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# Smoothing Filters

While the output of a rectifier is a pulsating DC, most electronic circuits require a substantially pure DC for proper operation. This type of output is provided by single or multisection filter circuits placed between the output of the rectifier and the load.

There are four basic types of filter circuits:

- Simple capacitor filter
- LC choke-input filter
- LC capacitor-input filter (pi-type)
- RC capacitor-input filter (pi-type)

The function of each of these filters will be covered in detail in this section.

Filtering is accomplished by the use of capacitors, inductors, and/or resistors in various combinations. Inductors are used as series impedances to oppose the flow of alternating (pulsating DC) current. Capacitors are used as shunt elements to bypass the alternating components of the signal around the load (to ground). Resistors are used in place of inductors in low current applications.

Let's briefly review the properties of a capacitor. First, a capacitor opposes
any change in voltage. The opposition to a change in voltage is called
capacitive reactance (*X*_{C}) and is measured in ohms. The
capacitive reactance is determined by the frequency (*f*) of the applied
voltage and the capacitance (*C*) of the capacitor:

From the formula, you can see that if frequency or capacitance is increased,
the *X*_{C} decreases.
Filter capacitors are placed in parallel with the load. The capacitor acts
as a short circuit for the AC component (ripple) of the voltage.
A low *X*_{C} will provide a smaller opposition to the AC component
and thus better filtering than a high *X*_{C}.
Therefore, to obtain the best possible filtering, you want the capacitor to be
as large as possible.

Another consideration in filter circuits are the charging and discharging times of the capacitor circuits. In filter circuits the capacitor is the common element to both the charge and the discharge paths (see the figure below). The other element of the charge path (solid line in the figure) is the internal resistance of the power supply, while the other element of the discharge path (dashed line in the figure) is the resistance of the load. So the charge and the discharge paths contains a resistance-capacitance circuit.

The time constant of the resistance-capacitance circuit is defined as the
time it takes a capacitor to charge to 63.2 percent of the applied voltage
or to discharge to 36.8 percent of its total voltage. The time constant
*τ* (RC time constant) can be expressed by the following equation:

where *R* represents the resistance of the charge or discharge path and
*C* represents the capacitance of the capacitor. It should be noted
that a capacitor is considered fully charged after five RC time constants.

To obtain a steady DC output, the capacitor must charge almost instantaneously to the value of applied voltage. The capacitor filter must have a short charge time constant. This can be accomplished by keeping the internal resistance of the power supply as small as possible. Once charged, the capacitor must retain the charge as long as possible. This can be accomplished by keeping the resistance of the load as large as possible (for a slow discharge time).

Now let's look at inductors and their application in filter circuits. Remember,
*an inductor opposes any change in current*. A change in current through an
inductor produces a changing electromagnetic field. The changing field, in turn,
cuts the windings of the wire in the inductor and thereby produces a back
electromotive force (BEMF). It is the BEMF that
opposes the change in circuit current. Opposition to a change in current at a
given frequency is called inductive reactance (*X*_{L}) and is
measured in ohms. The inductive reactance (*X*_{L}) of an inductor is
determined by the applied frequency and the inductance of the inductor.

Mathematically,

If frequency or inductance is increased, the *X*_{L} increases.
Since inductors are placed in series with the load, the larger the
*X*_{L}, the larger the AC voltage developed across the load.

Now that you have read about the components in filter circuits, the different types of filter circuits will be discussed.

## The Capacitor Filter

The simple capacitor filter is the most basic type of power supply filter.
The capacitor (*C*) shown in the figure below is a
simple capacitor filter connected across the output of the rectifier
in parallel with the load. The value of the capacitor is usually fairly large,
thus it presents a relatively low reactance to the pulsating current.

When this filter is used, the charge time of the filter capacitor
(*C*) must be short and the discharge time must be long
to eliminate ripple action. In other words, the capacitor must charge up fast,
preferably with no discharge at all. Better filtering also results when the
input frequency is high; therefore, the full-wave rectifier output is easier
to filter than that of the half-wave rectifier because of its
higher frequency.

The rate of charge for the capacitor is limited by the resistance of the
conducting diode which is relatively low. Therefore, the charge time of the
circuit is relatively short. As a result, when the pulsating voltage is first
applied to the circuit, the capacitor charges rapidly and almost reaches the peak
value of the rectified voltage within the first few cycles. The capacitor
attempts to charge to the peak value of the rectified voltage anytime a diode
is conducting, and tends to retain its charge when the rectifier output falls
to zero. (The capacitor cannot discharge immediately.) The capacitor slowly
discharges through the load resistance (*R*_{L}) during the time
the rectifier is nonconducting.

The rate of discharge of the capacitor is determined by the value of capacitance and the value of the load resistance. If the capacitance and load-resistance values are large, the discharge time for the circuit is relatively long.

Now, let's consider a complete cycle of operation using a half-wave rectifier,
a capacitive filter (*C*), and a load resistor
(*R*_{L}). As shown in view A of the figure below, the capacitive
filter (*C*) is assumed to be large enough to ensure a small
reactance to the pulsating rectified current. The resistance of
*R*_{L} is assumed to be much greater than the reactance of
*C* at the input frequency. When the circuit is energized, the
diode conducts on the positive half cycle and current flows through the circuit,
allowing *C* to charge. *C* will charge to
approximately the peak value of the input voltage. (The charge voltage is less than
the peak value because of the voltage drop across the diode
*D*). In view A of the figure, the charging of *C*
is indicated by the heavy solid line on the waveform.
After the capacitor has charged to its peak value, it tends to retain its charge
when the rectifier output falls to zero. Since the fall of this voltage
on the anode is considerably more rapid than the decrease of the capacitor
voltage, the cathode quickly become more positive than the anode, and the
diode ceases to conduct. The capacitor will start to discharge
through the load resistor *R*_{L}.

As illustrated in view B, the diode cannot conduct on the negative half cycle
because the anode of *D* is negative with respect to the
cathode. During this interval, *C* discharges through the load
resistor (*R*_{L}). The discharge of *C* produces
the downward slope as indicated by the solid line on the waveform in view B. In
contrast to the abrupt fall of the applied AC voltage from peak value to zero,
the voltage across *C* (and thus across *R*_{L})
during the discharge period gradually decreases until the time of the next half
cycle of rectifier operation. Keep in mind that for good
filtering, the filter capacitor should charge up as fast as possible and
discharge as little as possible.

Since practical values of *C* and *R*_{L} ensure
a more or less gradual decrease of the discharge voltage, a substantial charge
remains on the capacitor at the time of the next half cycle of operation. As
a result, no current can flow through the diode until the rising AC input
voltage at the anode of the diode exceeds the voltage remaining
on *C*. The voltage on *C* is the cathode
potential of the diode. When the potential on the anode exceeds the potential
on the cathode (the voltage on *C*), the diode again conducts,
and *C* begins to charge to approximately the peak value of
the applied voltage.

A comparison of a rectifier circuit with a capacitor and
one without a capacitor is illustrated in view B of the figure above.
With no capacitor connected across the
output of the rectifier circuit, the output waveform (dashed line) has
a large pulsating component (ripple) compared with the average or DC component
(*V*_{avg} = 0.318×*V*_{peak}).
When a capacitor is connected across the output, the average value of
output voltage (*V*_{avg}) is increased and the ripple is decreased
due to the filtering action of capacitor *C*.

Operation of the simple capacitor filter using a full-wave rectifier is basically the same as that discussed for the half-wave rectifier. Referring to the figure below, you should notice that because one of the diodes is always conducting on either alternation, the filter capacitor charges and discharges during each half cycle. (Note that each diode conducts only for that portion of time when the peak secondary voltage is greater than the voltage across the capacitor.)

Another thing to keep in mind is that the ripple component (*V*_{r})
of the output voltage is an AC voltage and the average output voltage
(*V*_{avg}) is the DC component of the output. Since the filter
capacitor offers a relatively low impedance to AC, the majority of the AC
component flows through the filter capacitor. The AC component is therefore
bypassed (shunted) around the load resistance, and the entire DC component (or
*V*_{avg}) flows through the load resistance. This statement
can be clarified by using the formula for *X*_{C} in a
half-wave and full-wave rectifier. First, you must establish some values
for the circuit.

**Half-wave rectifier**:

Frequency at rectifier output: 60 Hz

Capacitance of filter capacitor: 30 µF

Load resistance: 10 kΩ

The capacitive reactance

**Full-wave rectifier**:

Frequency at rectifier output: 120 Hz

Capacitance of filter capacitor: 30 µF

Load resistance: 10 kΩ

The capacitive reactance

As you can see from the calculations, by doubling the frequency of the
rectifier, you reduce the impedance of the capacitor by one-half. This
allows the AC component to pass through the capacitor more easily. As a
result, a full-wave rectifier output is much easier to filter than that of
a half-wave rectifier. Remember, the smaller the *X*_{C} of the
filter capacitor with respect to the load resistance, the better the
filtering action. Since

the largest possible capacitor will provide the best filtering. Remember,
also, that the load resistance is an important consideration. If load
resistance is made small, the load current increases, and the average value
of output voltage (*V*_{avg}) decreases. The discharge time
constant is a direct function of the value of the load resistance; therefore,
the rate of capacitor voltage discharge is a direct function of the current
through the load. The greater the load current, the more rapid the discharge of
the capacitor, and the lower the average value of output voltage. For this
reason, the simple capacitive filter is seldom used with rectifier
circuits that must supply a relatively large load current. Using the simple
capacitive filter in conjunction with a full-wave (bridge) rectifier provides
improved filtering because the increased ripple frequency
decreases the capacitive reactance of the filter capacitor.

## LC Choke-Input Filter

The LC choke-input filter is used primarily in power supplies where voltage regulation is important and where the output current is relatively high and subject to varying load conditions. This filter is used in high power applications such as those found in radars and communication transmitters.

Notice in the figure below that this filter consists of an input inductor
(*L*), or filter choke, and an output filter capacitor (*C*).
Inductor is placed at the input to the filter and is in series with the
output of the rectifier circuit. Since the action of an inductor is to
oppose any change in current flow, the inductor tends to keep a constant
current flowing to the load throughout the complete cycle of the applied
voltage. The reactance of the inductor (*X*_{L}) reduces the
amplitude of ripple voltage without reducing the DC output voltage by an
appreciable amount. (The DC resistance of the inductor is usually just a few
ohms.) The larger the value of the filter inductor, the better the
filtering action.

The shunt capacitor (*C*) charges and discharges at the ripple
frequency rate. The reactance of the shunt capacitor (*X*_{C})
presents a low impedance to the ripple component existing at the output
of the filter, and thus shunts the ripple component around the load.
The capacitor attempts to hold the output voltage relatively constant.

The value of the filter capacitor (*C*) must be relatively large to
present a low opposition (*X*_{C}) to the
pulsating current and to store a substantial charge. The larger the
value of the filter capacitor, the better the filtering action. However,
because of physical size, there is a practical limitation to the
maximum value of the capacitor.

Now look at the figure above which illustrates a complete cycle of operation
for a full-wave rectifier circuit used to supply the input voltage to the
filter. The rectifier voltage is developed across the capacitor
(*C*). The ripple voltage at the output of the filter is the alternating
component of the input voltage reduced in amplitude by the filter section.
Each time the anode of a diode goes positive with respect to the
cathode, the diode conducts and *C* charges. Conduction occurs twice
during each cycle for a full-wave rectifier. For a 60-hertz supply, this
produces a 120-hertz ripple voltage. Although the diodes alternate
(one conducts while the other is nonconducting), the filter input voltage
is not steady. As the anode voltage of the conducting diode increases
(on the positive half of the cycle), capacitor *C* charges - the
charge being limited by the impedance of the secondary transformer winding,
the diode's forward (anode-to-cathode) resistance, and the back electromotive
force developed by the choke. During the nonconducting interval (when the anode
voltage drops below the capacitor charge voltage), *C* discharges
through the load resistor (*R*_{L}).
*C* will only partially discharge, as the components in the
discharge path have a long time constant *τ* (discharge time
constant).
As the charge time constant is normally short compared to the discharge time
constant, the capacitor *C* discharges more slowly than it charges.

The filtering action is also illustrated in the figure below.
The inductor (*L*) and the capacitor (*C*) form a voltage divider
for the AC component (ripple) of the applied input voltage. This is shown in
view A of the figure below. As far as the ripple component is concerned,
the inductor offers a high impedance (*Z*) and the capacitor offers a low
impedance (view B). As a result, the ripple component (*V*_{r})
appearing across the load resistance is greatly attenuated (reduced).

Now that you have read how the LC choke-input filter functions, it will be discussed with actual component values applied. For simplicity, the input frequency at the primary of the transformer will be 117 volts 60 hertz. Both half-wave and full-wave rectifier circuits will be used to provide the input to the filter.

Starting with the half-wave configuration shown in the figure below, the basic
parameters are:

With 117 volts AC (RMS value) applied to the *Tr* primary,
165 volts AC peak is available at the secondary (117 × 1.414 = 165 V).
You should recall that the ripple frequency of this half-wave rectifier is
60 hertz. Therefore, the capacitive reactance of *C* is:

This means that the capacitor offers 265 ohms of opposition to the ripple
current. Note, however, that the capacitor offers an infinite impedance to
direct current. The inductive reactance of *L* is:

The above calculation shows that *L* offers a relatively high opposition
(3.8 kilohms) to the ripple in comparison to the opposition offered by *C*
(265 ohms). Thus, more ripple voltage will be dropped across
*L* than across *C*. In other words, the ripple voltage
appearing across the load resistance is attenuated.
In addition, the impedance of *C* (265 ohms)
is relatively low with respect to the resistance of the load (10 kilohms).
Therefore, more ripple current flows through *C* than the load. In
other words, *C* shunts most of the AC component around the load.
You can further increase the ripple voltage across *L* by increasing
the inductance.

Now let's discuss the DC component of the applied voltage. Remember, a
capacitor offers an infinite (∞) impedance to the flow of direct current. The DC
component, therefore, must flow through *R*_{L} and *L*.
As far as the DC is concerned, the capacitor does not exist. The choke and the
load are therefore in series with each other. The DC resistance of a filter
choke is very low (can be about 50 ohms). Consequently, most of
the DC component is developed across the load and a very small amount of the DC
voltage is dropped across the choke, as shown in the figure below.

As you may have noticed, both the AC and the DC components flow through *L*.
Because it is frequency sensitive, the choke provides a large resistance to AC
and a small resistance to DC. In other words, the choke opposes any change
in current. This property makes the choke a highly desirable filter component.

Note that the filtering action of the LC choke-input filter is improved when
the filter is used in conjunction with a full-wave rectifier, as shown in the
figure below. This is due to the decrease in the *X*_{C} of
the filter capacitor and the increase in the *X*_{L} of the choke.
Remember, ripple frequency of a full-wave rectifier is twice that of a
half-wave rectifier. For 60-hertz input, the ripple will be 120 hertz. The
*X*_{C} of *C* and the *X*_{L} of *L* are
calculated as follows:

When the *X*_{C} of a filter capacitor is decreased, it provides
less opposition to the flow of AC. The greater the AC flow through the
capacitor, the lower the flow through the load. Conversely, the larger the
*X*_{L} of the choke, the greater the amount of AC ripple
developed across the choke; consequently, less ripple is developed across the
load and better filtering is obtained.

## Resistor-Capacitor (RC) Filters

The single capacitor filter is suitable for many noncritical, low-current applications. However, when the load resistance is very low or when the percent of ripple must be held to an absolute minimum, the capacitor value required must be extremely large. While electrolytic capacitors are available in sizes up to 10 mF or greater, the large sizes are quite expensive. A more practical approach is to use a more sophisticated filter that can do the same job but that has lower capacitor values, such as the RC filter. The RC capacitor-input filter is limited to applications in which the load current is small. This type of filter is used in power supplies where the load current is constant and voltage regulation is not necessary.

The figure below shows an RC capacitor-input filter and associated waveforms.
Both half-wave and full-wave rectifiers are used to provide the inputs.
The waveforms shown in view A of the figure represent the
unfiltered output from typical rectifier circuits. The average value of
output voltage *V*_{avg} for the half-wave rectifier is less than
half (approximately 0.318) the amplitude of the voltage
peaks. The average value of output voltage for the full-wave rectifier is
greater than half (approximately 0.637), but is still much less than, the peak
amplitude of the rectifier-output waveform. With no filter circuit connected
across the output of the rectifier circuit (unfiltered), the waveform has
a large value of pulsating component (ripple) as compared to the average
(or DC) component.

The RC filter in the figure above consists of an input filter capacitor
(*C*_{1}), a series resistor (*R*_{1}), and an
output filter capacitor (*C*_{2}). (This filter is sometimes
referred to as an RC pi-section filter because its schematic symbol resembles
the Greek letter π).

*C*_{1} performs exactly the same function as it did in the
single capacitor filter. It is used to reduce the percentage of ripple to
a relatively low value (view B). This voltage is passed on to the
*R*_{1}-*C*_{2} network, which reduces the ripple
even further.

*C*_{2} offers an infinite impedance to the DC component of the
output voltage. Thus, the DC voltage is passed to the load, but reduced in
value by the amount of the voltage drop across *R*_{1}. However,
*R*_{1} is generally small compared to the load resistance.
Therefore, the drop in the DC voltage by *R*_{1} is not a
drawback.

Component values are designed so that the resistance of *R*_{1}
is much greater than the reactance (*X*_{C}) of
*C*_{2} at the ripple frequency. Therefore, most of the ripple
voltage is dropped across *R*_{1}. Only a trace of the ripple
voltage can be seen across *C*_{2} and the load (view C). In extreme cases
where the ripple must be held to an absolute minimum, a second stage of RC
filtering can be added. In practice, the second stage is rarely required. The
RC filter is extremely popular because smaller capacitors can be used with
good results.

The RC filter has some disadvantages. First, the voltage drop across
*R*_{1} takes voltage away from the load. Second, power is wasted
in *R*_{1} and is dissipated in the form of unwanted heat.
Finally, if the load resistance changes, the voltage across the load will
change. Even so, the advantages of the RC filter overshadow these disadvantages
in many cases.

## LC Capacitor-Input Filter

The LC capacitor-input filter is one of the most commonly used filters. This type of filter is used in any type of power supply where the output current is low and the load current is relatively constant.

The figure below shows an LC capacitor-input filter and associated waveforms,
which are similar to the case of RC filter shown above.
Both half-wave and full-wave rectifier circuits are used to provide the input.
The waveforms shown in view A of the figure represent the unfiltered output
from a typical rectifier circuit. Note that the average value of output
voltage (*V*_{avg}) for the half-wave rectifier is less than half
the amplitude of the voltage peaks. The average value of output voltage for
the full-wave rectifier is greater than half, but is still much less than
the peak amplitude of the rectifier-output waveform. With no filter
connected across the output of the rectifier circuit (which results in
unfiltered output voltage), the waveform has a large value of pulsating
component (ripple) as compared to the average (or DC) component.

*C*_{1} reduces the ripple to a relatively low level (view B).
*L*_{1} and *C*_{2} form the LC filter, which reduces
the ripple even further. *L*_{1} is a large value iron-core
inductor (choke). *L*_{1} has a high value of inductance an
therefore, a high value of *X*_{L} which offers a high reactance
to the ripple frequency. At the same time, *C*_{2}
offers a very low reactance to AC ripple. *L*_{1} and
*C*_{2} form an AC voltage divider and, because the reactance of
*L*_{1} is much higher than that of *C*_{2}, most of
the ripple voltage is dropped across *C*_{1}. Only a slight
trace of ripple appears across *C*_{2} and the load (view C).

While the *L*_{1}-*C*_{2} network greatly reduces AC
ripple it has little effect on DC. You should recall that an inductor offers
no reactance to DC. The only opposition to current flow is the resistance
of the wire in the choke. Generally, this resistance is very low and the DC
voltage drop across the inductor is minimal. Thus, the LC filter overcomes the
disadvantages of the RC filter.

The LC filter provides good filtering action over a wide range of currents. The capacitor filters are best when the load is drawing little current. Thus, the capacitor discharges very slowly and the output voltage remains almost constant. On the other hand, the inductor filters are best when the current is highest. The complementary nature of these two components ensures that good filtering will occur over a wide range of currents.

The LC filter has two disadvantages. First, it is more expensive than the RC filter because an iron-core choke costs more than a resistor. The second disadvantage is size. The iron-core choke is bulky and heavy, a fact which may render the LC filter unsuitable for many applications.