1. www.bzupages.com Introduction  The fundamental passive linear circuit elements are the  resistor (R),  capacitor (C)  inductor (L).  These circuit elements can be combined to form an electrical circuit in four distinct ways:  the RC circuit,  the RL circuit,  the LC circuit  the RLC circuit  with the abbreviations indicating which components are used.  These circuits exhibit important types of behaviour that are fundamental to analogue electronics.

2. www.bzupages.com RC circuit The simplest RC circuit is a capacitor and a resistor in series. The simplest RC circuit is a capacitor and a resistor in series. When a circuit composes of only a charged capacitor and a resistor, When a circuit composes of only a charged capacitor and a resistor, then the capacitor would discharge its energy into the resistor. then the capacitor would discharge its energy into the resistor. This voltage across the capacitor over time could be found through KCL, where the current coming out of the capacitor must equal the current going through the resistor. This results in the linear differential equation This voltage across the capacitor over time could be found through KCL, where the current coming out of the capacitor must equal the current going through the resistor. This results in the linear differential equation

3. www.bzupages.com Natural response The simplest RC circuit is a capacitor and a resistor in series. The simplest RC circuit is a capacitor and a resistor in series. When a circuit composes of only a charged capacitor and a resistor, When a circuit composes of only a charged capacitor and a resistor, then the capacitor would discharge its energy into the resistor. then the capacitor would discharge its energy into the resistor. This voltage across the capacitor over time could be found through KCL, This voltage across the capacitor over time could be found through KCL, where the current coming out of the capacitor must equal the current going through the resistor. where the current coming out of the capacitor must equal the current going through the resistor. This results in the linear differential equation This results in the linear differential equation.

4. www.bzupages.com Complex impedance The equivalent resistance of a capacitor increases in relation to the amount of charge stored on the capacitor. The equivalent resistance of a capacitor increases in relation to the amount of charge stored on the capacitor. If a capacitor is subjected to an alternating current voltage source, If a capacitor is subjected to an alternating current voltage source, then the voltage of the capacitor would flip to the frequency of the AC voltage source. then the voltage of the capacitor would flip to the frequency of the AC voltage source. The faster the voltage of the AC voltage source flips, the less time charge would allowed to be stored on the capacitor, The faster the voltage of the AC voltage source flips, the less time charge would allowed to be stored on the capacitor, therefore reducing the capacitor's equivalent resistance. therefore reducing the capacitor's equivalent resistance. This explains the inverse relationship the equivalent resistance of a capacitor has with the frequency of the voltage source. This explains the inverse relationship the equivalent resistance of a capacitor has with the frequency of the voltage source.

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6. The resistance, also known as the complex impedance, ZC (in ohms) of a capacitor with capacitance C (in farads) is The resistance, also known as the complex impedance, ZC (in ohms) of a capacitor with capacitance C (in farads) is The angular frequency s is, in general, a complex number, The angular frequency s is, in general, a complex number, where where j represents the imaginary unit: j represents the imaginary unit: j2 = − 1 j2 = − 1 is the exponential decay constant (in radians per second), and is the exponential decay constant (in radians per second), and is the sinusoidal angular frequency (also in radians per second). is the sinusoidal angular frequency (also in radians per second).

7. www.bzupages.com USES RC circuits are among the most useful, simple and robust passive electric circuits, RC circuits are among the most useful, simple and robust passive electric circuits, and play integral roles in everyday equipment such as traffic lights, pacemakers and and play integral roles in everyday equipment such as traffic lights, pacemakers and audio equipment. While their applications are numerous and varied, they are mostly used audio equipment. While their applications are numerous and varied, they are mostly used for their signal filtering capabilities and surprisingly precise timing abilities. for their signal filtering capabilities and surprisingly precise timing abilities.

8. www.bzupages.com A resistor-inductor circuit (RL circuit),  A resistor-inductor circuit (RL circuit),   or RL filter  or RL network , is one of the simplest analogue infinite impulse response electronic filters.  It consists of a resistor and an inductor, either in series or in parallel, driven by a voltage

9. www.bzupages.com  In particular, they are able to act as passive filters.  This article considers the RL circuit in both series and parallel as shown in the diagrams.  In practice, however, capacitors (and RC circuits) are usually preferred to inductors since they can be more easily manufactured and are generally physically smaller, particularly for higher values of components.  This article relies on knowledge of the complex impedance representation of inductors and on knowledge of the frequency domain representation of signals.

10. www.bzupages.com RLC An RLC circuit (also known as a resonant circuit, An RLC 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 harmonic oscillator. connected in series or in parallel. This configuration forms a harmonic oscillator. Tuned circuits have many applications particularly for oscillating circuits and in radio and communication engineering. Tuned circuits have many applications particularly for oscillating circuits and in radio and communication engineering.

11. www.bzupages.com They can be used to select a certain narrow range of frequencies from the total spectrum of ambient radio waves. They can be used to select a certain narrow range of frequencies from the total spectrum of ambient radio waves. For example, AM/FM radios with analog tuners typically use an RLC circuit to tune a radio frequency. For example, AM/FM radios with analog tuners typically use an RLC circuit to tune a radio frequency. Most commonly a variable capacitor is attached to the tuning knob, which allows you to change the value of C in the circuit and tune to stations on different frequencies. Most commonly a variable capacitor is attached to the tuning knob, which allows you to change the value of C in the circuit and tune to stations on different frequencies.

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

13. www.bzupages.com The Capacitor A capacitor is a device that can store electrical charge. The simplest kind is a "parallel plate" A capacitor is a device that can store electrical charge. The simplest kind is a "parallel plate" capacitor that consists of two metal plates that are separated by an insulating material such as dry capacitor that consists of two metal plates that are separated by an insulating material such as dry air, plastic or ceramic. Such a device is shown schematically below air, plastic or ceramic. Such a device is shown schematically below

14. www.bzupages.com An LC circuit An LC circuit is a variety of resonant circuit or tuned circuit and consists of an inductor, An LC circuit is a variety of resonant circuit or tuned circuit and consists 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 current can alternate between them at the circuit's resonant frequency: When connected together, an electric current can alternate between them at the circuit's resonant frequency:

15. www.bzupages.com Configurations Every RLC circuit consists of two components: a power source and resonator. There are two types of power sources – Every RLC circuit consists of two components: a power source and resonator. There are two types of power sources – Thévenin and Norton. Thévenin and Norton. Likewise, there are two types of resonators – series Likewise, there are two types of resonators – series LC and parallel LC. LC and parallel LC. As a result, there are four configurations of RLC circuits: As a result, there are four configurations of RLC circuits:

16. www.bzupages.com Series LC with Thévenin power source Series LC with Thévenin power source Series LC with Norton power source Series LC with Norton power source Parallel LC with Thévenin power source Parallel LC with Thévenin power source Parallel LC with Norton power source. Parallel LC with Norton power source.

17. www.bzupages.com Similarities and differences between series and parallel circuits The expressions for the bandwidth in the series and parallel configuration are inverses of each other. The expressions for the bandwidth in the series and parallel configuration are inverses of each other. This is particularly useful for determining whether a series or parallel configuration is to be used for a particular circuit design. This is particularly useful for determining whether a series or parallel configuration is to be used for a particular circuit design.

18. www.bzupages.com However, in circuit analysis, usually the reciprocal of the latter two variables is used to characterize the system instead. However, in circuit analysis, usually the reciprocal of the latter two variables is used to characterize the system instead. They are known as the resonant frequency and the Q factor respectively. They are known as the resonant frequency and the Q factor respectively.

19. www.bzupages.com It is relatively easy to show that each of the two series configurations can be transformed into the other using elementary network transformations – It is relatively easy to show that each of the two series configurations can be transformed into the other using elementary network transformations – specifically, by transforming the Thévenin power source to the equivalent Norton power source, specifically, by transforming the Thévenin power source to the equivalent Norton power source, or vice versa. Likewise, each of the two parallel configurations can be transformed into the other using the same network transformations. or vice versa. Likewise, each of the two parallel configurations can be transformed into the other using the same network transformations. Finally, the Series/Thévenin and the Parallel/Norton configurations are dual circuits of one another. Likewise, the Series/Norton and the Parallel/Thévenin configurations are also dual circuits. Finally, the Series/Thévenin and the Parallel/Norton configurations are dual circuits of one another. Likewise, the Series/Norton and the Parallel/Thévenin configurations are also dual circuits.

20. www.bzupages.com where L is the inductance in henries, where L is the inductance in henries, and C is the capacitance in farads. and C is the capacitance in farads. The angular frequency has units of radians per second. The angular frequency has units of radians per second. LC circuits are used either for generating signals at a particular frequency, LC circuits are used either for generating signals at a particular frequency, or picking out a signal at a particular frequency from a more complex signal. or picking out a signal at a particular frequency from a more complex signal. They are key components in many applications such as oscillators, filters, tuners and frequency mixers. They are key components in many applications such as oscillators, filters, tuners and frequency mixers.

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