# Single Phase AC Circuits

### Contents

**Describe the equation of the alternating voltages and currents**1**Explain the difference between DC and AC waveforms**2**List different Periodic Function Waveforms**3**Describe Amplitude, Frequency, Time period, and Phase**4**Explain Peak value, Instantaneous value, Phase, Average value, RMS value, Crest(or amplitude or peak) Factor, and Form factor**5**Describe vector representation of alternating quantities**5**Explain vector addition and vector subtraction of alternating signals**6**Explain significance of j-operator**7**Discuss various forms or methods of representing vector quantities**8**Explain behaviour of AC through pure resistive, pure inductive, and pure capacitive circuit**9**Explain power in pure resistive, pure inductive, and pure capacitive circuit**10**Explain behaviour of AC through series R-L circuit**11**Explain behaviour of AC through series R-C circuit**12**Explain Power, its components, and Power Factor in series R-L, R-C circuits**13**Explain behaviour of AC through series R-L-C circuit**14**Explain behaviour of AC through parallel R-L circuits**15**Explain behaviour of AC through parallel R-C circuits**16**Explain the Series equivalent of Parallel circuit**18**Explain the Parallel equivalent of Series circuit**18

Describe the equation of the alternating voltages and currents

Alternating voltages and currents can be described using sinusoidal functions. A sinusoidal function is a mathematical function that describes a wave with a repeating pattern, such as the oscillations of a pendulum or the fluctuations of an AC voltage or current.

The equation for a sinusoidal alternating voltage can be written as:

V(t) = V_{m} sin(ωt + φ)

where V_{m} is the maximum or peak value of the voltage, ω is the angular frequency (which is equal to 2πf, where f is the frequency in Hertz), t is time, and φ is the phase angle.

Similarly, the equation for a sinusoidal alternating current can be written as:

I(t) = I_{m} sin(ωt + φ’)

where I_{m} is the maximum or peak value of the current, ω is the angular frequency, t is time, and φ’ is the phase angle.

Both equations describe a sinusoidal wave that oscillates with a frequency of ω/2π Hertz (or f Hertz), where the wave reaches its maximum value of V_{m} or I_{m} at time t = 0, and oscillates between positive and negative values.

The phase angle φ in the voltage equation and the phase angle φ’ in the current equation represent the relative position of the voltage and current waves with respect to each other. If the two waves are in phase (i.e., the peaks and troughs of the waves occur at the same time), then φ – φ’ = 0. If the two waves are out of phase (i.e., the peaks and troughs of the waves occur at different times), then φ – φ’ ≠ 0.

Explain the difference between DC and AC waveforms

DC and AC waveforms are two different types of electrical signals that differ in their shape and properties. The formulae for DC and AC waveforms are different.

DC (direct current) is a type of electrical signal that flows in one direction only and has a constant amplitude over time. The formula for DC waveform is:

V(t) = V_{dc}

where V_{dc} is the constant voltage or potential difference of the DC signal. The voltage of a DC signal is constant over time, and therefore, the waveform of a DC signal is a straight horizontal line.

AC (alternating current) is a type of electrical signal that periodically reverses its direction and amplitude over time. The formula for AC waveform is:

V(t) = V_{p} sin(2πft + φ)

where V_{p} is the peak voltage or amplitude of the AC signal, f is the frequency of the signal, t is time, and φ is the phase angle of the signal. The voltage of an AC signal changes over time, and therefore, the waveform of an AC signal is a sinusoidal curve.

The key difference between the two waveforms is that DC has a constant voltage or potential difference, while AC has a voltage that changes periodically over time. Additionally, AC has a frequency and phase angle, which are not present in DC signals

List different Periodic Function Waveforms

There are several different periodic function waveforms, some of which are:

**Sinusoidal waveform:**This waveform is the most common type of periodic function and is used to represent many natural phenomena such as sound waves, alternating current (AC) voltages, and electromagnetic waves. The waveform is characterized by a smooth, repetitive oscillation that follows a sine function.**Square waveform:**This waveform has a fixed voltage level for a certain period of time, followed by an abrupt change to a different fixed voltage level. The waveform repeats in this manner, creating a square-shaped pattern.**Triangle waveform:**This waveform has a linear rise and fall time and is similar in shape to a ramp. The waveform repeats in a periodic manner, creating a triangular-shaped pattern.**Sawtooth waveform:**This waveform is similar to a triangle waveform, but the rise time is much faster than the fall time, creating a sawtooth-shaped pattern. The waveform repeats in a periodic manner.**Pulse waveform:**This waveform has a fixed voltage level for a short duration of time, followed by a period of zero voltage. The waveform repeats in a periodic manner, creating a pulse-shaped pattern.**Sine wave with amplitude modulation:**This waveform is a sine wave that has been modulated with a lower frequency waveform. The amplitude of the sine wave changes in accordance with the modulation waveform, creating a complex waveform that repeats periodically.**Sine wave with frequency modulation:**This waveform is a sine wave that has been modulated with a lower frequency waveform. The frequency of the sine wave changes in accordance with the modulation waveform, creating a complex waveform that repeats periodically.**Noise waveform:**This waveform is a random, unpredictable signal that does not follow any specific pattern. The waveform may have some periodic characteristics, but the signal is generally unpredictable and does not repeat in a regular manner.

Describe Amplitude, Frequency, Time period, and Phase

Amplitude, frequency, time period, and phase are all important characteristics of periodic waveforms.

**Amplitude:**Amplitude refers to the maximum displacement or value of a waveform from its equilibrium or zero point. In the case of a sinusoidal waveform, the amplitude is the maximum value of the waveform, which occurs at the peaks of the wave. The amplitude is usually measured in volts (V) or amperes (A) for electrical waveforms or in meters (m) for mechanical waveforms.**Frequency:**Frequency is the number of complete cycles of a waveform that occur per unit time. It is usually measured in Hertz (Hz), which represents one cycle per second. For example, a waveform with a frequency of 50 Hz completes 50 cycles per second. The frequency of a sinusoidal waveform can be determined by measuring the time between two consecutive peaks or troughs and taking the inverse of that time.**Time period:**Time period is the time taken for one complete cycle of a waveform to occur. It is the inverse of frequency and is usually measured in seconds (s). For example, a waveform with a frequency of 50 Hz has a time period of 1/50 = 0.02 s. The time period of a sinusoidal waveform can be determined by measuring the time between two consecutive peaks or troughs.**Phase:**Phase refers to the position of a waveform in its cycle relative to a reference waveform. It is usually measured in degrees or radians and is used to describe the relationship between two or more waveforms. In the case of sinusoidal waveforms, phase is often measured relative to a reference waveform at zero degrees or radians. Two sinusoidal waveforms with the same frequency and amplitude but different phase angles will be out of phase with each other and will produce interference when combined.

In summary, amplitude refers to the maximum value of a waveform, frequency refers to the number of cycles per unit time, time period refers to the time taken for one complete cycle to occur, and phase refers to the position of a waveform in its cycle relative to a reference waveform. These characteristics are essential for understanding and analyzing periodic waveforms in various fields such as electronics, acoustics, and optics.

Explain Peak value, Instantaneous value, Phase, Average value, RMS value, Crest(or amplitude or peak) Factor, and Form factor

In electrical systems, several important values are used to describe the properties of waveforms, including:

**Peak value:**The peak value of a waveform is the highest voltage or current value that the waveform reaches over a complete cycle.- Instantaneous value: The instantaneous value of a waveform is its value at a specific moment in time.
**Phase:**The phase of a waveform is a measure of its position in time relative to a reference waveform.**Average value:**The average value of a waveform is its average voltage or current value over one complete cycle. It is often used to describe the average power delivered by a waveform.**RMS value:**The RMS (root mean square) value of a waveform is its equivalent DC voltage or current that would produce the same heating effect as the AC waveform. The RMS value is a more accurate representation of the effective value of a waveform for power calculations.**Crest (or amplitude or peak) factor:**The crest factor of a waveform is the ratio of its peak value to its RMS value. It is a measure of the waveform’s peakiness, with a higher crest factor indicating a more peaked waveform.**Form factor:**The form factor of a waveform is the ratio of its RMS value to its average value. It is a measure of the waveform’s symmetry, with a form factor of 1 indicating a perfect sine wave.

These values are used to describe and analyse the behavior of waveforms in electrical systems. For example, the RMS value is used to determine the power delivered by a waveform, while the crest factor is used to determine the level of distortion in a waveform. The form factor is used to determine the shape of a waveform and its deviation from a perfect sine wave.

Describe vector representation of alternating quantities

Vector representation of alternating quantities is a method used to represent the amplitude and phase of alternating current (AC) or voltage waveforms. It is a graphical representation that uses vectors to represent the magnitude and phase of the AC waveform.

The vector representation of an AC waveform is done using a complex number or phasor, which consists of a real part and an imaginary part. The real part represents the amplitude of the waveform, while the imaginary part represents the phase angle. The complex number or phasor can be represented as a vector in a two-dimensional space, where the horizontal axis represents the real part or amplitude, and the vertical axis represents the imaginary part or phase angle.

To create a vector representation of an AC waveform, the amplitude and phase angle of the waveform must be determined. The amplitude can be measured directly using a multimeter or oscilloscope, while the phase angle can be determined using a phase meter or by measuring the time difference between the waveform and a reference waveform.

Once the amplitude and phase angle are determined, a complex number or phasor can be created by combining the amplitude and phase angle using the following formula:

phasor = A ∠ θ

where A is the amplitude of the waveform, and θ is the phase angle in radians. The symbol ∠ represents the angle between the phasor and the real axis.

The phasor can then be represented as a vector in a two-dimensional space, where the length of the vector represents the amplitude or magnitude of the waveform, and the angle between the vector and the real axis represents the phase angle. The vector is usually drawn in a clockwise direction, starting from the positive end of the real axis.

The vector representation of AC waveforms is particularly useful in circuit analysis, as it allows complex AC circuits to be analyzed using simple mathematical operations such as addition, subtraction, and multiplication. By representing AC waveforms as vectors, it is possible to determine the resulting waveform that would be produced when multiple AC waveforms are combined in a circuit.

Explain vector addition and vector subtraction of alternating signals

Vector addition and vector subtraction are important concepts in the vector representation of alternating quantities. These operations allow us to combine multiple waveforms into a single waveform, and to analyse the behavior of complex waveforms in electrical systems.

**Vector addition:**Vector addition is the process of adding two or more vectors to produce a single resultant vector. In the vector representation of alternating quantities, vector addition can be used to add two or more waveforms together to produce a single waveform that represents the sum of the individual waveforms.

For example, if two waveforms with phasors A and B are added together, the resulting waveform can be represented by a phasor C, which is the sum of phasors A and B. This can be done graphically by placing the two phasors head to tail and drawing a vector from the tail of the first phasor to the head of the second phasor.

**Vector subtraction:**Vector subtraction is the process of subtracting one vector from another to produce a single resultant vector. In the vector representation of alternating quantities, vector subtraction can be used to subtract one waveform from another to produce a single waveform that represents the difference between the individual waveforms.

For example, if two waveforms with phasors A and B are subtracted, the resulting waveform can be represented by a phasor C, which is the difference between phasors A and B. This can be done graphically by placing the two phasors head to tail and drawing a vector from the head of the first phasor to the tail of the second phasor.

Vector addition and subtraction are useful tools for analysing complex waveforms in electrical systems. For example, they can be used to determine the net voltage or current in a circuit, or to analyse the behavior of multiple waveforms in a power system.

Explain significance of j-operator

The “j-operator” is a mathematical operator that is used in electrical engineering to represent the imaginary unit. In mathematics, the imaginary unit is usually denoted by “i”, but in electrical engineering, “j” is used to avoid confusion with the symbol “i” that is used to represent current.

The j-operator is defined as:

j = √(-1)

In other words, j represents the square root of the negative one. It is used to represent the imaginary part of a complex number, which consists of a real part and an imaginary part.

The significance of the j-operator in electrical engineering is that it allows us to represent AC waveforms and circuits using complex numbers or phasors. In a complex number, the real part represents the DC or steady-state component of the signal, while the imaginary part represents the AC or time-varying component of the signal. By using complex numbers or phasors, it is possible to perform mathematical operations such as addition, subtraction, and multiplication on AC waveforms and circuits.

For example, if we have two AC waveforms with different frequencies, we can represent them as complex numbers or phasors and add them together using simple mathematical operations. This allows us to determine the resulting waveform that would be produced when the two AC waveforms are combined in a circuit.

In summary, the j-operator is a mathematical operator that is used in electrical engineering to represent the imaginary unit. It is significant because it allows us to represent AC waveforms and circuits using complex numbers or phasors, which makes it possible to perform mathematical operations on these waveforms and circuits.

Discuss various forms or methods of representing vector quantities

Vector quantities are physical quantities that have both magnitude and direction. There are several ways of representing vector quantities, including graphical representation, component form, unit vector form, and magnitude-angle form.

**Graphical Representation:**One of the simplest and most intuitive ways of representing vector quantities is through a graphical representation. This involves drawing a vector arrow in a two-dimensional or three-dimensional space, where the length of the arrow represents the magnitude of the vector, and the direction of the arrow represents the direction of the vector. This method is particularly useful for visualising vector quantities in physics and engineering.**Component Form:**Another common way of representing vector quantities is through the component form. This method involves breaking down the vector into its component parts along a set of coordinate axes. For example, in a two-dimensional space, a vector can be represented as (x,y), where x and y are the horizontal and vertical components of the vector, respectively. This method is useful for performing mathematical operations on vector quantities, such as addition, subtraction, and multiplication.**Unit Vector Form:**The unit vector form is a normalised version of the component form, where the vector is represented as a unit vector along a set of coordinate axes. A unit vector has a magnitude of 1 and is used to indicate the direction of a vector quantity. In a two-dimensional space, a unit vector can be represented as (cos θ, sin θ), where θ is the angle between the vector and the positive x-axis. This method is useful for representing the direction of a vector quantity.**Magnitude-Angle Form:**The magnitude-angle form is similar to the unit vector form, but instead of representing the vector as a unit vector, it is represented as the product of its magnitude and a unit vector along its direction. In a two-dimensional space, a vector can be represented as (r, θ), where r is the magnitude of the vector and θ is the angle between the vector and the positive x-axis. This method is useful for performing mathematical operations on vector quantities, such as addition, subtraction, and multiplication.

In summary, there are several ways of representing vector quantities, including graphical representation, component form, unit vector form, and magnitude-angle form. Each method has its advantages and is useful in different situations. Graphical representation is intuitive and useful for visualising vector quantities, while component form and unit vector form are useful for performing mathematical operations on vector quantities. Magnitude-angle form is useful for representing vector quantities in polar coordinates.

Explain behaviour of AC through pure resistive, pure inductive, and pure capacitive circuit

The behavior of alternating current (AC) through pure resistive, pure inductive, and pure capacitive circuits can be described as follows:

**Pure resistive circuit:**In a pure resistive circuit, the current and voltage are in phase, meaning that they reach their maximum and minimum values at the same time. The resistance in the circuit causes the voltage to drop across the resistor, and the current flowing through the circuit is proportional to the voltage. The impedance of a pure resistive circuit is purely real and is equal to the resistance of the circuit.**Pure inductive circuit:**In a pure inductive circuit, the voltage leads the current by 90°, meaning that the current reaches its maximum value after the voltage has reached its maximum value. The inductance in the circuit causes the current to lag behind the voltage, and the current flowing through the circuit is proportional to the rate of change of the voltage. The impedance of a pure inductive circuit is given by jLω, where L is the inductance and ω is the angular frequency of the waveform.**Pure capacitive circuit:**In a pure capacitive circuit, the current leads the voltage by 90°, meaning that the current reaches its maximum value before the voltage has reached its maximum value. The capacitance in the circuit causes the voltage to lag behind the current, and the current flowing through the circuit is proportional to the rate of change of the voltage. The impedance of a pure capacitive circuit is given by -j/Cω, where C is the capacitance and ω is the angular frequency of the waveform.

In each of these circuits, the impedance affects the magnitude and phase relationship between the voltage and current, and the specific behavior of the circuit depends on the type and value of the impedance. In practice, most electrical circuits are composed of a combination of resistive, inductive, and capacitive components, and the behavior of the circuit can be analysed using techniques such as phasor analysis or complex impedance analysis.

In summary, the behavior of alternating current through pure resistive, pure inductive, and pure capacitive circuits can be characterised by the phase relationship between the voltage and current and the impedance of the circuit. The behavior of a circuit depends on the specific components in the circuit and the waveform being considered.

Explain power in pure resistive, pure inductive, and pure capacitive circuit

The power in a pure resistive, pure inductive, and pure capacitive circuit can be described as follows:

**Pure resistive circuit:**In a pure resistive circuit, the voltage and current are in phase, meaning that they reach their maximum and minimum values at the same time. The power in a pure resistive circuit is given by the equation P = VI, where V is the voltage across the resistor and I is the current flowing through the resistor. The power in a pure resistive circuit is purely real and is equal to the product of the voltage and current.**Pure inductive circuit:**In a pure inductive circuit, the voltage leads the current by 90°, meaning that the current reaches its maximum value after the voltage has reached its maximum value. The power in a pure inductive circuit is given by the equation P = VI, where V is the voltage across the inductor and I is the current flowing through the inductor. The power in a pure inductive circuit is alternating and contains both real and reactive components. The real power in a pure inductive circuit is zero, and the reactive power is given by the equation Q = VIcos(Φ), where Φ is the phase difference between the voltage and current.**Pure capacitive circuit:**In a pure capacitive circuit, the current leads the voltage by 90°, meaning that the current reaches its maximum value before the voltage has reached its maximum value. The power in a pure capacitive circuit is given by the equation P = VI, where V is the voltage across the capacitor and I is the current flowing through the capacitor. The power in a pure capacitive circuit is alternating and contains both real and reactive components. The real power in a pure capacitive circuit is zero, and the reactive power is given by the equation Q = VIcos(Φ), where Φ is the phase difference between the voltage and current.

In each of these circuits, the power is affected by the magnitude and phase relationship between the voltage and current, and the specific behavior of the circuit depends on the type and value of the components in the circuit. In practice, most electrical circuits are composed of a combination of resistive, inductive, and capacitive components, and the power in the circuit can be analysed using techniques such as phasor analysis or complex impedance analysis.

In summary, the power in a pure resistive, pure inductive, and pure capacitive circuit can be characterised by the magnitude and phase relationship between the voltage and current and the type of components in the circuit. The power in a circuit depends on the specific components in the circuit and the waveform being considered.

Explain behaviour of AC through series R-L circuit

The behavior of an alternating current (AC) through a series R-L circuit can be described as follows:

A series R-L circuit consists of a resistor and an inductor connected in series. The current flowing through the circuit is common to both components and is determined by the voltage across the circuit and the impedance of the circuit.

At low frequencies, the impedance of the inductor is primarily inductive, meaning that the impedance is proportional to the frequency. At these frequencies, the voltage across the inductor is in phase with the current, and the power absorbed by the inductor is purely reactive. The voltage across the resistor is in phase with the current, and the power absorbed by the resistor is purely real.

At high frequencies, the impedance of the inductor is primarily resistive, meaning that the impedance is independent of the frequency. At these frequencies, the voltage across the inductor is in phase with the current, and the power absorbed by the inductor is purely real. The voltage across the resistor is in phase with the current, and the power absorbed by the resistor is purely real.

At intermediate frequencies, the impedance of the inductor is partially resistive and partially inductive, meaning that the impedance depends on both the frequency and the value of the inductor. At these frequencies, the voltage across the inductor leads the current by some angle, and the power absorbed by the inductor is a combination of reactive and real power. The voltage across the resistor is in phase with the current, and the power absorbed by the resistor is purely real.

In summary, the behavior of an AC through a series R-L circuit depends on the frequency of the AC waveform and the value of the components in the circuit. The voltage and current in the circuit can be analysed using techniques such as phasor analysis or complex impedance analysis to determine the power absorbed by the circuit and the behavior of the circuit at different frequencies.

Explain behaviour of AC through series R-C circuit

The behavior of an alternating current (AC) through a series R-C circuit can be described as follows:

A series R-C circuit consists of a resistor and a capacitor connected in series. The current flowing through the circuit is common to both components and is determined by the voltage across the circuit and the impedance of the circuit.

At low frequencies, the impedance of the capacitor is primarily capacitive, meaning that the impedance is proportional to the reciprocal of the frequency. At these frequencies, the voltage across the capacitor leads the current by 90 degrees, and the power absorbed by the capacitor is purely reactive. The voltage across the resistor is in phase with the current, and the power absorbed by the resistor is purely real.

At high frequencies, the impedance of the capacitor is primarily resistive, meaning that the impedance is independent of the frequency. At these frequencies, the voltage across the capacitor leads the current by 90 degrees, and the power absorbed by the capacitor is purely reactive. The voltage across the resistor is in phase with the current, and the power absorbed by the resistor is purely real.

At intermediate frequencies, the impedance of the capacitor is partially resistive and partially capacitive, meaning that the impedance depends on both the frequency and the value of the capacitor. At these frequencies, the voltage across the capacitor leads the current by some angle less than 90 degrees, and the power absorbed by the capacitor is a combination of reactive and real power. The voltage across the resistor is in phase with the current, and the power absorbed by the resistor is purely real.

In summary, the behaviour of an AC through a series R-C circuit depends on the frequency of the AC waveform and the value of the components in the circuit. The voltage and current in the circuit can be analysed using techniques such as phasor analysis or complex impedance analysis to determine the power absorbed by the circuit and the behaviour of the circuit at different frequencies.

Explain Power, its components, and Power Factor in series R-L, R-C circuits

In an AC circuit, power and its components can be analysed in terms of the voltage and current in the circuit. In a series R-L or R-C circuit, the power absorbed by the circuit can be separated into real power and reactive power. The power factor of the circuit is a measure of how effectively the circuit uses the power supplied to it. Real power, also known as active power, is the power that is actually used to perform work in the circuit. It is measured in watts (W) and is equal to the product of the voltage and current in the circuit, multiplied by the cosine of the phase angle between the voltage and the current. Reactive power, on the other hand, is the power that is stored and returned in the circuit and is not used to perform work. It is measured in volt-amperes reactive (VAR) and is equal to the product of the voltage and current in the circuit, multiplied by the sine of the phase angle between the voltage and the current.The power factor of the circuit is a measure of how effectively the power supplied to the circuit is used. It is defined as the ratio of the real power to the apparent power, which is the magnitude of the vector sum of the real power and the reactive power. The power factor can be expressed as a value between 0 and 1, with a power factor of 1 indicating that all of the power supplied to the circuit is being used effectively and a power factor of 0 indicating that none of the power is being used effectively. In a series R-L circuit, the inductive component of the circuit causes the current to lag behind the voltage, resulting in a power factor that is less than 1. In a series R-C circuit, the capacitive component of the circuit causes the voltage to lead the current, also resulting in a power factor that is less than 1. Improving the power factor of an AC circuit can be accomplished by adding capacitors or inductors in appropriate combinations to counteract the effects of the resistive components.

In summary, the power absorbed by an AC circuit in a series R-L or R-C circuit can be separated into real power and reactive power, with the power factor of the circuit providing a measure of how effectively the circuit uses the power supplied to it.

Explain behaviour of AC through series R-L-C circuit

In an AC circuit that contains both resistive, inductive, and capacitive components, the behaviour of the circuit can be more complex than in a series R-L or R-C circuit. The voltage and current in the circuit will exhibit both magnitude and phase relationships that are influenced by the individual impedance of each component.

The impedance of a series R-L-C circuit is the combined opposition to the flow of AC current in the circuit and is a complex quantity that can be expressed in terms of its magnitude and phase. The magnitude of the impedance is a function of the resistance, inductance, and capacitance in the circuit and is a measure of the overall opposition to the flow of AC current in the circuit. The phase angle of the impedance represents the phase relationship between the voltage and the current in the circuit.

The magnitude of the impedance of the circuit determines the degree to which the voltage and current are affected by the individual components in the circuit. When the impedance of the circuit is high, the voltage and current in the circuit will be smaller, and when the impedance of the circuit is low, the voltage and current will be larger. The phase angle of the impedance determines the relative phase relationship between the voltage and the current in the circuit, which in turn affects the power factor of the circuit.

In a series R-L-C circuit, the voltage and current in the circuit will oscillate at the same frequency as the applied AC voltage, but the relative magnitude and phase of the voltage and current will depend on the individual components in the circuit. The relative magnitude and phase of the voltage and current in the circuit can be determined using vector analysis techniques and can be used to calculate the power absorbed by the circuit and the power factor of the circuit.

In summary, the behaviour of an AC circuit that contains both resistive, inductive, and capacitive components can be complex and is influenced by the magnitude and phase of the impedance of the circuit. The relative magnitude and phase of the voltage and current in the circuit can be determined using vector analysis techniques, which can be used to calculate the power absorbed by the circuit and the power factor of the circuit.

Explain behaviour of AC through parallel R-L circuits

In a parallel R-L circuit, the behaviour of the circuit can be understood by analysing the individual impedances of the resistive and inductive components. The impedance of an inductor is proportional to its inductance and the frequency of the applied AC voltage. The impedance of a resistor is simply its resistance.

In a parallel R-L circuit, the current in each branch is equal and is determined by the total impedance of the circuit. The total impedance of a parallel circuit can be calculated using the formula:

1/Z_{t} = 1/Z_{1} + 1/Z_{2} + … + 1/Z_{n}

where Z_{t} is the total impedance, Z_{1}, Z_{2}, … Z_{n} are the impedances of each branch in the circuit.

Since the impedance of an inductor is proportional to the frequency of the applied AC voltage, the impedance of the inductor will increase as the frequency of the AC voltage increases. This means that as the frequency of the AC voltage increases, the current in the inductive branch will decrease, while the current in the resistive branch will remain constant.

The voltage across each branch in a parallel R-L circuit is proportional to the impedance of each branch, with the voltage across the inductive branch leading the voltage across the resistive branch by 90 degrees. This phase relationship between the voltage and the current in the inductive branch results in the stored energy in the magnetic field of the inductor alternately increasing and decreasing over time.

In summary, the behaviour of a parallel R-L circuit is determined by the relative magnitude and phase of the impedances of the resistive and inductive components. The current in each branch is equal and is determined by the total impedance of the circuit, while the voltage across each branch is proportional to the impedance of each branch. The phase relationship between the voltage and current in the inductive branch results in the stored energy in the magnetic field of the inductor alternately increasing and decreasing over time.

Explain behaviour of AC through parallel R-C circuits

In a parallel R-C circuit, the behaviour of the circuit can be understood by analysing the individual impedances of the resistive and capacitive components. The impedance of a capacitor is proportional to the reciprocal of its capacitance and the frequency of the applied AC voltage. The impedance of a resistor is simply its resistance.

In a parallel R-C circuit, the voltage across each branch is equal and is determined by the applied AC voltage. The current in each branch, however, is determined by the impedance of each branch. The total impedance of a parallel circuit can be calculated using the formula:

1/Z_{t} = 1/Z_{1} + 1/Z_{2} + … + 1/Z_{n}

where Z_{t} is the total impedance, Z_{1}, Z_{2}, … Z_{n} are the impedances of each branch in the circuit.

Since the impedance of a capacitor is proportional to the reciprocal of the frequency of the applied AC voltage, the impedance of the capacitor will decrease as the frequency of the AC voltage increases. This means that as the frequency of the AC voltage increases, the current in the capacitive branch will increase, while the current in the resistive branch will remain constant.

The current in each branch in a parallel R-C circuit is proportional to the impedance of each branch, with the current in the capacitive branch leading the current in the resistive branch by 90 degrees. This phase relationship between the voltage and the current in the capacitive branch results in the stored energy in the electric field of the capacitor alternately increasing and decreasing over time.

In summary, the behaviour of a parallel R-C circuit is determined by the relative magnitude and phase of the impedances of the resistive and capacitive components. The voltage across each branch is equal and is determined by the applied AC voltage, while the current in each branch is determined by the impedance of each branch. The phase relationship between the voltage and current in the capacitive branch results in the stored energy in the electric field of the capacitor alternately increasing and decreasing over time.

Explain behaviour of AC through parallel R-L-C circuits

The behavior of an AC circuit with a combination of resistive, inductive, and capacitive components in a parallel configuration can be understood by analysing the individual impedances of each component. The impedance of a resistor is simply its resistance, while the impedance of an inductor is proportional to its inductance and the frequency of the applied AC voltage. The impedance of a capacitor is proportional to the reciprocal of its capacitance and the frequency of the applied AC voltage.

In a parallel R-L-C circuit, the voltage across each branch is equal and determined by the applied AC voltage. The current in each branch, however, is determined by the impedance of each branch. The total impedance of a parallel circuit can be calculated using the formula:

1/Z_{t} = 1/Z_{1} + 1/Z_{2} + … + 1/Z_{n}

where Z_{t} is the total impedance, Z_{1}, Z_{2}, … Z_{n} are the impedances of each branch in the circuit.

Since the impedance of a capacitor is proportional to the reciprocal of the frequency of the applied AC voltage, the impedance of the capacitor will decrease as the frequency of the AC voltage increases. On the other hand, the impedance of an inductor is proportional to its inductance and the frequency of the AC voltage, so the impedance of the inductor will increase as the frequency of the AC voltage increases.

The result of these relationships is that at some specific frequency, the impedance of the capacitor will be equal and opposite to the impedance of the inductor, leading to the resonant frequency of the circuit. At the resonant frequency, the impedance of the circuit is at a minimum, and the current in the circuit is at a maximum.

The behaviour of the current in a parallel R-L-C circuit is complex, and it is dependent on the relative values of the impedances of the resistive, inductive, and capacitive components, as well as the frequency of the applied AC voltage. The current in each branch in a parallel R-L-C circuit will have a different phase relationship with the voltage across each branch, with the current in the capacitive branch leading the current in the resistive branch by 90 degrees, and the current in the inductive branch lagging the current in the resistive branch by 90 degrees.

In summary, the behaviour of a parallel R-L-C circuit is determined by the relative magnitudes and phase relationships of the impedances of the resistive, inductive, and capacitive components, as well as the frequency of the applied AC voltage. The voltage across each branch is equal and determined by the applied AC voltage, while the current in each branch is determined by the impedance of each branch. The phase relationships between the voltage and current in each branch can be complex, and the total impedance of the circuit is dependent on the frequency of the applied AC voltage.

Explain the Series equivalent of Parallel circuit

The series equivalent of a parallel circuit refers to the equivalent impedance or resistance of the circuit that behaves as a single impedance in a series circuit. The series equivalent of a parallel circuit is obtained by adding the impedances of the individual branches in a parallel circuit. The impedance of each branch is found by considering it as a single impedance in a series circuit with the other branches being short-circuited. This equivalent impedance can be found by using the reciprocal of the sum of the reciprocals of the individual branch impedances. The formula for the series equivalent impedance of a parallel circuit is given by:

Z_{eq} = 1/(1/Z_{1} + 1/Z_{2} + 1/Z_{3} + … + 1/Z_{n})

where Z_{eq} is the equivalent impedance, Z_{1}, Z_{2}, Z_{3}, … Z_{n} are the impedances of the individual branches, and n is the number of branches in the circuit. This equivalent impedance can then be used to calculate the total current, voltage, and power in the circuit.

Explain the Parallel equivalent of Series circuit

In electrical circuits, a series circuit consists of components connected one after another, creating a single path for the current to flow. In contrast, a parallel circuit consists of components connected in multiple branches, where each component has its own separate path for the current.

The parallel equivalent of a series circuit refers to a simplified representation of a series circuit as an equivalent parallel circuit with equivalent properties. This is useful for analyzing and solving complex circuits more easily.

To determine the parallel equivalent of a series circuit, you need to consider two main aspects: the equivalent resistance and the equivalent voltage sources (if present).

- Equivalent Resistance:

- For resistors connected in series, their resistances simply add up to give the total resistance of the series circuit. If you have resistors R1, R2, R3, …, connected in series, then the equivalent resistance (Req) is given by Req = R1 + R2 + R3 + …
- In the parallel circuit equivalent, the resistors are replaced by a single resistor (let’s call it Rpar) connected in parallel. This equivalent resistance is calculated using the reciprocal of the sum of the reciprocals of the individual resistances. For resistors R1, R2, R3, …, connected in parallel, the equivalent resistance is given by 1/Rpar = 1/R1 + 1/R2 + 1/R3 + …

- Equivalent Voltage Sources (if present):

- In some cases, a series circuit may include voltage sources (such as batteries) connected in series. If you have multiple voltage sources V1, V2, V3, …, connected in series, then the equivalent voltage (Veq) is simply the sum of the individual voltages: Veq = V1 + V2 + V3 + …
- In the parallel circuit equivalent, the voltage sources are replaced by a single voltage source (let’s call it Vpar) connected in parallel. The equivalent voltage source in a parallel circuit is the same as the voltage across each branch.

It’s important to note that other circuit elements, such as capacitors or inductors, would also require appropriate conversions to determine their parallel equivalents.

By finding the parallel equivalent of a series circuit, you can simplify the circuit analysis and apply parallel circuit rules and formulas to determine the overall behavior of the circuit.