a series or parallel resonant circuit is given by: $$f_0 = \frac12\pi \sqrtLC$$ where L is the inductance and C is the capacitance. At resonance, the impedance of a series resonant circuit becomes minimum and the current becomes maximum. Conversely, the impedance of a parallel resonant circuit becomes maximum and the current becomes minimum.

A filter can be designed to have a desired center frequency, bandwidth, and quality factor by choosing appropriate values for the components. For example, a multiple-feedback band-pass filter can be constructed using an op-amp and three passive components as shown in Figure . The center frequency, bandwidth, and quality factor of this filter are given by: $$f_0 = \frac12\pi R_2 C_2$$ $$B = \frac12\pi R_1 C_1$$ $$Q = \fracR_2R_1$$ where R1, R2, C1, and C2 are the component values.

Figure : A multiple-feedback band-pass filter.

## Chapter 3: Polyphase AC Circuits

The third chapter discusses the advantages and disadvantages of single-phase and polyphase AC systems. It also explains how to generate and connect three-phase AC sources and loads, such as wye and delta configurations. Finally, it shows how to measure and balance the power and currents in three-phase AC systems.

### Single-Phase and Polyphase AC Systems

A single-phase AC system is a system that has only one sinusoidal voltage source. A single-phase AC system can be used to power simple devices such as lamps, heaters, fans, etc. However, a single-phase AC system has some limitations, such as:

It cannot produce a rotating magnetic field, which is required for operating induction motors.

It has low efficiency and high losses due to the use of transformers and transmission lines.

It has poor voltage regulation and power factor due to the presence of harmonics and reactive loads.

A polyphase AC system is a system that has more than one sinusoidal voltage source with different phases. The most common polyphase AC system is the three-phase AC system, which has three sinusoidal voltage sources with 120 phase difference. A three-phase AC system has some advantages over a single-phase AC system, such as:

It can produce a rotating magnetic field, which is ideal for operating induction motors.

It has high efficiency and low losses due to the elimination of transformers and transmission lines.

It has better voltage regulation and power factor due to the cancellation of harmonics and reactive loads.

A three-phase AC system can be either balanced or unbalanced. A balanced three-phase AC system is a system that has equal magnitudes and phase differences for all three voltages and currents. An unbalanced three-phase AC system is a system that has unequal magnitudes or phase differences for some or all of the voltages and currents. An unbalanced three-phase AC system can cause problems such as overheating, vibration, noise, etc.

### Three-Phase AC Sources and Loads

A three-phase AC source is a device that generates three sinusoidal voltages with 120 phase difference. A three-phase AC source can be either star-connected (wye-connected) or delta-connected. A star-connected source has one common point (called the neutral) that connects all three phases. A delta-connected source has no common point but connects each phase to the next in a loop.

A three-phase AC load is a device that consumes power from a three-phase AC source. A three-phase AC load can also be either star-connected or delta-connected. A star-connected load has one common point (called the neutral) that connects all three phases. A delta-connected load has no common point but connects each phase to the next in a loop.

The relationship between the line and phase voltages and currents for a star-connected source or load is given by: $$V_L = \sqrt3 V_P$$ $$I_L = I_P$$ where VL is the line voltage, VP is the phase voltage, IL is the line current, and IP is the phase current. The relationship between the line and phase voltages and currents for a delta-connected source or load is given by: $$V_L = V_P$$ $$I_L = \sqrt3 I_P$$ where VL, VP, IL, and IP are as defined above.

Figure : A star-connected (wye-connected) source or load.

Figure : A delta-connected source or load.

### Power and Current Measurement and Balance in Three-Phase AC Systems

The power in a three-phase AC system is the sum of the powers in each phase. The power in each phase can be calculated by using the following formulas:

The average power (P) is the product of the rms voltage and the rms current times the power factor (cos φ): $$P = V_rms I_rms \cos \phi$$ where Vrms is the rms phase voltage, Irms is the rms phase current, and φ is the phase angle between the voltage and the current.

The apparent power (S) is the product of the rms voltage and the rms current: $$S = V_rms I_rms$$ where Vrms and Irms are as defined above. The apparent power is measured in volt-amperes (VA).

The reactive power (Q) is the product of the rms voltage and the rms current times the sine of the phase angle: $$Q = V_rms I_rms \sin \phi$$ where Vrms, Irms, and φ are as defined above. The reactive power is measured in volt-amperes reactive (VAR).

The total power in a three-phase AC system can be calculated by multiplying the power in each phase by three: $$P_T = 3P$$ $$S_T = 3S$$ $$Q_T = 3Q$$ where PT is the total average power, ST is the total apparent power, QT is the total reactive power, and P, S, and Q are as defined above.

The current in a three-phase AC system can be measured by using different methods, such as:

A three-wattmeter method, which uses three wattmeters to measure the power in each phase and then calculates the current from the power.

A two-wattmeter method, which uses two wattmeters to measure the power in two phases and then calculates the current from the power.

A one-wattmeter method, which uses one wattmeter to measure the power in one phase and then calculates the current from the power.

The balance in a three-phase AC system can be checked by comparing the magnitudes and phase angles of the voltages and currents of each phase. A balanced three-phase AC system has equal magnitudes and phase differences for all three voltages and currents. An unbalanced three-phase AC system has unequal magnitudes or phase differences for some or all of the voltages and currents. An unbalanced three-phase AC system can be caused by various factors, such as:

A fault in one or more phases, such as a short circuit, an open circuit, or a ground fault.

A load imbalance in one or more phases, such as a single-phase load connected to a three-phase source.

A source imbalance in one or more phases, such as a voltage drop or a frequency variation.

The balance in a three-phase AC system can be improved by using different methods, such as:

A symmetrical components method, which converts a three-phase unbalanced system into two sets of balanced phasors and a set of single-phase phasors, or symmetrical components. These sets of phasors are called the positive, negative-, and zero-sequence components. The symmetrical components method can be used to analyze and correct various types of faults and imbalances in three-phase AC systems.

A load balancing method, which adjusts the load distr