mod 3 rg 3.14 resistive inductive and capacitive circuits

Electrical circuits which contain capacitors and inductors are found to oppose alternating current in two different ways.

Firstly, when passing direct current, capacitors and inductors possess electrical resistance, and some electrical energy is dissipated as heat in these components due to this resistance.

Secondly, capacitors and inductors also possess reactance. Reactance, like resistance, is measured in ohms

but is not the same as resistance.

there are two types of reactance, “capacitive reactance” Xc and “inductive reactance” Xl. Reactance opposes the flow of Alternating Current (AC) but, rather than being dissipated as heat,

energy is stored by the components and returned to the circuit later in the AC cycle.

In circuits with reactance, the AC voltage and AC current are found to be

out of phase with each other. This means that they do not reach their peak values at the same time. This is an important property of such circuits and it is possible to both predict and make use of this phase difference.

Circuits containing combinations of inductors, capacitors, and resistors are sometimes referred to a

“LCR circuits” and are essential to many electrical and electronic devices. LCR circuits are fundamental to the design and operation of radio transmitters and receivers.

A resistance (a filament bulb rated at 230 V and 60 W) is connected in series with an ammeter to a DC voltage source of 230 V. At the same time, a similar bulb with an equal resistance is connected in series with an ammeter to an AC voltage source of 230 V with a frequency of 50 Hz. As a result, both light bulbs glow with the same brightness and dissipate electrical energy at the same rate. Thus,

with both DC and AC, the bulbs have the same resistance and produce the same effect.

Resistance is sometimes called a “real load”, to distinguish it from

he imaginary load of a circuit's reactance.

A real load produces a measurable effect, i.e.,

it converts electrical energy into another form of energy (heat, light, mechanical energy).

An imaginary load stores

energy in one part of the cycle and then returns the energy to the circuit in the next part of the cycle.

Using an oscilloscope to measure the current and voltage in an AC circuit with resistance, and plotting the results on a graph, would show that:

• The sinusoidal voltage causes a sinusoidal current.

• The zero passages and peak values of voltage and current would coincide; therefore, both graphs would follow the same course.

In a real load, current and voltage are

in phase

in an AC circuit, resistance can be calculated from the

RMS values of voltage and current by applying Ohm’s law.

In an AC circuit with resistance, Ohm’s law is valid:

resistance = R= Vr/Ir

Where Vr is the resistive voltage in ohms and Ir is the resistive current in amps

A heating device has a measured potential difference of 230 V across it and draws a current of 2 A. What is the resistance of the heating device?

resistance = R= Vr/Ir

R=230/2

R=115 ohms

A coil with a winding resistance of 50 Ω is connected in series with an ammeter and a DC voltage source of 230 V. An identical coil is connected in series with an ammeter and an AC voltage source of 230 V with a frequency of 50 Hz.

The AC current through the coil is now found to be

smaller than the DC current.

In addition to the resistance (winding resistance) R,a coil in an AC circuit presents another kind of opposition to the current, called

“inductive reactance” XL

The inductive reactance results from the AC in a coil causing a

self-induced voltage. This self-induced voltage opposes the applied voltage, thereby reducing the overall voltage across the coil, and therefore, restricting the current.

The magnetic field repeatedly builds up in one direction, breaks down, and builds again in the other direction. This causes the coil to draw energy, store it briefly in the magnetic field, and then transfer it back to the circuit as the magnetic field collapses. As this cycle repeats, energy is pushed back and forth without being dissipated as heat by the inductor or doing any work. This energy is called

“reactive energy”.

An ideal coil can be approximated by using very thick wire, with a resistance. The wire has some resistance but the

value is very small and it can be ignored.

The induced voltage across the coil is at a maximum when the rate of change of current is at a maximum. This occurs when

the current is zero (where the gradient of the current-time graph is steepest).

Therefore, in a circuit with only(pure) inductive reactance, the current lags behind the voltage by

one quarter of a cycle, which is a phase angle of 90° or pie/2 radians.

Phase shift = 90°

The phase shift is caused by the influence of the coil’s self-induction.

phasor diagram is an alternative way of representing sinusoidal variables. Each quantity is represented by a vector that rotates at a steady angular speed (w) measured in either degrees per second or radians per second. The instantaneous value of each quantity is represented by the vertical component of the corresponding vector. For example, in the animation above, the voltage is initially zero at 0° on the graph, because the red, voltage phasor is horizontal and has no vertical component. Whereas the current phasor

tarts at its maximum, negative value because it is pointing vertically downwards. As the phasors rotate, the graph plots the height of the tip of each phasor on the vertical axis of the graph. The quantity (wt) is the angular speed multiplied by the elapsed time, and gives the phase angle (in either degrees or radians). As the two vectors rotate anticlockwise, the phase angle (phi) between them remains constant.

The self-induced voltage opposes the applied voltage, which delays

the initial build-up of current.

In a purely inductive circuit, the inductive reactance can be determined from

the RMS values of voltage and current by analogy to Ohm’s law.

XL = vRMS/iRMS

Introducing an iron core to the coil increases the

inductive reactance.

The value of inductive reactance can be calculated using the following formula.

XL=2pie f L

were XL is measured in ohms (Ω)

f is frequency measured in hertz (Hz)

L is inductance measured in henrys (H)

A capacitor with a capacitance C of 10µF is connected in series, with an ammeter, to a DC voltage source of 230 V. An identical capacitor is connected to an AC voltage source of 230 V.As already mentioned, the capacitor blocks DC (except for the short initial current impulse). A capacitor has a very high resistance, so

no current flows in a DC circuit after this initial pulse, which charges the capacitor.

In an AC circuit, the capacitor does not oppose the flow of current to the same extent, i.e., the capacitor operates in a similar way to a resistor in a DC circuit. No current actually passes through the capacitor, instead

charging and discharging cycles repeat rapidly, and the associated charging and discharging currents flow in the rest of the circuit. However, it is conventional to refer to AC current as “passing through” a capacitor.

In an AC circuit, a capacitor opposes but does not prevent the flow of electric current. This opposition is referred to as

“capacitive reactance” Xc

Every time the polarity in an AC circuit changes, the capacitor is charged and discharged. Therefore,

the current repeatedly rises and falls

The capacitor in an AC circuit behaves like a resistor in a DC circuit because

the current is restricted by the potential difference, which periodically builds up across the capacitor

However, unlike resistance, the energy associated with capacitive reactance is not dissipated as heat. This is because

the energy is transferred to the capacitor (reactive energy) during the charging process and stored briefly in the electric field between its plates. During the discharging process, the stored energy is transferred back to the circuit. The energy is transferred back and forth, without any transfer of energy through work or heat due to the capacitance.

Using an oscilloscope to measure the current and voltage of an AC circuit with capacitive reactance, and plotting the results on a graph, shows that:

• The sinusoidal AC voltage causes a sinusoidal current.

• The zero passages of current and voltage are out of phase.

• In a circuit with only capacitive reactance (ideal capacitor), the current leads the applied voltage by 90° (i.e., ¼ cycle).phase shift of 90

n an AC circuit, a capacitor generates a phase shift opposite to that of a

coil (inductor).

Applying a DC voltage to a capacitor generates an initial current (charging current) for a short time. Increasing the voltage also generates a brief charging current; decreasing the voltage causes a

current to flow in the opposite direction (discharging current).

in a capacitive circuit The current has reached its peak value, when the highest change in voltage occurs, i.e.,at the zero passage of the voltage. If the supplied voltage does not change, the current is 0; this happens at the peak values of the voltage.Thus, at the same time:

• Voltage is zero when current is at its peak value.

• Voltage is at its peak value when current is zero.

he capacitive reactance can be calculated from the

RMS values of the supplied voltage and the current, and analogous to Ohm’s law, capacitive reactance

Xc = Vrms/Irms

The value of capacitive reactance can be calculated using the following formula.

Xc = 1/2pie f C

Where XC is capacitive reactance, measured in ohms (Ω)

f is frequency, measured in hertz (Hz)

C is capacitance, measured in farads (F)

The capacitive reactance of a capacitor decreases as the capacitance increases. This is because

the higher the capacitance, the more easily it can be charged with a certain voltage, i.e., the higher the current and, therefore, the lower the reactance.

The capacitive reactance of a capacitor decreases as the frequency increases. This is because,

the higher the frequency, the less time there is available for the capacitor to charge, i.e., the higher the current and, therefore, the lower the reactance.

There are two types of reactance: capacitive reactance (Xc) and inductive reactance (XL). Reactance opposes the flow of Alternating Current (AC) but, rather than being dissipated as heat, energy is stored by the components and returned to the circuit later in the AC cycle. The energy stored in the electric field of a capacitor or in the magnetic field of an inductor, during a cycle of AC, is known as “reactive energy” and the rate at which this energy is transferred is known as the

“reactive power”.

The power formula for DC circuits P = IV is also valid for instantaneous values in AC circuits.

Therefore:

P(t) = I(t)V(t)

V(t) and I(t) are both time-dependent quantities, so P(t) also depends on the time.

the variation of voltage and current with time in a circuit with a purely resistive load i.e. with no phase shift. Multiplying the related instantaneous values of voltage and current gives the instantaneous values of the power this kind of power is referred to as

“true power” (also known as “real power” or “effective power”).

The true power of the AC varies constantly between

zero and its maximum value. At frequencies well below 50 Hz, this can lead to visible light fluctuation in light bulbs, although this is still hard to detect with the naked eye.

True power can be calculated using the formula:

P=VIr

P = power in watts (W)

V = voltage in volts (V)

Ir = resistive current in amps (A)

The unit for true power is the watt (W). The true power of a resistance is the product of

the RMS values of voltage across the resistance and the current flowing through it.

In a pure inductor, the current lags the voltage by 90°. Power can be analysed by plotting a similar graph as previously, multiplying the instantaneous values of current and voltage. Again, the power graph is a sinusoidal waveform with twice the frequency of the AC, but now consists of

positive and negative alternations (for a resistive load, the power has no negative alternation). Positive power represents the power supplied to the inductor by the circuit; negative power represents the power returned to the circuit by the inductor.

The energy briefly transferred from the circuit to the inductor is stored in a magnetic field. When the magnetic field collapses, the stored energy

is transferred from the magnetic field, back to the circuit.

Power is transmitted back and forth in this way twice during each cycle of AC.

This power is not dissipated as heat, light, or mechanical energy; it is called “inductive reactive power”. The true power is

zero. The average value of the inductive reactive power is also zero over a whole cycle but can be measured by calculating an RMS value in the following way. This quantity is given the symbol QL

QL =

VIL

QL= is inductive reactive power, measured in volt-ampere reactive (VAr).

V is RMS voltage, measured in volts (V)

IL is RMS inductive reactive current, measured in amps (A).

Volt ampere reactive (VAr) and Watts (W) are both products of voltage and current. VAr identifies

reactive power allowing for the phase shift between the voltage and current in circuits with reactance (AC). Watts is the result of the product of voltage and current in DC circuits.

For a purely capacitive load, the current leads the voltage

by 90°. The relationship between power and time can again be plotted by multiplying the instantaneous values of current and voltage. Again, the power graph has a sine waveform with twice the frequency, consisting of positive and negative alternations.

The energy briefly transferred from the circuit to the capacitor is stored in an electric field. When the capacitor discharges

stored energy is transferred from the electric field, back to the circuit.

Power is transmitted back and forth in this way twice during each cycle of AC.

This power is not dissipated as heat, light, or mechanical energy; it is called “capacitive reactive power”. The true power is zero. The average value of the inductive reactive power is also zero over a whole cycle, but it can be measured by calculating an RMS value in the following way.

Qc = VIc

Qc= capacitive reactive power, measured in volt-ampere-reactive (VAr).

V= is RMS voltage, measured in volts (V)

Ic = is RMS inductive capacitive current, measured in ampere (A)

A real coil has some resistance. To analyse a real coil it can be represented as a pure inductor in series with a pure resistor, as shown in the graphic below. A coil with significant resistance like this is sometimes referred to as a

“coil with loss” because some power is dissipated by the intrinsic resistance of the wire in the coil.

a pure inductor in series with a pure resistor the current lags the voltage in this example, by a phase angle of

60°

Multiplying the instantaneous values of current and voltage in a pure inductor in series with a pure resistor shows the variation of power with time. Again, the power graph has a sinusoidal waveform with twice the frequency of the AC. The graph mainly runs above the time axis. It has positive and negative values, but the overall positive power is greater over one cycle. This means that

more power is delivered to the circuit than is returned to the supply, because some power is dissipated by the load, and some is returned to the supply.

If measured with an AC ammeter and AC voltmeter, the circuit above appears to have a power consumption of the RMS voltage multiplied by the RMS current this is not the true power, because

current and voltage are not in phase with each other, but the meters do not indicate this). This quantity is called the “apparent power” and given the symbol S

S=VI

S apparent power measured in volt-ampere (VA)

V is RMS voltage, measured in volts (V)

I is RMS current, measured in ampere (A)

The unit for apparent power is the volt-ampere (VA), which is the same unit as the watt (W) but

is used to clearly identify a quantity as apparent power.

In a purely resistive circuit, the opposition to current flow is called Resistance R. However, in an a.c. circuit, inductors and capacitors also oppose current flow. This is called

'Reactance', as the components 'react' to an AC current and is given the unit X

n the case of Inductive reactance, it is given

XL

in the case of a capacitivecircuit is given

Xc

The resistance R, capacitive reactance X_C, and inductive reactance X_L of a circuit can be combined to produce an overall measure of the opposition of the circuit to AC, known as

impedance', Z. However, this is not a straightforward addition of these quantities, and due to the nature of the phase relationship between them, vector addition (geometric addition) must be used.

To determine unknown values of power or current, it is necessary to either apply

Pythagoras’s theorem to the phasor diagram, if two of the phasors are known, or to apply trigonometric functions, if the phase angle is known.

he graph in the graphic below summarises the effect of frequency on resistance R inductive reactance X_L,and capacitive reactance X_C.

Resistance is not dependent on frequency, and so is represented by

a horizontal line

inductive reactance is directly proportional to

frequency, XL = 2piefL and therefore, it increases with increasing frequency.

capacitive reactance is inversely proportional to

frequency, (Xc) = 1/ 2piefc and therefore, it decreases with increasing frequency.

In theory, the frequency has no influence on the resistance value of a resistor; however, in practice,

there can be slight differences.

At high frequencies, the resistance of a conductor can increase due to the skin effect, where a higher current is observed to flow along the outside 'skin' of a wire than in its core. This is caused by

eddy currents within the conducting wire itself, which tend to restrict the flow of AC current in the centre of the conductor, effectively reducing the cross-sectional area of a wire, and therefore, effectively increasing its resistance.

Effective resistances have constant values in the range of 0 Hz to approximately 1000 Hz. If the frequency is higher than 1000 Hz, the resistance rises. Therefore,

effective resistance is not considered frequency dependent below 1000 Hz.

Impedance is a measure of

the opposition to AC in a circuit and takes both the resistance and reactance of the circuit into account.

The impedance of a coil in an AC circuit comprises the resistance of the wire in the windings and the inductive reactance of the coil. Inductive reactance is not the same as resistance, because

no energy is dissipated due to reactance, but it still impedes the current in a similar way to resistance.

The total resistance and reactance of a circuit is called

‘impedance’ Z, measured in ohms 

Analogous to Ohm’s law, impedance can be determined from the RMS values of AC voltage and AC current.

Z = Vrms/Irms

Making Irms the subject of the equation gives the equivalent to Ohm’s law for AC circuits:

Irms = Vrms/Z

This formula can be used to determine RMS current, RMS voltage, or impedance in AC circuits.

An AC voltage of 230 V is applied to a coil with both resistance and reactance, which draws a current of 0.5 A. What is the impedance of the coil?

Z = Vrms/Irms

Z= 230/0.5

Z = 460

Only the Root Mean Square (RMS) values of current and voltage are used in this section on the series connection of a resistor and inductor. All references to current (I) and voltage (V), refer to the

RMS values.

In practice, coils cannot be considered to be ideal and will always present some resistance. A coil can be represented as

an ideal coil in series with a resistor, the resistance and reactance together comprising the total impedance (note that this is not a simple addition, as can be seen below).

For series connections, the current is the reference phasor, because

it is the same for all components.

The voltage phasor V_R (resistive voltage) is in phase with

the current phasor I.

The voltage phasor V_L (reactive voltage) is

perpendicular to the current phasor, aligned vertically upwards (reactive voltage V_L leading the current I by 90°).

In a phasor diagram, resistance vectors and reactance vectors are always

perpendicular.

Since phasors are vectors, to find the sum of two phasors,

geometrical addition (vector addition) must be applied.

for the addition of voltage phasors a parallelogram is used. The resulting diagonal line represents a geometrical sum. This sum can be used to

determine both the size and the direction of the resultant voltage i.e. the voltage across the real coil that this circuit represents, the applied voltage V.

The geometric addition of the phasors V_R and V_L gives the applied voltage V. The phasor length indicates the

peak value of voltage V. The position of the phasor indicates the phase angle between the applied voltage V and the current I.

The current I lags the voltage V with a phase angle between 0 ° and 90 °.

In the phasor diagram of the series connection shown above, the three phasors rotate about the origin at the same angular speed, keeping them in the same position, relative to each other. The resultant voltage V also rotates about the origin at the same angular speed. To solve by vector addition,

this diagram can be simplified by moving the reactive voltage, phasor V_L, to the right, so that it begins where the resistive voltage phasor V_R ends. This forms a right-angled triangle that can be solved either by applying Pythagoras’s theorem or trigonometrical functions.

Since resistive and reactive vectors are always perpendicular to each other, this triangle is always

right-angled.

In a right-angled triangle, each side can be separately calculated using Pythagoras’ theorem.

From the diagram of geometric addition, one can write:

V² = Vr² + VL²

voltage = square root of Vr² + VL²

resistive voltage Vr= square root V² + VL²

reactive voltage VL= square root V² + Vr²

Another way to calculate the sides, and therefore the angles, of a right-angled triangle, is to use trigonometric functions. In the right-angled triangle for the vector addition above, we can apply either of the following equations to determine the phase angle.

cos angle = Vr/v

sin angle = Vl/v

resistive voltage = V cos angle

reactive voltage = V sin angle

were V is the applied voltage

In a series connection consisting of a resistance and an inductive reactance, each voltage can be divided by the common current I to calculate:

Z = V/I

resistance:

R=Vr/I

inductive reactance:

XL=VL/I

These three values are proportional to their associated voltages. Therefore, dividing all sides of the triangle previously analysed for voltage by the current I results in a right-angled triangle for impedance of the same proportions as that for voltage.

If X_L and R are known, impedance Z can be determined either by drawing the impedance triangle to scale and measuring the hypotenuse; or by calculating the length of the hypotenuse Z using Pythagoras's theorem or trigonometric functions:

Impedance:

Z = square root R² + x²L

cos angle = R/Z

sin angle = XL/Z

If V and I are both known (the supplied potential difference and current), the impedance can be determined by simply calculating their quotient:

Z = V/I

Determine the impedance of a coil with a resistance of 30 ohms, and an inductive reactance of 90 ohms?

R= 30 XL=90

Construct a scale drawing and measure the length Z  as shown in the diagram below. Here, a scale of 1 mm to 1 Ω was used to draw the resistance and reactance vectors, then the hypotenuse was drawn and measured.

OR

Z= sqaure root of R² + XL²

Z= square root of 30² + 90²

z= square root of 9000

z= 95

If a resistor and capacitor are placed in series with an AC supply, the same sinusoidal current flows through both components. This current is due to

a resistive voltage V_R across the resistor and a reactive voltage V_C across the capacitor.

The voltage V_R is in phase with the current I. In the capacitor, the same current I leads the voltage V_C by 90°, therefore

the voltage V_C lags the current by 90°.

The reference phasor I is placed parallel to the horizontal axis as before. The resistive voltage phasor V_R points in the same direction as I; the reactive voltage phasor V_C is perpendicular to the current phasor I and points downwards (lagging the current I by 90°). For a series connection, the applied voltage V is obtained from the geometric addition of the phasors V_R and V_C. Therefore,

the current I leads the voltage V by a phase angle theta where theta is between 0° and 90° (remember thatI and V are the RMS current and voltage supplied to the series connection).

To solve for an unknown value, the phasors are combined by vector addition by forming a triangle where V_C is added to V_R to give V. Since resistive and reactive values are always perpendicular to each other in the phasor diagram, this triangle is always right-angled.

In a right-angled triangle, each side can be separately calculated using Pythagoras’ theorem.

From the diagram of geometric addition one can write:

V² = VR² + VC²

voltage = square root VR² + VC²

resistive voltage:

VR = square root V² - VC²

reactive voltage

VC V² - VR²

Another way to calculate the resistive and reactive capacitive voltages and therefore the angles, of a right-angled triangle, is to use trigonometric functions. In the right-angled triangle for the vector addition above, we can apply either of the following equations to determine the phase angle.

cos angle = VR/V

sin angle = VR/V

resistive voltage: VR = V cos angle

reactive voltage VC = V sin angle

were V is the applied voltage

in a series connection consisting of a resistance and a capacitive reactance, each voltage can be divided by the common current I to calculate:

impedance Z= V/I

resistance R= Vr/I

capacitive Xc =Vc/I

These three values are proportional to their associated voltages. Therefore, dividing all sides of the triangle previously analysed for voltage by the current I results in a right-angled triangle for impedance of the same proportions as that for voltage.

impedance: Z = square root R² + Xc²

resistance: R = Z cos angle

capacitive Xc = Z sin angle

Real capacitors are not ideal and have a measurable resistance. This resistance permits a small amount of current to flow between the plates of the capacitor. This current is sometimes referred to as

'lost' current, and the capacitor as a capacitor with 'loss'.

A capacitor with loss can be modelled by an equivalent circuit, containing an ideal capacitor with its resistance connected in

parallel.

For parallel connections, there is a common voltage across each component but

the current through each can be different.