# Inductors in series and parallel pdf

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## Series and parallel circuits

A resistor—inductor circuit RL circuit , or RL filter or RL network , is an electric circuit composed of resistors and inductors driven by a voltage or current source. A first-order RL circuit is composed of one resistor and one inductor and is the simplest type of RL circuit. A first order RL circuit is one of the simplest analogue infinite impulse response electronic filters. It consists of a resistor and an inductor, either in series driven by a voltage source or in parallel driven by a current source.

The fundamental passive linear circuit elements are the resistor R , capacitor C and 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 and the RLC circuit with the abbreviations indicating which components are used. These circuits exhibit important types of behaviour that are fundamental to analogue electronics.

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.

Both RC and RL circuits form a single-pole filter. Depending on whether the reactive element C or L is in series with the load, or parallel with the load will dictate whether the filter is low-pass or high-pass. The complex impedance Z L in ohms of an inductor with inductance L in henrys is. The complex frequency s is a complex number ,. The complex-valued eigenfunctions of any linear time-invariant LTI system are of the following forms:.

From Euler's formula , the real-part of these eigenfunctions are exponentially-decaying sinusoids:. Sinusoidal steady state is a special case in which the input voltage consists of a pure sinusoid with no exponential decay. As a result,. By viewing the circuit as a voltage divider , we see that the voltage across the inductor is:. The transfer function to the inductor voltage is.

The transfer functions have a single pole located at. In addition, the transfer function for the inductor has a zero located at the origin. These expressions together may be substituted into the usual expression for the phasor representing the output:. The impulse response for each voltage is the inverse Laplace transform of the corresponding transfer function.

It represents the response of the circuit to an input voltage consisting of an impulse or Dirac delta function. The zero-input response ZIR , also called the natural response , of an RL circuit describes the behavior of the circuit after it has reached constant voltages and currents and is disconnected from any power source.

It is called the zero-input response because it requires no input. These are frequency domain expressions. Analysis of them will show which frequencies the circuits or filters pass and reject. This analysis rests on a consideration of what happens to these gains as the frequency becomes very large and very small. This shows that, if the output is taken across the inductor, high frequencies are passed and low frequencies are attenuated rejected.

Thus, the circuit behaves as a high-pass filter. If, though, the output is taken across the resistor, high frequencies are rejected and low frequencies are passed. In this configuration, the circuit behaves as a low-pass filter. Compare this with the behaviour of the resistor output in an RC circuit , where the reverse is the case. The range of frequencies that the filter passes is called its bandwidth.

The point at which the filter attenuates the signal to half its unfiltered power is termed its cutoff frequency. This requires that the gain of the circuit be reduced to.

Clearly, the phases also depend on frequency, although this effect is less interesting generally than the gain variations. The most straightforward way to derive the time domain behaviour is to use the Laplace transforms of the expressions for V L and V R given above. Assuming a step input i. Partial fractions expansions and the inverse Laplace transform yield:. Thus, the voltage across the inductor tends towards 0 as time passes, while the voltage across the resistor tends towards V , as shown in the figures.

This is in keeping with the intuitive point that the inductor will only have a voltage across as long as the current in the circuit is changing — as the circuit reaches its steady-state, there is no further current change and ultimately no inductor voltage.

Kirchhoff's voltage law implies that the voltage across the resistor will rise at the same rate. When the voltage source is then replaced with a short circuit, the voltage across the resistor drops exponentially with t from V towards 0. Note that the current, I , in the circuit behaves as the voltage across the resistor does, via Ohm's Law. The delay in the rise or fall time of the circuit is in this case caused by the back-EMF from the inductor which, as the current flowing through it tries to change, prevents the current and hence the voltage across the resistor from rising or falling much faster than the time-constant of the circuit.

Since all wires have some self-inductance and resistance, all circuits have a time constant. The rise instead takes several time-constants to complete. If this were not the case, and the current were to reach steady-state immediately, extremely strong inductive electric fields would be generated by the sharp change in the magnetic field — this would lead to breakdown of the air in the circuit and electric arcing , probably damaging components and users.

These results may also be derived by solving the differential equation describing the circuit:. The first equation is solved by using an integrating factor and yields the current which must be differentiated to give V L ; the second equation is straightforward. The solutions are exactly the same as those obtained via Laplace transforms. For short circuit evaluation, RL circuit is considered. The more general equation is:. Which can be solved by Laplace transform :. In case the source voltage is a Heaviside step function DC :.

The parallel RL circuit is generally of less interest than the series circuit unless fed by a current source. This is largely because the output voltage V out is equal to the input voltage V in — as a result, this circuit does not act as a filter for a voltage input signal.

The parallel circuit is seen on the output of many amplifier circuits, and is used to isolate the amplifier from capacitive loading effects at high frequencies. Because of the phase shift introduced by capacitance, some amplifiers become unstable at very high frequencies, and tend to oscillate. This affects sound quality and component life especially the transistors , and is to be avoided.

From Wikipedia, the free encyclopedia. Electrical circuit consisting of resistive and inductive elements, with no capacitive elements. This article does not cite any sources. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.

Network synthesis filters. Image impedance filters. Constant k filter m-derived filter General image filters Zobel network constant R filter Lattice filter all-pass Bridged T delay equaliser all-pass Composite image filter mm'-type filter. Simple filters. Categories : Analog circuits Electronic filter topology. Hidden categories: Articles with short description Short description with empty Wikidata description Articles lacking sources from December All articles lacking sources.

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## RL circuit

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## Series and parallel circuits

A resistor—inductor circuit RL circuit , or RL filter or RL network , is an electric circuit composed of resistors and inductors driven by a voltage or current source. A first-order RL circuit is composed of one resistor and one inductor and is the simplest type of RL circuit. A first order RL circuit is one of the simplest analogue infinite impulse response electronic filters. It consists of a resistor and an inductor, either in series driven by a voltage source or in parallel driven by a current source. The fundamental passive linear circuit elements are the resistor R , capacitor C and inductor L.

To understand why this is so, consider the following: the definitive measure of inductance is the amount of voltage dropped across an inductor for a given rate of current change through it. If inductors are connected together in series thus sharing the same current , and seeing the same rate of change in current , then the total voltage dropped as the result of a change in current will be additive with each inductor, creating a greater total voltage than either of the individual inductors alone. Greater voltage for the same rate of change in current means greater inductance. The formula for calculating the series total inductance is the same form as for calculating series resistances:.