slide 2 / 69 lrc series ac circuits sources of alternating

Alternating Current

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Topics to be coveredSources of alternating EMF

AC Circuits and Impedance

LRC Series AC Circuits

Resonance in AC Circuit Oscillations

Transformers

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Sources of Alternating EMF

Faraday's discovery of electro-magnetic induction played a very important role in the development of the electric generator. A generator transforms mechanical energy into electrical energy.The generator presented by the diagram to the left can be found in every high school physics lab.

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Sources of Alternating EMFA simple generator consists of many coils of wire wound on an armature that can rotate in a magnetic field created by a permanent magnet. The axle is turned by a hand. In real life it can be any mechanical means (falling water, steam flow, car motor belt...). An EMF is induced in the coil because of constant change in the magnetic flux through the coil. The turning coil is connected to the external circuit by two slip rings and brushes connected to them.

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Sources of Alternating EMFNow it is time to look at the AC current production in more detail. When a single loop of wire is placed in a uniform magnetic field the magnetic flux is determine by the following formula:

Where θ is an angle between magnetic field B and the normal to the loop.

When the loop rotates the angle changes with time θ=ωt. It proves that the magnetic flux varies with time even when the field stays constant.

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1 Which of the following is the unit of the magnetic flux?

A T.m

B Wb.m

C Wb/m2

D T.m2

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2 A rectangular loop of wire with a size of 10x20 cm is placed in a uniform magnetic field of 2 T. Find the magnetic flux through the loop when the angle between the field and the normal to the loop is 60o.

A 0.05 Wb

B 0.02 Wb

C 0.01 Wb

D 0.04 Wb

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3 A rectangular loop of wire with a size of 10x20 cm is placed in a uniform magnetic field of 2 T. Find the magnetic flux through the loop when the angle between the field and the normal to the loop is 0o.

A 0.05 Wb

B 0.02 Wb

C 0.01 Wb

D 0.04 Wb

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Sources of Alternating EMFAccording to Faraday's Law the induced emf is proportional to the rate of change of magnetic flux.

The induced emf varies sinusoidally with time. When an external circuit is connected to terminals ab in the diagram above, the electric current caused by the emf is also sinusoidal which we call an alternating current or ac current.

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4 The magnetic flux through a coil of wire changes from 0.5 Wb to 2.5 Wb in 0.1 s. What emf is induced in the coil?

A 50 V

B 40 V

C 20 V

D 10 V

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5 A square coil of wire with 10 turns and an area of 0.5 m2 is placed in a parallel uniform magnetic field of 0.75 T. The coil is turned so it is now perpendicular to the magnetic field. This action takes 0.15 s to complete. What is the emf induced in the coil?

A 25 V

B 20 V

C 15 V

D 10 V

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This equation is valid for any shape loop. When a single loop is replaced with a coil consisting of N loops the formula looks slightly different.

Since ω is measured in radians per second (rad/s), we can write ω=2πf, where f is the frequency.

The United States and Canada use generators operating at the frequency of 60 Hz, although 50 Hz is used in many countries.

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6 An AC generator consists of 60 loops of wire of area 0.4 m2. What is the induced emf generated by the loops if they rotate at a constant angular velocity of 15 rad/s in an uniform magnetic field of 0.15 T?

A 100 V

B 140 v

C 180 v

D 200 v

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7 An AC generator has a coil with 10 loops of wire and an area of 0.08 m2. The coil rotates at a constant rate of 60 rev/s in a uniform magnetic field of 0.4 T. What is the maximum induced emf in the coil?

A 90.7 V

B 100.4 V

C 110.2 V

D 120.6 V

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8 An AC generator consists of 200 turns of wire of area 0.25 m2 and total resistance of 25 Ω. The generator rotates at a constant rate of 60 rev/s in a uniform magnetic field of 0.04 T. Find the maximum induced current.

A 10.2 A

B 30.1 A

C 25.7 A

D 44.2 A

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9 An AC generator consists of 100 turns of wire of area 0.4 m2 and total resistance of 30 Ω. The generator rotates at a constant rate of 60 rev/s in a uniform magnetic field of 0.05 T. Find the maximum induced current.

A 10.2 A

B 20.1 A

C 25.7 A

D 31.4 A

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These two graphs show the time variation of the magnetic flux through the loop and the resulting EMF at terminals ab. At time t = 0 the angle θ =90o.

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It is hard to imagine our life without ac current. It became so popular because of its simple production and, most importantly, its long-distance transmission through electrical lines. With the combination of an ac transformer it is easy to minimize i2R energy losses in the cables.

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TransformersA transformer is a device for increasing or decreasing an ac voltage. Transformer are found everywhere: in TV sets, on utility poles, in converters and chargers.

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TransformersA transformer consists of two coils of wire known as primary and secondary coils. The coils are linked by a soft iron core which is laminated to prevent eddy-currents losses. In the iron core all the magnetic flux produced in the primary coil at the same time passes through the secondary coil.

In our discussion we ignore energy losses in the resistance of the coils and due to eddy-currents. It is a good approximation for real transformers, which provide more than 99% of efficiency.

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TransformersWhen an ac voltage is applied to the primary coil, the changing magnetic field it produces will induce an ac voltage of the same frequency in the secondary coil. The magnitude of the secondary voltage depends on the number of turns in each coil. From Faraday's Law the secondary voltage is:

The input of the primary voltage also depends on the rate of change of flux:

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TransformersThe ratio between the secondary and primary voltage is called the transformer equation:

If Ns is greater than Np, we have a step-up transformer.The secondary voltage is greater than the primary voltage.

If Ns is less than Np, we have a step-down transformer.The secondary voltage is less than the primary voltage.

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10 A step-down transformer has 100 turns in the primary coil and 10 turns in the secondary coil. What is the voltage in the secondary coil if 110 V applied to the primary coil?

A 11

B 10

C 110

D 100

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11 A step-up transformer is design to increase voltage from 12 V to 120 V. What is the number of turns is in the secondary coil if the primary coil has 20 turns?

A 40

B 100

C 140

D 200

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TransformersA well-designed transformer can have efficiency greater than 99%. The power input equals the power output.

When a step-up transformer increases an ac voltage at the same time it decreases an ac current by the same number. When a step-down transformer decreases an ac voltage at the same time it increases an ac current.

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12 A 4 A current flows through a primary coil of a transformer. The primary coil has 50 turns. How many turns must be in the secondary coil in order to produce 28 A of current in it?

A 14

B 28

C 32

D 48

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13 A transformer has 100 turns in the primary coil and 400 turns in the secondary coil. What is the current in the primary coil if 5 A flows through the secondary coil?

A 20

B 15

C 10

D 5

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Transformers

The diagram above demonstrates the importance of step-up and step-down transformers in the transmission of electricity.

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AC Circuits and ImpedanceIn this part of the chapter we will examine, one at a time, how a resistor, a capacitor, and an inductor behave when connected to a source of alternating emf. We assume in each case that the emf gives rise to a current:

Where Io is the peak current (maximum value).We must know that all ac meters are design to measure Irms and Vrms (root-mean-square) current and voltage instead of peak current and voltage. The formulas below show the relationships between them.

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14 An AC current in a circuit with a resistance R is given by the following formula . If the peak current is 1.41 A what is the rms current in the circuit?

A 5 A

B 3 A

C 2 A

D 1 A

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15 An AC voltage is applied to a circuit with a total resistance R. What is the rms voltage if the peak voltage is 170 V?

A 50 V

B 70 V

C 120 V

D 170 V

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AC Circuits and ImpedanceResistor When a resistor is connected to an

ac source , the current increases and decreases with voltage in phase. They reach maximum and minimum values at the same time. Average power dissipated in the resistor:

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16 A 2000 Ω resistor is connected to an AC circuit with Irms= 0.25 A? What is the average power in dissipated in the resistor?

A 100 W

B 200 W

C 400 W

D 500 W

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17 What average power is produced in a 500 Ω resistor connected to an AC power supply with Vrms =120 V?

A 72 W

B 94 W

C 123 W

D 187 W

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Inductor Now, we replace the resistor with a pure inductor with inductance L and zero resistance. We assume that the current varies with time according the following equation:

Changing current gives rise to a self-induced emf:

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InductorAC Circuits and Impedance

The previous equation explains that if the current in the inductor is positive (from a to b) and is increasing the induced emf is negative (directed form b to a). Point a has a higher potential than point b. Thus the voltage across the inductor is:

When an inductor is connected to an ac circuit the current is behind voltage by a quarter-cycle or 90o.

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AC Circuits and ImpedanceThe maximum value of voltage across of an inductor is:

Form this equation we can defined the inductive reactance or impedance XL of an inductor as

(in the form similar for a resistor V=IR)

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18 A coil with an inductance of 2 mH is connected to an AC source operating at a frequence of 80 Hz. What is the inductive reactance of the coil?

A 5 ΩB 6 ΩC 4 ΩD 1 Ω

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19 A coil with an inductance of 15 mH is connected to an AC source operating at a frequence of 100 Hz. What is the inductive reactance of the coil?

A 9.4 ΩB 6.2 ΩC 4.7 ΩD 1.8 Ω

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20 Find the operating frequency of an AC power supply if a 40 mH coil has an inductive reactance of 15 Ω.

A 6 Hz

B 60 Hz

C 600 Hz

D 6000 Hz

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21 Find the operating frequency of an AC power supply if a 50 mH coil has an inductive reactance of 75 Ω.

A 2 Hz

B 22 Hz

C 220 Hz

D 440 Hz

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AC Circuits and ImpedanceFinally, we connect a capacitor C to an ac source producing a current I through the capacitor.

To find voltage across the capacitor we show q as the charge on the capacitor. The current is related to the charge by the equation:

By integrating this equation, we get charge as a function of time:

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Voltage across the capacitor equals charge divided by capacitance.

To find the voltage across the capacitor we have combined two previous formulas.

Current leads the voltage by 90o.

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AC Circuits and ImpedanceThe maximum value of voltage across the capacitor is:

When we compare this formula and V=IR we can find the capacitance reactance or impedance XC:

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22 What is capacitance reactance of a 4 μF capacitor connected to an AC power supply operating at the frequency of 60 Hz?A 66 ΩB 167 ΩC 356 ΩD 663 Ω

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23 What is capacitance reactance of a 6 μF capacitor connected to an AC power supply operating at the frequency of 120 Hz?A 21 ΩB 221 ΩC 316 ΩD 463 Ω

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24 What capacitance is needed to have a 150 Ω capacitive reactance at 60 Hz?

A 17.7 μFB 28.5 μFC 36.4 μFD 43.5 μF

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25 What capacitance is needed to have a 500 Ω capacitive reactance at 100 Hz?

A 7 μFB 2 μFC 3 μFD 4 μF

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LRC Series AC CircuitsMany ac circuits used in electronic devices involve resistance, inductive reactance, and capacitive reactance. A simple example is a combination of a resistor, capacitor, and inductor connected in series.

The voltage across each of the elements will follow the phase relations we discussed in the last section. That is, VR will be in phase with the current, VL will lead the current by 90o, and VC will be behind the current by 90o.Since, we have a series circuit the current in each of the elements is in the same phase.

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LRC Series AC CircuitsThe difference in phases of voltage across each element gives rise to a more complex discussion. We cannot simply add to each other the rms voltages (actually measured by ac voltmeters).

It is convenient to analyze an LRC circuit using a phasor diagram. Where arrows (like vectors) are drawn in an x-y coordinate system to represent each voltage. The length of each vector represents the maximum (peak) voltage across each element.

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LRC Series AC CircuitsThe phasor diagram represents the voltage vectors across each element including the phase shift: VLO is ahead of VRO by 90o and VCO is behind VRO by 90o.

The resultant vector VO equals the vector sum of all three VRO,VLO, and VCO.

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LRC Series AC CircuitsLet's replace voltages with current and corresponding resistance or reactance.

Total impedance for the circuit.

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26 What is the impedance of an AC circuit with 25 Ω resistance, 20 Ω inductive reactance, 40 Ω capacitance reactance?

A 12 ΩB 23 ΩC 32 ΩD 46 Ω

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27 What is the impedance of an AC circuit with 75 Ω resistance, 65 Ω inductive reactance, 10 Ω capacitance reactance?

A 99 ΩB 83 ΩC 62 ΩD 46 Ω

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LRC Series AC Circuits

In the phasor diagram the angle φ between the voltage and the current phasors is the phase angle of the source voltage v with respect to the current i, it is the angle by which the source voltage leads the current. From the diagram:

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28 What is the phase angle at 1000 Hz if a 500 Ω resistor, 40 mH coil, and 8 μF capacitor are connected in series?

A 5 o

B 15 o

C 25 o

D 35 o

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29 What is the phase angle at 2000 Hz if a 1500 Ω resistor, 80 mH coil, and 6 μF capacitor are connected in series?

A 15.8 o

B 5.9 o

C 25.1 o

D 35.6 o

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LRC Series AC CircuitsThe average power in the LRC circuit is

From the phasor diagram we can relate R and Z as

Therefore, or

Where cosφ is the power factor. For a pure resistor cosφ =1, and P=IrmsVrms. For a capacitor, φ=±90o, respectively cosφ=0. No energy is dissipated.

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Resonance in AC CircuitsOscillations

The rms current in an LRC series circuit is given by:

It shows that the current in an RLC circuit depends on the frequency ω of the source. The current will reach its maximum value when:

orWhere ω is the resonant frequency.

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30 What is the resonant frequency (rad/s) in a circuit containing 20 mH coil and 4 μF capacitor?

A 2145.6 rad/s

B 3535.5 rad/s

C 4394.4 rad/s

D 5687.3 rad/s

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31 What is the resonant frequency (Hz) in a circuit containing 20 mH coil and 8 μF capacitor?

A 1259 rad/s

B 2535 rad/s

C 3394 rad/s

D 4687 rad/s

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When an RLC circuit is "tuned" to the resonant frequency the current reaches its maximum value for a given source voltage amplitude. At the resonant frequency voltage amplitudes VLO and VCO are equal.

We can replace ωo with fo, since

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An electromagnetic resonance plays a very important role in different practical fields such as: medicine (MRI), radio (producing and receiving e/m signals), particle accelerators, and solar energy transformation.

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When R is very small we can consider an electric circuit with a capacitor and an inductor which is know as an LC circuit. We will discuss the situation when a capacitor is initially charged and disconnected from the source of emf. Such fully charged capacitor is then connected to an inductor. As the charge on the capacitor decreases to zero the current in the inductor reaches its maximum value in the same time interval (self-inductance). The presence of these two devises causes charge and current oscillations.

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All these five diagrams explain in details e/m oscillations. Also they are the analogy to the mass-spring oscillating system.

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From an energy standpoint, during e/m oscillations electric energy stored in the capacitor transforms into magnetic energy in the inductor and back.

LC circuits are commonly used in oscillators, which are devices that put out an e/m signal of a particular frequency.

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32 An AC circuit consists of a 20 mH coil and 12 μF capacitor. When the circuit is set to oscillations the current reaches its maximum value of 10 A. What is maximum magnetic energy stored in the coil?

A 10 J

B 20 J

C 5 J

D 1 J

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33 An AC circuit consists of a 20 mH coil and 12 μF capacitor. When the circuit is set to oscillations the current reaches its maximum value of 10 A. What is the maximum value of the electric charge in the capacitor?

A 5 mC

B 2 mC

C 7 mC

D 1 mC

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slide 2 / 69 lrc series ac circuits sources of alternating

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