What is the voltage of the AC single-phase supply?
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240 Vrms.
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What is the voltage of the AC single-phase supply?
240 Vrms.
What is the phase difference (θ) in a resistive component?
0 degrees.
What is the term used to distinguish AC resistance from DC resistance?
Reactance.
What is the opposition to current flow in a DC circuit called?
Resistance.
In a purely resistive circuit, what is the reactive component of impedance?
Always zero.
What is the reactance of an inductor called?
Inductive Reactance (X_L).
What is the formula for calculating inductive reactance?
X_L = 2πƒL.
What components are connected in series in the described circuit?
A pure non-inductive resistance, R, and a pure inductance, L.
How is the voltage across a pure ohmic resistor related to the current flowing through it?
It is linearly proportional, as defined by Ohm’s Law.
What is the role of an inductor in a time-varying circuit?
It acts as an impedance limiting the amount of time-varying current flowing in the coil.
How do you convert peak values to RMS values in phasor relationships?
By dividing the peak value by √2.
What laws can be used for calculating voltage, current, impedance, and power in purely resistive AC circuits?
Ohm’s Law and Kirchhoff’s Law.
What direction is considered a positive rotation for phasors?
Anti-clockwise direction.
What values are usually used when working with circuit rules in AC circuits?
RMS values.
What effect does increasing inductance have on inductive reactance?
Increasing inductance also increases the overall inductive reactance value.
How is the inductive voltage represented in the vector diagram?
Line OC shows the inductive voltage, which is 90 degrees in front of the current.
What is the phasor sum of balanced three-phase voltages?
V_a + V_b + V_c = 0.
What does the quantity Z represent in a circuit?
Impedance of the circuit.
What defines a balanced three-phase system?
The individual sinusoidal voltages have a fixed relationship of 120 degrees between each other.
What is the expression for the sinusoidal voltage in the circuit?
V(t) = V max sin(ωt).
What is the voltage drop across each component in an RL circuit?
Calculated as V_R = I * R and V_L = I * XL.
What corresponds to the maximum voltage value of self-induced emf in an inductor?
The maximum rate of current change.
What happens to the current when the switch is closed?
The current rises from zero to its maximum value.
In a pure inductance circuit, how does the current relate to the voltage?
The current lags the voltage.
What is the relationship between frequency and the rate of change of flux in an inductor?
As frequency increases, the rate of change of flux within the coil also increases.
What is meant by 'impure inductance'?
An inductive coil that has resistance in series with inductance.
What does impedance depend on in circuits with reactive components?
The frequency of the circuit (ω).
What does phasor notation define?
The effective (rms) magnitude of voltages and currents.
What does V represent in the context of the circuit?
The r.m.s value of the applied voltage.
What is impedance in an AC circuit?
The total opposition to current, consisting of resistance and reactance.
What is the vertical component of voltage V1?
17.32 volts.
What is the analytical method for adding two voltages?
Finding their vertical and horizontal components to calculate the resultant 'r vector', VT.
What is the power factor in a purely resistive AC circuit?
One (1.0).
How is the magnitude of voltage VT determined?
Using Pythagoras’s Theorem.
What is the term used for electrical resistance in an AC circuit?
Impedance.
What colors are typically used to identify the three phases in a three-phase system?
Red, Yellow, and Blue.
How does the current behave when a 100Hz supply is applied to the same inductor?
The current is still delayed by 90 degrees but reaches a lower maximum value due to reduced time to peak.
What does a phasor represent compared to a vector?
A phasor represents RMS values, while a vector represents peak values.
What law determines the self-induced emf in an inductor?
Faraday’s Law.
What is the significance of the phase angle in AC circuits?
It indicates the time difference between the current and voltage waveforms.
How does phasor subtraction compare to phasor addition?
Phasor subtraction involves taking away the vector components instead of adding them.
How are phasors defined in relation to a reference phasor?
Phasors are defined relative to a reference phasor that always points to the right along the x-axis.
What is generated when three identical coils are placed at an electrical angle of 120 degrees to each other?
A three-phase voltage supply.
What is the inductance of the coil in the example?
0.5 H.
What is the phase difference between current and voltage in a purely resistive AC circuit?
There is no phase difference (θ = 0).
What is the vertical component of voltage V2?
0 volts.
What is the normal sequence of rotation for a three-phase supply?
Red followed by Yellow followed by Blue (R, Y, B).
What does line OA represent in the vector diagram?
The voltage across the resistive component, which is in-phase with the current.
What are the units for impedance in AC circuits?
Ohm's (Ω), the same as for DC.
What is the formula for calculating the rms value of the supply voltage?
V_rms = I * Z, where I is the current and Z is the impedance.
What are the two key characteristics of a phasor?
Magnitude and Direction.
What is the rectangular expression for voltage V2?
30 + j0 volts.
What does the 'in-phase' effect represent in a phasor diagram?
It shows that voltage and current are aligned with no phase difference.
What does a phase difference represent?
The angular difference between two sinusoidal waveforms of the same frequency.
How does inductive reactance change with frequency?
Inductive reactance increases as frequency increases.
What happens to the current when a 50Hz supply is connected to an inductor?
The current is delayed by 90 degrees and reaches a peak value before the voltage reverses polarity.
In an RL series circuit, what is common to both the resistance and the inductance?
The current.
How does the blue phase voltage relate to the yellow phase voltage?
The blue phase voltage lags V_YN by 120 degrees.
What do phasor diagrams represent?
The magnitude and directional relationship between two or more alternating quantities.
What is the horizontal component of voltage V1?
10 volts.
What is the sum of the horizontal components of V1 and V2?
40 volts.
What is the sum of the vertical components of V1 and V2?
17.32 volts.
What does resistance represent in the complex domain?
A real number only, with no imaginary component ('j').
What is the reference phase voltage in the given three-phase system?
The red phase voltage, V_RN.
What is the current flowing through the coil?
4 amps.
What is the average power consumed by an AC resistance defined by?
Using Ohm’s Law.
What is a rotating vector also known as?
A phase vector.
In a pure inductor, how does the current relate to the voltage?
The current lags the voltage by 90°.
What is the reference source in a series AC circuit?
The current.
What does the symbol ω represent in the context of phasors?
The angular velocity in radians per second.
What does line OD represent in the vector diagram?
The resulting supply voltage.
What type of connection can three-phase phasors represent?
Star or delta connected systems.
What is the instantaneous alternating current flowing through a resistor?
i = I_m sin(ωt).
What is the impedance of a resistor connected to a time-varying supply?
It is simply its AC resistance, as the current is in-phase with the voltage.
What is the polar form of the rms voltage across the coil?
VL = 230 ∠ -30° (volts).
What happens to the designation of phasors when the current waveform crosses the horizontal zero axis?
The current phasor can be used as a new reference, making the voltage phasor 'leading' the current phasor by angle Φ.
What happens when the switch is closed in an AC circuit with a resistor?
An AC voltage is applied, causing a current to flow that rises and falls sinusoidally.
What is the phase angle between voltage and current in a purely resistive AC circuit?
Zero.
What does Ohm's Law state in the context of an AC circuit?
The maximum voltage across the resistor is proportional to the current flowing through it.
What does cos(0°) equal in terms of power factor?
1.0.
What is the phase difference (θ) between voltage and current when they are in-phase?
θ = 0.
How does the yellow phase voltage relate to the red phase voltage?
The yellow phase voltage lags V_RN by 120 degrees.
What does line OB represent in the vector diagram?
The horizontal current reference.
What do phasor diagrams represent?
The phase of an alternating quantity at any instant in time.
How does the blue phase voltage relate to the red phase voltage?
The blue phase voltage, V_BN, leads the red phase voltage, V_RN, by 120 degrees.
What is the behavior of inductive reactance as frequency approaches zero?
Inductive reactance decreases to zero, acting like a short circuit.
What is the point of origin in phasor diagrams?
The fixed zero point around which the vector pivots.
What is the current drawn from the supply for the heating element?
4 A (240 V / 60 Ohms).
How can the phasor sum V1 + V2 be found graphically?
By drawing the two phasors to scale on graph paper and measuring the diagonal line known as the 'resultant r-vector'.
What is the expression for current in a purely resistive AC circuit?
I(t) = Im x sin(ωt + θ), where Im is the maximum amplitude and θ is the phase angle.
What is the Rectangular Form in phasor analysis?
A method that divides the phasor into a real part (x) and an imaginary part (y), represented as Z = x ± jy.
How are the lengths of the phasors in the diagram determined?
They are proportional to the values of voltage (V) and current (I) at that instant in time.
What terms are commonly used to describe the relationship between sinusoidal waveforms?
Lead, lag, in-phase, and out-of-phase.
What is the relationship between inductive reactance (X_L) and frequency (ƒ)?
Inductive reactance is proportional to frequency (X_L α ƒ).
What occurs to the current and voltage values in an AC resistance circuit?
They achieve their maximum, minimum, and zero values simultaneously.
What is the relationship between DC resistance and AC impedance for a resistor?
DC resistance = AC impedance, or R = Z.
How is the impedance of the circuit calculated?
Using the formula Z = √(R² + (XL)²), where XL = 2πfL.
What types of AC circuits typically consist of pure ohmic resistance?
Heating elements and lamps.
What is the phase relationship between current and voltage in a purely inductive coil?
The current lags the voltage by 90°.
How is inductive reactance related to frequency?
Inductive reactance is directly proportional to frequency.
What does 'purely inductive coil' imply?
It has no ohmic resistance and therefore no I²R losses.
How is vector V_R represented in the vector diagram for a resistive circuit?
Superimposed to scale onto the current vector.
What forms can phasors represent mathematically?
Rectangular, Polar, or Exponential form.
What does a phasor diagram represent?
The relationship between voltage and current in terms of their magnitudes and phase angles.
What is meant by the term 'in-phase' in an AC circuit?
Current and voltage reach their maximum and minimum values simultaneously.
What geometric shape is used to represent phasor subtraction?
A parallelogram, where the vector difference is the other diagonal.
What points are the three windings of the coils connected to in a three-phase system?
Points a1, b1, and c1 for a common neutral connection.
How can the cosine wave waveform be expressed in terms of a sine waveform?
As sin(ωt + 0° + 90°).
How is the addition of two vectors A and B expressed in phasor analysis?
Using the generalised expression for phasor addition.
How are the voltage and current vectors represented in a phasor diagram when in-phase?
They are drawn superimposed along the same reference axis.
What is the frequency of the supply voltage in the example?
50 Hz.
What components make up the total voltage in an RL series circuit?
The component voltages V_R and V_L.
What does the length of a rotating vector represent?
An AC quantity that has both magnitude (peak amplitude) and direction (phase).
What is the significance of the phasor domain in AC circuits?
It allows for the representation of voltage and current as rotating vectors.
What is the significance of the transformation from sinusoidal time-domain to phasor-domain?
It allows for easier analysis of AC circuits using RMS values.
What is the instantaneous alternating voltage applied to a resistor?
v = V_m sin(ωt).
Why is it impossible to have a purely inductive coil in the real world?
All coils have some resistance due to the resistivity of copper wire.
What does it mean when the current phasor lags the voltage phasor?
It means the current phasor is behind the voltage phasor by the angle Φ.
What must be considered when representing the relationship between two waveforms in phasor notation?
The phase difference, Φ, of the waveform.
What does d/dt represent in the context of current?
The rate of change of current with respect to time.
What is the phase relationship between voltage and current in a pure inductor?
Voltage leads current by 90 degrees, or current lags voltage by 90 degrees.
What does the Rectangular Form represent?
Both the magnitude and the phase of the sinusoidal voltage.
What does a phase difference indicate in sinusoidal waveforms?
The angle by which one waveform lags or leads another.
What is the general sinusoidal expression in time-domain form?
A(t) = Am sin(ωt ± Φ).
What is the AC resistance of the heating element?
60 Ohms.
What is the horizontal component of voltage V2?
30 volts.
In which direction do the phasors for a three-phase system rotate?
In an anti-clockwise direction.
How do you calculate the current drawn from the supply?
Using Ohm's Law: I = V/R.
What is the formula for total impedance Z in a circuit with resistance and reactance?
Z = R + jX.
How does a coil behave at maximum current flow?
It acts more like a short circuit.
How is the active power consumed by the AC resistance calculated?
P = V * I.
What is the active power consumed by the heating element?
960 W (240 V * 4 A).
What does the change in current induce within the coil?
A magnetic field.
What is the assumed angular velocity of a sine wave in phasor diagrams?
ω in rad/sec.
What does the tip of the rotating vector represent at 0°, 180°, and 360°?
The angles at which the vector is horizontal.
How can Ohm's Law be applied in AC circuits?
It can be used to calculate voltages and currents in circuits containing AC resistance.
What is the rectangular expression for voltage V1?
10 + j17.32 volts.
What is the phase relationship of current lagging voltage in an inductive circuit?
Current lags voltage by 90 degrees.
What type of current flows through an inductive coil (L)?
Sinusoidal current.
How does the alternating current behave in a purely resistive circuit?
It varies in proportion to the applied voltage, following the same sinusoidal pattern.
What angle does voltage V1 lead voltage V2 by?
60 degrees.
How is impedance symbolized in AC circuits?
By the letter Z.
What does θ represent in the context of voltage and current waveforms in an inductor?
The phase difference or phase angle, which is +90°.
How does the effective power consumed by an AC resistance compare to that in a DC circuit?
They are equal.
How is the inductive reactance represented in polar form?
X_L ∠ 90°.
What occurs to the back emf in a coil at maximum steady state?
It decays to zero.
What is the general expression for impedance in a purely resistive circuit?
Z = R + j0 = R Ω.
What is the supply voltage defined in the example?
V(t) = 325 sin(314t - 30°).
How does current flow in an AC circuit compared to a DC circuit?
It behaves differently due to AC inductance.
What factors determine the opposition to current in an AC circuit?
Inductance of the coil and frequency of the applied voltage waveform.
How is vector V_L represented in the vector diagram for an inductive circuit?
Drawn 90 degrees in front of the current vector.
What are the units of impedance?
Ohms (Ω).
What can be inferred from the phasor diagram?
The relative phase difference and amplitude of current and voltage.
What is the phase difference when the current lags the voltage in the example provided?
30 degrees.
What does the phasor diagram represent at time zero?
The relationship between voltage and current.
What is the relationship between voltage and current phasors in a pure resistance within an AC circuit?
It is the same as in a DC circuit, showing a linear relationship.
What characterizes a balanced three-phase voltage supply?
Three individual sinusoidal voltages that are equal in magnitude and frequency, out-of-phase by 120 degrees.
What is the relationship between V_MAX and I_MAX in an AC inductance?
V_MAX = ωLI_MAX = √2V_RMS.
What is the horizontal component of Voltage V2 of 30 volts pointing along the horizontal zero axis?
30 cos 0° = 30 volts.
What is the total resistance for resistive elements connected in series?
R1 + R2 + R3, etc.
What happens to the current in a coil after five time constants?
It reaches its maximum steady state condition.
How is a complete sine wave constructed in phasor representation?
By a single vector rotating anti-clockwise at an angular velocity of ω = 2πƒ.
What is the effect of self-induction in an inductor?
It produces an emf that counteracts the changing current.
What type of circuit is described in the text?
A pure inductance circuit.
What is self-inductance?
The effect where the magnetic field created by current opposes any change in current.
What is the main difference between vectors and phasors?
A vector's magnitude is the peak value of the sinusoid, while a phasor's complex magnitude is the rms value of the sinusoid.
What remains the same for both vectors and phasors?
The phase angle, direction, and angular velocity.
In which direction do the current and voltage phasors rotate?
They rotate in an anticlockwise direction.
What is the voltage supply in the sinusoidal voltage example?
V(t) = 100 x cos(ωt + 30°).
What is the relationship between the scales of V_R and V_L in the vector diagram?
They are drawn to the same scale.
What is the purpose of finding a common component in vector diagrams?
To accurately represent the relationships between voltages and currents.
How is the phase angle between current and supply voltage calculated?
Using the relationship between the phasors of current and voltage.
What is the downside of the graphical method for phasor addition?
It is time-consuming and may produce errors if not drawn accurately or to scale.
What is the relationship between phase voltages in a three-phase system?
They are equal in magnitude but differ in phase angle.
What is the resistance of the coil in the given example?
30 Ω.
What mnemonic helps remember the phase relationship in a pure inductor circuit?
ELI, which stands for Electromotive force (E) leads Inductance (L) before Current (I).
How is instantaneous power consumed by a resistor calculated in a purely resistive circuit?
It is given by the product of voltage and current since they are in-phase.
What is the significance of the angle Φ in phasor diagrams?
It represents the phase difference between two sinusoidal quantities.
What happens to the current flowing through an inductor as frequency increases?
The current flowing through the inductor reduces in value.
What happens to inductive reactance when frequency increases?
Inductive reactance increases, approaching infinity as frequency approaches infinity.
How is the total reactance for inductive elements connected in series calculated?
X1 + X2 + X3, etc.
How are the voltage and current in a resistive circuit related?
They are in phase.
What law do vectors in phasor diagrams obey?
The parallelogram law of addition and subtraction.
What is the inductance value given in the example?
L = 2.2 H.
What is the equivalent rms voltage from the peak voltage of 325V?
230V.
How is the total impedance of a coil represented?
As a resistance in series with an inductance.
What symbol is used to represent reactance?
X.
What is the resistance connected to the sinusoidal voltage supply?
50 Ohms.
How do you determine the impedance in a purely resistive circuit?
Impedance Z = R.
What is the relationship between current and voltage in a pure ohmic AC resistance?
They are both 'in-phase' with no phase difference.
What is impedance in an AC circuit?
The effective resistance to current flow, measured in Ohms, including both AC resistance and AC reactance.
What law is used to find the current flowing through the coil?
Ohm's Law.
What can sinusoidal waveforms be transformed into for analysis?
The spatial or phasor-domain.
What does inductive reactance depend on?
It depends linearly on the supply frequency.
Why might we need to know the position of a phasor?
To compare two different waveforms on the same axis.
If V1 has a peak voltage of 20 volts and V2 has a peak voltage of 30 volts, with V1 leading V2 by 60 degrees, what is the first step to find the total voltage?
Draw a phasor diagram representing the two vectors.
What does I represent in the context of the circuit?
The r.m.s. value of the series current.
What does impedance (Z) consist of in a pure ohmic resistance?
A real part (the actual AC resistance value, R) and a zero imaginary part (j0).
How can voltage and current phasors be represented in an AC circuit?
Using complex numbers to represent a complex quantity.
What can be said about the current when using voltage as a reference?
The current lags the voltage by 90 degrees.
What geometric shape represents the resultant phasor diagram for the RL circuit?
A right-angled voltage triangle (OAD).
What is the effect of the applied voltage reaching its maximum value in relation to the current?
The applied voltage reaches its maximum a quarter of a cycle earlier than the current.
What is the formula for impedance in terms of resistance and reactance?
(Impedance)² = (Resistance)² + (j Reactance)².
How does the magnetic field affect the change in current?
It opposes or restricts the change in current.
What does impedance depend on?
The frequency (ω) of the circuit.
What does V_L represent?
The I.X_L voltage drop across the inductance, which leads the current by 90 degrees.
What does the time axis of the waveform represent?
The angle in degrees or radians through which the phasor has moved.
How do you find the peak value of the current flowing through the circuit?
Using Ohm's Law: I_peak = V_peak / R.
What method is used to determine the Resultant Phasor when adding out-of-phase voltages?
The parallelogram law is used.
What does 'j' represent in the impedance formula?
The 90° phase shift.
What is the significance of drawing vector diagrams in AC circuits?
To visualize the relationship between voltage and current.
By how much is the current delayed in a purely inductive circuit?
90 degrees (or 1/4 of a cycle).
What does the voltage do in relation to the current in a purely inductive circuit?
The voltage leads the current by 90 degrees.
What is the impedance of the circuit with a 50 Ohm resistance?
50 Ohms.
What does back emf do in a purely inductive coil?
It opposes the rise and fall of the current.
What does the RL Impedance Triangle represent?
The relationship between resistance, reactance, and impedance.
How do inductors store energy?
In the form of a magnetic field created when a voltage is applied across the terminals.
What is the relationship between voltage and the component voltages in an AC circuit with resistance and inductance?
The voltage is the phasor sum of the component voltages, V_R and V_L.
What is a key use of phasors in circuit analysis?
They are used for summing sinusoids of the same frequency.
How do you add two in-phase alternating waveforms?
They can be added together like DC values to find the algebraic sum.
What relationship can be constructed using phasor diagrams?
The phasor voltage-current relationship.
How can the resultant voltage across the RL circuit be mathematically found?
Using Pythagoras' theorem.
What phase angle is assumed in the waveform starting at time t = 0?
A corresponding phase angle in either degrees or radians.
How is the inductive reactance of the coil calculated?
XL = ωL = 314 x 2.2 = 690 Ω.
What happens to the phase difference when resistance is added to an inductive circuit?
The current still lags the voltage, but by an amount less than 90°.
What symbol represents the phase angle of the circuit?
The Greek symbol phi, Φ.
How is impedance represented mathematically?
Z = R + jX_L, where R is resistance and X_L is reactance.
What happens when an inductive coil is connected to an AC supply?
It produces a self-induced emf opposing the emf that initially set up the current.
What is the relationship between voltage and current in a purely resistive circuit?
The voltage across the resistance is the same as the supply voltage.
What does V_R represent?
The I.R voltage drop across the resistance, which is in-phase with the current.
What do the vertical positions of the vector represent?
The positive peak value at 90° and the negative peak value at 270°.
What is the opposition to current flow through an AC inductor called?
Inductive Reactance.
What visual representation can illustrate the phase relationship in a purely inductive circuit?
A phasor diagram.
What is the peak value of the current flowing through the circuit?
2 A (100 V / 50 Ohms).
What are inductors and chokes made of?
Coils or loops of wire wound around a hollow tube (air cored) or ferromagnetic material (iron cored).
What is the relationship between current and voltage in a pure inductance?
The current lags the voltage by exactly 90 degrees.
What does a phasor represent at a specific point in time?
A scaled voltage or current value of a rotating vector.
How is reactance represented in terms of magnitude and direction?
As a vector.
What is the significance of the angle Φ in phasor relationships?
It indicates the phase difference between the current and voltage phasors.
What does the symbol 'ω' represent in relation to inductive reactance?
ω equals 2πƒ (angular frequency).
What must be considered when adding two out-of-phase voltages?
The phase angle between them must be taken into account.
What type of supply is considered when analyzing AC resistance?
A single-phase AC supply.
What does the positive phase angle θ indicate?
The phase difference between voltage and current.
What is the result of adding two in-phase voltages of 50 volts and 25 volts?
The result is a total voltage of 75 volts (50 + 25).
Is impedance a phasor?
No, it is the result of two or more phasors combined.
What is the formula for the applied voltage in the RL circuit?
The applied voltage is the vector sum of V_R and V_L.
What determines the growth of current flowing through an inductor?
The inductor's own self-induced or back emf value.
What is the relationship between back emf voltage (V_L) and current in an inductor?
V_L is proportional to the rate of change of the current flowing through it.
