Harmonics of the Input Current

10.5.1.4 Harmonics of the Input Current

In general, the total harmonic distortion (THD) of an input FIGURE 10.25 Equivalent circuit for input ac filter. current is defined as

2 The power factor of the circuit shown in Fig. 10.22 can be THD s =

(10.71) improved by installing an ac filter between the source and the

I s1

rectifier, as shown in Fig. 10.24.

Considering only the harmonic components, the equiva- where I s is the rms value of the input current and I s1 and the lent circuit of the rectifier given in Fig. 10.24 can be found as

rms value of the fundamental component of the input current. shown in Fig. 10.25. The rms value of the nth harmonic cur- The THD can also be expressed as

rent appearing in the supply can then be obtained using the current-divider rule,

2 I rn (10.75)

L i C i where I sn is the rms value of the nth harmonic component of where I rn is the rms value of the nth harmonic current of the

the input current.

rectifier.

Moreover, the input power factor is defined as Applying Eq. (10.73) and knowing I rn /I r1 = 1/n from

Eq. (10.74), the THD of the rectifier with input filter shown in PF =

I s1

cos φ

(10.73) Fig. 10.24 can be found as

where φ is the displacement angle between the fundamental ! 1 1 components of the input current and voltage.

Filtered THD = 2 2 (10.76)

Assume that inductor L f of the circuit shown in Fig. 10.22

has an infinitely large inductance. The input current is then The important design parameters of typical single-phase

a square wave. This input current contains undesirable higher and three-phase rectifiers with inductor-input dc filter are harmonics that reduce the input power factor of the system. listed in Table 10.5. Note that, in a single-phase half-wave rec- The input current can be easily expressed as

tifier, a freewheeling diode is required to be connected across the input of the dc filters such that the flow of load current

4I m ! 1

(10.74) can be maintained during the negative half-cycle of the supply π n

sin 2nπf i t

=1,3,5, n

voltage.

The rms values of the input current and its fundamental √

10.5.2 Capacitive-input DC Filters

component are I m and 4I m /(π

2) respectively. Therefore, the

THD of the input current of this circuit is 0.484. Since the Figure 10.26 shows a full-wave rectifier with capacitor-input √ displacement angle φ = 0, the power factor is 4/(π 2) = 0.9.

dc filter. The voltage and current waveforms of this rectifier

10 Diode Rectifiers 165 TABLE 10.5 Important design parameters of typical rectifier circuits with inductor-input dc filter

Full-wave rectifier

bridge rectifier

star rectifier

bridge rectifier double-star rectifier

center-tapped

with inter-phase

transformer Peak repetitive reverse voltage V RRM

transformer

3.14V dc 1.57V dc 2.09V dc 1.05V dc 2.42V dc RMS input voltage per transformer leg V s

1.11V dc 1.11V dc 0.885V dc 0.428V dc 0.885V dc Diode average current I

F (AV) 0.5I dc 0.5I dc 0.333I dc 0.333I dc 0.167I dc Peak repetitive forward current I FRM

3.00I F (AV) Diode rms current I

2.00I

F (AV)

2.00I

F (AV)

3.00I

F (AV)

3.00I

F (AV)

F (RMS) 0.707I dc 0.707I dc 0.577I dc 0.577I dc 0.289I dc Form factor of diode current I

F (RMS) /I

F (AV)

1.73 1.73 1.73 Transformer rating primary VA 1.11P dc 1.11P dc 1.21P dc 1.05P dc 1.05P dc Transformer rating secondary VA 1.57P dc 1.11P dc 1.48P dc 1.05P dc 1.48P dc

Output ripple frequency f r

Ripple component V r at (a) fundamental,

0.667V dc 0.667V dc 0.250V dc 0.057V dc 0.057V dc

(b) second harmonic,

0.133V dc 0.133V dc 0.057V dc 0.014V dc 0.014V dc

(c) third harmonic of the ripple frequency

0.057V dc 0.057V dc 0.025V dc 0.006V dc 0.006V dc

R inrush

V r (pp) v =V m sin wt

FIGURE 10.26 Full-wave rectifier with capacitor-input dc filter. wt

are shown in Fig. 10.27. When the instantaneous voltage of the secondary winding v s is higher than the instantaneous value

D 2 conducts

wt capacitor C is charged up from the transformer. When the

of capacitor voltage v L , either D 1 or D 2 conducts, and the

3p instantaneous voltage of the secondary winding v s falls below

p/2

2p

D 1 conducts the instantaneous value of capacitor voltage v L , both the diodes

D 1 conducts

are reverse biased and the capacitor C is discharged through load resistance R. The resulting capacitor voltage v L varies between a maximum value of V m and a minimum value of FIGURE 10.27 Voltage and current waveforms of the full-wave rectifier

V m as shown in Fig. 10.27. (V is the peak-to-peak −V with capacitor-input dc filter. r(pp) r(pp) ripple voltage.) As shown in Fig. 10.27, the conduction angle θ c of the diodes becomes smaller when the output-ripple volt- age decreases. Consequently, the power supply and the diodes

Therefore, the average output voltage V dc is given by suffer from high repetitive surge currents. An LC ac filter, as

shown in Fig. 10.24, may be required to improve the input

V dc =V m 1

(10.78) 2f r RC

power factor of the rectifier.

In practice, if the peak-to-peak ripple voltage is small, it can

be approximated as The rms output ripple voltage V ac is approximately given

(10.79) where f r is the output ripple frequency of the rectifier.

V ac = √

2 2f r RC

166 Y. S. Lee and M. H. L. Chow The ripple factor RF can be found from

(which is known as forced turn-off). The temporary short circuit during the reverse recovery period may result in large

1 reverse current, excessive ringing, and large power dissipation, RF = √

2 r RC −1

all of which are highly undesirable.

The forward recovery time of a diode may be understood as the time a non-conducting diode takes to change to the fully- on state when a forward current is suddenly forced into it (which is known as forced turn-on). Before the diode reaches

the fully-on state, the forward voltage drop during the for- The resistor R inrush in Fig. 10.26 is used to limit the inrush ward recovery time can be significantly higher than the normal

10.5.2.1 Inrush Current

current imposed on the diodes during the instant when the on-state voltage drop. This may cause voltage spikes in the rectifier is being connected to the supply. The inrush current circuit. can be very large because capacitor C has zero charge initially.

It should be interesting to note that, as far as circuit opera- The worst case occurs when the rectifier is connected to the tion is concerned, a diode with a long reverse recovery time is

supply at its maximum voltage. The worst-case inrush current similar to a diode with a large parasitic capacitance. A diode can be estimated from

with a long forward recovery time is similar to a diode with a large parasitic inductance. (Spikes caused by the slow forward

I inrush = (10.81) recovery of diodes are often wrongly thought to be caused

R sec +R ESR

by leakage inductance.) Comparatively, the adverse effect of a long reverse recovery time is much worse than that of a long

where R sec is the equivalent resistance looking from the sec- forward recovery time. ondary transformer and R ESR is the equivalent series resistance

Among commonly used diodes, the Schottky diode has the (ESR) of the filtering capacitor. Hence the employed diode shortest forward and reverse recovery times. Schottky diodes

should be able to withstand the inrush current for a half cycle are therefore most suitable for high-frequency applications. of the input voltage. In other words, the Maximum Allow- However, Schottky diodes have relatively low reverse break- able Surge Current (I FSM ) rating of the employed diodes must down voltage (normally lower than 200 V) and large leakage

be higher than the inrush current. The equivalent resistance current. If, due to these limitations, Schottky diodes cannot associated with the transformer windings and the filtering

be used, ultra-fast diodes should be used in high-frequency capacitor is usually sufficient to limit the inrush current to converter circuits.

an acceptable level. However, in cases where the transformer Using the example of a forward converter, the operations is omitted, e.g. the rectifier of an off-line switch-mode sup- of a forward rectifier diode, a flywheel diode, and a clamping

ply, resistor R inrush must be added for controlling the inrush diode will be studied in Subsection 10.6.1. Because of the dif- current.

ficulties encountered in the full analyses taking into account Consider as an example, a single-phase bridge rectifier, parasitic/stray/leakage components, PSpice simulations are

which is to be connected to a 120-V–60-Hz source (with- extensively used here to study the following: out transformer). Assume that the I FSM rating of the diodes is 150 A for an interval of 8.3 ms. If the ESR of the filter-

• The idealized operation of the converter. ing capacitor is zero, the value of the resistor for limiting

• The adverse effects of relatively slow rectifiers (e.g. the so- called ultra-fast diodes, which are actually much slower

using Eq. (10.81).

than Schottky diodes). • The improvement achievable by using high-speed recti-

fiers (Schottky diodes). • The effects of leakage inductance of the transformer.

10.6 High-frequency Diode Rectifier

• The use of snubber circuits to reduce ringing.

Circuits

• The operation of a practical converter with snubber

circuits.

In high-frequency converters, diodes perform various func- tions, such as rectifying, flywheeling, and clamping. One

Using the example of a flyback converter, the operations special quality a high-frequency diode must possess is a fast of a flyback rectifier diode and a clamping diode will also be switching speed. In technical terms, it must have a short reverse studied in Subsection 10.6.2. recovery time and a short forward recovery time.

The design considerations for high-frequency diode rectifier The reverse recovery time of a diode may be understood as circuits will be discussed in Subsection 10.6.3. Some precau- the time a forwardly conducting diode takes to recover to a tions which must be taken in the interpretation of computer blocking state when the voltage across it is suddenly reversed simulation results are briefed in Subsection 10.6.4.

10 Diode Rectifiers 167

10.6.1 Forward Rectifier Diode, Flywheel Diode,

The switch M 1 is turned on at t = 0.

and Magnetic-reset Clamping Diode in a

The voltage at node 3, denoted as V(3), is

Forward Converter

V (3) = 0 for 0 < t < DT (10.82) Figure 10.28 shows the basic circuit of a forward converter.