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IntroductionIntroduction

The fundamental passive linear circuit elements are the The fundamental passive linear circuit elements are the resistor (R), resistor (R), capacitor (C) capacitor (C) inductor (L). inductor (L).

These circuit elements can be combined to form an electrical These circuit elements can be combined to form an electrical circuit in four distinct ways:circuit in four distinct ways:

the RC circuit, the RC circuit, the RL circuit, the RL circuit, the LC circuit the LC circuit the RLC circuit the RLC circuit with the abbreviations indicating which components are used. with the abbreviations indicating which components are used. These circuits exhibit important types of behaviour that are These circuits exhibit important types of behaviour that are

fundamental to analogue electronics. fundamental to analogue electronics.

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RC circuitRC circuit

• The simplest RC circuit is a capacitor and a The simplest RC circuit is a capacitor and a resistor in series. resistor in series.

• When a circuit composes of only a charged When a circuit composes of only a charged capacitor and a resistor, capacitor and a resistor,

• then the capacitor would discharge its then the capacitor would discharge its energy into the resistor. energy into the resistor.

• This voltage across the capacitor over time This voltage across the capacitor over time could be found through KCL, where the could be found through KCL, where the current coming out of the capacitor must current coming out of the capacitor must equal the current going through the resistor. equal the current going through the resistor. This results in the linear differential equationThis results in the linear differential equation

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Natural responseNatural response

• The simplest RC circuit is a capacitor and a resistor in The simplest RC circuit is a capacitor and a resistor in

series. series. • When a circuit composes of only a charged capacitor and a When a circuit composes of only a charged capacitor and a

resistor, resistor, • then the capacitor would discharge its energy into the then the capacitor would discharge its energy into the

resistor. resistor. • This voltage across the capacitor over time could be found This voltage across the capacitor over time could be found

through KCL, through KCL, • where the current coming out of the capacitor must equal where the current coming out of the capacitor must equal

the current going through the resistor. the current going through the resistor. • This results in the linear differential equationThis results in the linear differential equation. .

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Complex impedanceComplex impedance

• The equivalent resistance of a capacitor increases in The equivalent resistance of a capacitor increases in relation to the amount of charge stored on the capacitor.relation to the amount of charge stored on the capacitor.

• If a capacitor is subjected to an alternating current voltage If a capacitor is subjected to an alternating current voltage source,source,

• then the voltage of the capacitor would flip to the then the voltage of the capacitor would flip to the frequency of the AC voltage source.frequency of the AC voltage source.

• The faster the voltage of the AC voltage source flips, the The faster the voltage of the AC voltage source flips, the less time charge would allowed to be stored on the less time charge would allowed to be stored on the capacitor, capacitor,

• therefore reducing the capacitor's equivalent resistance. therefore reducing the capacitor's equivalent resistance. • This explains the inverse relationship the equivalent This explains the inverse relationship the equivalent

resistance of a capacitor has with the frequency of the resistance of a capacitor has with the frequency of the voltage source.voltage source.

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• The resistance, also known as the complex The resistance, also known as the complex impedance, impedance, ZCZC (in ohms) of a capacitor with (in ohms) of a capacitor with capacitance capacitance CC (in farads) is (in farads) is

• The angular frequency The angular frequency ss is, in general, a complex is, in general, a complex number,number,

• wherewhere• jj represents the imaginary unit: represents the imaginary unit: • jj2 = − 1 2 = − 1 • is the exponential decay constant (in radians per is the exponential decay constant (in radians per

second), and second), and • is the sinusoidal angular frequency (also in is the sinusoidal angular frequency (also in

radians per second). radians per second).

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USESUSES

• RC circuits are among the most useful, RC circuits are among the most useful, simple and robust passive electric circuits,simple and robust passive electric circuits,

• and play integral roles in everyday and play integral roles in everyday equipment such as traffic lights, pacemakers equipment such as traffic lights, pacemakers andand

• audio equipment. While their applications are audio equipment. While their applications are numerous and varied, they are mostly usednumerous and varied, they are mostly used

• for their signal filtering capabilities and for their signal filtering capabilities and surprisingly precise timing abilities.surprisingly precise timing abilities.

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A A resistor-inductor resistor-inductor circuitcircuit (RL circuit)(RL circuit),,

A A resistor-inductor circuitresistor-inductor circuit (RL circuit)(RL circuit),, or or RL filterRL filter

or or RL networkRL network

, is one of the simplest analogue infinite , is one of the simplest analogue infinite impulse response electronic filters.impulse response electronic filters.

It consists of a resistor and an inductor, either It consists of a resistor and an inductor, either in series or in parallel, driven by a voltage in series or in parallel, driven by a voltage

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In particular, they are able to act as passive filters.In particular, they are able to act as passive filters. This article considers the RL circuit in both series and This article considers the RL circuit in both series and

parallel as shown in the diagrams.parallel as shown in the diagrams. In practice, however, capacitors (and RC circuits) are In practice, however, capacitors (and RC circuits) are

usually preferred to inductors since they can be more easily usually preferred to inductors since they can be more easily manufactured and are generally physically smaller, manufactured and are generally physically smaller, particularly for higher values of components.particularly for higher values of components. This article relies on knowledge of the complex impedance This article relies on knowledge of the complex impedance

representation of inductors and on knowledge of the frequency representation of inductors and on knowledge of the frequency domain representation of signalsdomain representation of signals. .

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RLCRLC

• An An RLC circuitRLC circuit (also known as a resonant circuit, (also known as a resonant circuit,• tuned circuit, tuned circuit, • or LCR circuit) or LCR circuit) • electrical circuit electrical circuit • consisting of a resistor (R),consisting of a resistor (R),• an inductor (L), an inductor (L), • capacitor (C), capacitor (C), • connected in series or in parallel. This configuration forms a connected in series or in parallel. This configuration forms a

harmonic oscillator.harmonic oscillator.• Tuned circuits have many applications particularly for Tuned circuits have many applications particularly for

oscillating circuits and in radio and communication oscillating circuits and in radio and communication engineering. engineering.

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• They can be used to select a certain They can be used to select a certain narrow range of frequencies from the narrow range of frequencies from the total spectrum of ambient radio waves.total spectrum of ambient radio waves.

• For example, AM/FM radios with analog For example, AM/FM radios with analog tuners typically use an RLC circuit to tune tuners typically use an RLC circuit to tune a radio frequency. a radio frequency.

• Most commonly a variable capacitor is Most commonly a variable capacitor is attached to the tuning knob, which attached to the tuning knob, which allows you to change the value of C in allows you to change the value of C in the circuit and tune to stations on the circuit and tune to stations on different frequencies.different frequencies.

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• An RLC circuit is called a An RLC circuit is called a second-second-orderorder circuit as any voltage or circuit as any voltage or current in the circuit can be current in the circuit can be described by a second-order described by a second-order differential equation for circuit differential equation for circuit analysis.analysis.

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The CapacitorThe Capacitor

• A capacitor is a device that can store A capacitor is a device that can store electrical charge. The simplest kind electrical charge. The simplest kind is a "parallel plate"is a "parallel plate"

• capacitor that consists of two metal capacitor that consists of two metal plates that are separated by an plates that are separated by an insulating material such as dryinsulating material such as dry

• air, plastic or ceramic. Such a device air, plastic or ceramic. Such a device is shown schematically belowis shown schematically below

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An An LC circuitLC circuit

• An An LC circuitLC circuit is a variety of is a variety of resonant resonant circuitcircuit or or tuned circuittuned circuit and consists and consists of an inductor,of an inductor,

• represented by the letter L, represented by the letter L,

• capacitor, represented by the letter C. capacitor, represented by the letter C.

• When connected together, an electric When connected together, an electric current can alternate between them at current can alternate between them at the circuit's resonant frequency:the circuit's resonant frequency:

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ConfigurationsConfigurations

• Every RLC circuit consists of two Every RLC circuit consists of two components: a components: a power sourcepower source and and resonatorresonator. . There are two types of power sources –There are two types of power sources –

• Thévenin and Norton.Thévenin and Norton.• Likewise, there are two types of resonators Likewise, there are two types of resonators

– series– series• LC and parallel LC. LC and parallel LC. • As a result, there are four configurations of As a result, there are four configurations of

RLC circuits:RLC circuits:

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• Series LC with Thévenin power Series LC with Thévenin power source source

• Series LC with Norton power source Series LC with Norton power source • Parallel LC with Thévenin power Parallel LC with Thévenin power

source source • Parallel LC with Norton power source. Parallel LC with Norton power source.

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Similarities and differences Similarities and differences between series and parallel between series and parallel circuitscircuits

• The expressions for the bandwidth in the The expressions for the bandwidth in the series and parallel configuration are series and parallel configuration are inverses of each other. inverses of each other.

• This is particularly useful for determining This is particularly useful for determining whether a series or parallel configuration whether a series or parallel configuration is to be used for a particular circuit design. is to be used for a particular circuit design.

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• However, in circuit analysis, usually the However, in circuit analysis, usually the reciprocal of the latter two variables is reciprocal of the latter two variables is used to characterize the system instead. used to characterize the system instead.

• They are known as the resonant They are known as the resonant frequency and the Q factor respectively.frequency and the Q factor respectively.

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• It is relatively easy to show that each of the two It is relatively easy to show that each of the two series configurations can be transformed into the series configurations can be transformed into the other using elementary network transformations – other using elementary network transformations –

• specifically, by transforming the Thévenin power specifically, by transforming the Thévenin power source to the equivalent Norton power source,source to the equivalent Norton power source,

• or vice versa. Likewise, each of the two parallel or vice versa. Likewise, each of the two parallel configurations can be transformed into the other configurations can be transformed into the other using the same network transformations. using the same network transformations.

• Finally, the Series/Thévenin and the Parallel/Norton Finally, the Series/Thévenin and the Parallel/Norton configurations are dual circuits of one another. configurations are dual circuits of one another. Likewise, the Series/Norton and the Parallel/Thévenin Likewise, the Series/Norton and the Parallel/Thévenin configurations are also dual circuits.configurations are also dual circuits.

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• where where LL is the inductance in henries, is the inductance in henries,• and and CC is the capacitance in farads. is the capacitance in farads. • The angular frequency has units of radians The angular frequency has units of radians

per second.per second.• LC circuits are used either for generating LC circuits are used either for generating

signals at a particular frequency, signals at a particular frequency, • or picking out a signal at a particular or picking out a signal at a particular

frequency from a more complex signal. frequency from a more complex signal. • They are key components in many They are key components in many

applications such as oscillators, filters, applications such as oscillators, filters, tuners and frequency mixers. tuners and frequency mixers.

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An LCAn LC

• An LC circuit is an idealized model An LC circuit is an idealized model since it assumes there is no since it assumes there is no dissipation of energy due to dissipation of energy due to resistance. For a model incorporating resistance. For a model incorporating resistance see RLC circuit.resistance see RLC circuit.