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Linear analog electronic filters 

Simple filters

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This article does not cite any references or sources. (December 2006) 
Linear analog electronic filters 

Simple filters

edit 
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 singlepole 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 lowpass or highpass.
Frequently RL circuits are used for DC power supplies to RF amplifiers, where the inductor is used to pass DC bias current and block the RF getting back into the power supply.
The complex impedance Z_{L} (in ohms) of an inductor with inductance L (in henries) is
The complex frequency s is a complex number,
where
The complexvalued eigenfunctions of any linear timeinvariant (LTI) system are of the following forms:
From Euler's formula, the realpart of these eigenfunctions are exponentiallydecaying 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,
and the evaluation of s becomes
By viewing the circuit as a voltage divider, we see that the voltage across the inductor is:
and the voltage across the resistor is:
The current in the circuit is the same everywhere since the circuit is in series:
The transfer function for the inductor is
Similarly, the transfer function for the resistor is
Both transfer functions have a single pole located at
In addition, the transfer function for the inductor has a zero located at the origin.
The gains across the two components are found by taking the magnitudes of the above expressions:
and
and the phase angles are:
and
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 impulse response for the inductor voltage is
where u(t) is the Heaviside step function and
is the time constant.
Similarly, the impulse response for the resistor voltage is
The Zero input response, 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 zeroinput response because it requires no input.
The ZIR of an RL circuit is:
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.
As :
As :
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 highpass 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 lowpass 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
Solving the above equation yields
or
which is the frequency that the filter will attenuate to half its original power.
Clearly, the phases also depend on frequency, although this effect is less interesting generally than the gain variations.
As :
As :
So at DC (0 Hz), the resistor voltage is in phase with the signal voltage while the inductor voltage leads it by 90°. As frequency increases, the resistor voltage comes to have a 90° lag relative to the signal and the inductor voltage comes to be inphase with the signal.
The most straightforward way to derive the time domain behaviour is to use the Laplace transforms of the expressions for and given above. This effectively transforms . Assuming a step input (i.e., before and then afterwards):
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 steadystate, there is no further current change and ultimately no inductor voltage.
These equations show that a series RL circuit has a time constant, usually denoted being the time it takes the voltage across the component to either fall (across L) or rise (across R) to within of its final value. That is, is the time it takes to reach and to reach .
The rate of change is a fractional per . Thus, in going from to , the voltage will have moved about 63% of the way from its level at toward its final value. So the voltage across L will have dropped to about 37% after , and essentially to zero (0.7%) after about . 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 shortcircuit, the voltage across R drops exponentially with t from towards 0. R will be discharged to about 37% after , and essentially fully discharged (0.7%) after about . Note that the current, , in the circuit behaves as the voltage across R does, via Ohm's Law.
The delay in the rise/fall time of the circuit is in this case caused by the backEMF 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 timeconstant of the circuit. Since all wires have some selfinductance and resistance, all circuits have a time constant. As a result, when the power supply is switched on, the current does not instantaneously reach its steadystate value, . The rise instead takes several timeconstants to complete. If this were not the case, and the current were to reach steadystate 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 ; the second equation is straightforward. The solutions are exactly the same as those obtained via Laplace transforms.
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 is equal to the input voltage —as a result, this circuit does not act as a filter for a voltage input signal.
With complex impedances:
This shows that the inductor lags the resistor (and source) current by 90°.
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.