1. diodes and rectifier circuits - university of cambridgeprp/3b3/3b3lec1-4.pdf · diodes and...

1. Diodes and Rectifier Circuits

1.1 The diode

1.1.1 *The p-n junction diode

A p-n junction diode is formed by bringing together p-type and n-type silicon.Note that n-type silicon is made by substituting some silicon atoms with atoms ofa Group V element. Four of the outer electrons are used in bonding but the fifthis relatively free. Consequently, at room temperature there are many electronsavailable for conduction in n-type silicon. Arsenic is a commonly used donorimpurity.

Similarly, p-type silicon is made by substituting silicon atoms with those of aGroup III element. The Group III atoms, having only three electrons in the outershell, readily accept electrons, leading to a large concentration of 'holes', orabsences of electrons, available for conduction. Boron is the normally usedacceptor impurity.

In the region of the junction, holes and electrons recombine to leave a regionwithout any free carriers, known as the depletion region. The charge distributionassociated with donor and acceptor atoms in the depletion region that have losttheir electrons or holes results in a potential field which acts as a barrier whichprevents further flow of holes and electrons into the depletion region.

With forward bias, that is p side positive, the applied field counteracts thepotential barrier and forces carriers through the depletion region. The depletionwidth narrows and current flows. The electrons ind holes are collectively termedcarriers.

Under reverse bias, that is p side negative, the majority carriers are repelled fromthe junction and the depletion width widens. The carriers generated thermally inthe depletion region result in a small leakage current.

* Not in exam

Lectures 1-4, Engineering IIA, 3B3 Switch-Mode Electronics

Remember that electrons are the majority carriers in the n-type material and holesare the majority carriers in p-type material. 1.1.2 The diode characteristic

The diode is a two terminal device which conducts quite freely in one directionbut not the other. We use p-n junction diodes for most applications, but Schottkydiodes (metal-silicon diodes) are increasingly important.

Practical silicon p-n junction power diodes, i.e. those with a forward currentrating of 1A, have a forward drop of about 1V over a wide range of current.Note that in a small signal diode, which normally has lower ratings, the forwarddrop is usually quoted as about 0.7V.

In the reverse direction, a small current flows, almost independent of reversevoltage, until the device breaks down.

Semiconductor physics leads to the diode equation relating current I and voltageV (forward is positive):

where Io and m are constants, T is the temperature, e is the electronic charge andk is Boltzmann's constant. The value of Io depends on the properties of thesemiconductor and the geometry of the device and m depends on the conditionsaround the junction during conduction.


Under reverse bias, I tends to -Io and is almost independent of the applied voltageas exp(eV/mkT) is small. With forward bias, I grows rapidly for small increases inV, doubling roughly every 25 to 40 mV. At very high currents ohmic resistancebecomes significant.

Diodes are made with ratings up to several thousand amps with sustainingvoltages of several thousand. The two key ratings are

1. Maximum Repetitive Reverse Voltage, VRRM

2. Maximum average forward current, Ifav

at the maximum permitted junction temperature.

1.1.3 *The relationship between diode ratings and diode construction

When the diode is reverse biased, the depletion layer grows and semiconductorphysics shows that the depletion width w is related to the applied bias V (reversebias is negative) and the doping densities, NA for the p-type material and ND forthe n-type.

Where VO is the built in potential of the junction, about 0.7V, ε is the permitivityof silicon and e is the electronic charge. Typically a maximum field strength of20MVm-1 (20V/um) can be sustained in silicon, so the depletion layer must be ofa certain width for a given voltage rating.

Very often for practical reasons one side of the junction is lightly doped sothat the depletion layer is much wider in that side. Then the width is propor-tional to the square root of the applied reverse voltage. The drawback of thisis that the lightly doped material has a higher resistivity, contributing seriesresistance to the diode. So, to handle large currents, it is also necessary toincrease the area of the diode.

* Not in exam


1.1.4 Piecewise linear model

It is often useful to have a simple model for the diode, for example as shownbelow

In most cases, take the reverse resistance as infinite, the forward resistance aszero and the forward drop as 1V.

1.1.5 Reverse recovery time

When the voltage on a diode reverses, conduction continues for a certain time,known as the reverse recovery time. The reverse recovery time reflects the speedat which the carriers related to the forward current can be removed from thevarious regions and the time taken to establish the depletion layer.

At mains frequency, 50 or 60 Hz, the reverse recovery time is not significant.However, in switching circuits, at high frequencies, fast recovery diodes areneeded. The reverse recovery time can be reduced by shortening the lifetimeof carriers, but beyond a certain point the resistivity of the silicon increasesunacceptably.


Schottky barrier devices are very fast, and have a relatively low forward dropof 0.5V. Unfortunately silicon devices are only available up to about 100Vrating.

1.2 Simple rectifier circuits

In the following circuits we use ideal diodes unless otherwise noted.

1.2.1 Half wave circuit

The output is unidirectional, but not continuous. The average voltage, VDC, foundby integrating over a half cycle, is low.

The half wave circuit is simple and cheap, but the output voltage has a highharmonic content (See Electrical Databook).


1.2.2 The bi-phase circuit

The average voltage, found by integration over a half cycle is

2 2V�

This is obviously twice the value from the half wave circuit, but with a lowerharmonic constant than the have wave circuit.

This circuit needs two anti-phase voltages. These may be obtained from acentre-tapped transformer, but each half secondary only carries current forhalf cycle, so utilisation of the windings is poor.


1.2.3 The bridge circuit

When terminal 1 of the generator is positive with respect to terminal 2, currentflows through D1 to the load, and back to the source via D2. When the polarityof the generator reverses, current flows to the load through D3 and returns viaD4. Current therefore always flows through the load in the same direction.

The average output voltage is as for the bi-phase circuit. 2 2V�

The bridge rectifier is very widely used and it avoids the need for a centretapped transformer. Note that 2 diodes are conducting at once and the 2Vdrop is a problem in low voltage circuits, e.g. 5V power supplies.


2. Capacitor Smoothing

2.1 Basic circuit

Consider a bridge rectifier circuit.

The load waveform is shown below

Passive smoothing circuits only use energy storage elements (capacitors andinductors) to reduce the ripple (harmonics) on the d.c. output to the requiredlevel. However, to obtain the low ripple usually needed for electroniccircuits, 0.1%, such smoothing circuits become cumbersome and expensive.Therefore electronic regulator circuits are used, preceded by a simplesmoothing circuit; These active circuits will be covered later.


2.2 Full analysis

In period A, D1 and D2 conduct and the load voltage follows the supply voltage; the capacitor charges. In period B, the diodes do not conduct and the loadcurrent is supplied from the capacitor.

Consider periods A and B separately. Period A is between ωt = θ1, where diodeconduction starts, to ωt = θ2 where conduction ceases. θ1 may be about 30O

from the top of the sine wave, and θ2 is a few degrees past the peak of the sinewave.

For period A

Capacitor current

Load current is

Diode current

At θ2 , i = 0.

In period B, from ωt = θ2 to ωt = θ1 + π there are no diodes conducting, andthe load current is supplied by the capacitor. Assuming RLC >> π/ω (10ms at50Hz), θ2 is close to π/2 and can normally be taken to be exactly π/2. Thus

Using a false time origin where t/ = 0 at π/2 (strictly θ2 ). The decayingexponential meets the rising sine wave, v = , so θ1 can be found by trialV̂ sin�tand error. There is no closed form of the expression.


The effect of the approximation above is to define the initial voltage forperiod B. This makes it an attractive and common approximation. Inpractice, simplifications are often used, as the analysis is often performedignoring the source impedance, capacitor tolerance etc. thus reducing theaccuracy of the result.

2.3 A simplified analysis

An estimate of the ripple is often all that is needed. An overestimate of the ripple∆V, adequate for most cases, is obtained by assuming that:

1. linear discharge of the capacitor (remember that RLC >> π/ω )

2. the discharge occurs over the whole half period, i.e. charging iseffectively instantaneous.

Hence, v = V̂ (1 − t�

RLC )

With t' = π/ω or 1/2f , the voltage at the end of the discharge is

The peak-to-peak ripple, ∆V , is then

�V =

If the ripple is small, the mean load current, I , is . SoV̂/RL

This is a commonly quoted result. A better estimate could be obtained byassuming a linear decay to q1 . However, to get a useful gain in accuracy, thesource impedance should be included. (e.g. The transformer windingresistance).


2.4 Mean Output Voltage

Because the capacitor changes the operation of the rectifier, reducing the time thediodes are on, the average voltage produced by the rectifier is no longer that ofthe simple rectifier with a resistive load. The mean output voltage is the peakvoltage minus half the ripple

Mean output voltage =

The mean load voltage will fall with increasing load current as the rippleincreases. This change in voltage is known as regulation. The impedance of thesupply will worsen the regulation.

2.5 Calculation of diode current

The capacitor charges through two diodes in period A. The larger the smoothingcapacitor is made to reduce the ripple, the shorter the conduction period and thehigher the diode current.

The current from θ1 to θ2 must be several times the load current, to average out.

The diode current, i, is approximately

Sketched:

The pulse nature of the current not only imposes stresses on the diodes and thesmoothing capacitor but also leads to a poor power factor for the circuit.


In practical circuits, the shape of the current at θ1 is significantly modified bythe source impedance, especially if the circuit is supplied through a trans-former, which contributes leakage reactance. The source impedance can alsobe augmented, often by adding inductance, to spread the pulses and reduceharmonic distortion, although this results in a lower dc voltage.

A further consequence of the pulse nature of the diode current is increasedcopper (I2R) losses in transformer windings. (The rms value of the ac currentdetermines the copper losses).

So far we have considered the repetitive current pulses arising in steady stateoperation. There is also an even larger current pulse on switch-on, moderated bythe supply impedance.

Electrolytic capacitors are generally used in smoothing circuits and have amaximum rated ripple current, usually quoted at 100/120 Hz. This ratingmay be marked on the capacitor. If not the manufacturer's data must beconsulted to ensure that the rating is not exceeded. The rated workingvoltage can be exceeded for short periods by a small amount, about 10%, butthe life of capacitors is severely reduced by large, repetitive overvoltages,especially at high operating temperatures. The switch-on surge also stressescomponents and means that circuits with capacitor smoothing must beprotected using ‘slow-blow’ fuses.

2.6 Further capacitor smoothed circuits

2.6.1 Half wave

The half wave rectifier from section 1.2 can be used with capacitor smoothing.The capacitor is only charged once per mains cycle so the ripple is approximatelytwice that of the bridge circuit for the same capacitance and load current.

The regulation is poor and the supply current waveform is very poor - it evencontains a dc component. However, this circuit is economical for applica-tions where a low load current is demanded.


2.6.2 The split rail supply

The bridge rectifier is often used in conjunction with a centre tapped transformerwinding to give two equal and opposite dc supplies. It can be considered as twobi-phase circuits connected in series.

Since the circuit is essentially two bi-phase rectifier circuits, the transformerwindings are fully utilised through each half cycle.


2.6.3 The voltage doubler circuit

The voltage doubler is an ingenious circuit for obtaining an output voltage ofnearly twice the peak of the supply voltage and is shown below.

The circuit can be thought of as two half wave circuits, one acting on the positivehalf cycles through diode 1, the other using the negative half cycles throughdiode 2.

The same comments about the general performance of the half wave rectifiersupply to this circuit also. Again it is a useful circuit where high voltages atlow current are needed, for example for the final anode of cathode ray tubes.Then it is operated at a high frequency to reduce the size of capacitor needed.


An alternative topology is shown below in which the generator and load have acommon earth.

These circuits are employed as charge pumps, which are useful for creatingadditional voltage rails in integrated circuits. The generator is often anon-chip oscillator.


2.7 Diode ratings in circuits with capacitor smoothing

Diodes are specified as having a maximum average forward current. As theforward characteristic of the diode is a nearly constant voltage drop of about 1V,it is not normally necessary to take into account the waveform of the diodecurrent.

The peak inverse voltage imposed on the diode varies with the circuit. In the halfwave circuit and the bi-phase circuits, the peak value is nearly twice the peaksupply voltage. The capacitor holds the cathode at nearly the peak of the supplyvoltage while the negative peak is applied to the anode.

In the bridge circuit each diode only experiences the peak voltage, which isan attractive feature of the bridge circuit in high voltage applications.

2.8 Summary

Capacitor smoothing is cheap and simple - it is widely used as large electrolyticcapacitors and supercapacitors are available, up to 1F. However, the high peakcurrents cause waveform distortion which is attracting attention from theregulatory authorities.


3. Inductor Smoothing

3.1 Basic circuits3.1.1 Half wave circuit

The current lags the voltage. The current is unidirectional, due to the diode, butnot continuous. The average voltage, VDC, found by integrating carefully.

1.1.1 Bridge rectifier

The inductor stores magnetic energy and uses it to maintain the current byproviding the voltage difference between v and vL.

The voltage waveform out of the rectifier is on the limit of continuous andthe inductance will keep the current flowing. Thus the current is continuous.


The mean output voltage of the rectifier is as in section 1.2.3.2V 2�

The inductor stores energy by integrating the voltage applied to it, setting upthe magnetic field.

3.1.3 The calculation of ripple by an approximate method

At θ1, as the supply voltage exceeds the load voltage, the current starts to grow.At θ2, the supply voltage falls below the load voltage and the current starts todecrease.

The circuit equation is

rewrite in integral form:

Note limits. The current ripple ∆i is

For L/R >> 1/2f , as is usual, the current ripple is small (< 10% of mean current).


Then the load volts, iRL , is almost constant at . Hence the limits, θ12 2 V�RL

and θ2 may be taken as approximately where the mean load voltage intersects thesupply voltage waveform.

Evaluating the integral gives . �i = 0.4 2 V�L

For example, for a supply giving 100 V dc at 10 A dc, using a 1000 mH inductor with a 50 Hz source, L/R is ten times the half period. Thecurrent ripple is 0.18 A peak-to-peak, corresponding to a voltage rippleof 1.8 V peak-to-peak, which is very small compared to the averageload voltage of 100 V.

3.1.4 Calculation of ripple by Fourier analysis

The Fourier analysis of the rectified waveform v (a series of half sine waves) is inthe data book

v = 2� V̂ + 4

3� V̂ sin(2� t) + ...

For the component at 2ω ,

The amplitude of the ripple across the load at 2ω is

If 2ωL >> RL , then


The peak to peak current ripple I2 is

Compare this with the result in Section 3.1.3.

If 2ω0L >> R, the amplitude of higher harmonics can be ignored as not onlyare their initial magnitudes small, but they are also more attenuated.

3.1.5 Supply current and components

Provided that the smoothing inductor is large, or L/R >>1/2f , the rectifiedcurrent is close to constant. Therefore the supply current is a square wave. Theharmonic content is reasonable (see data book) and the power factor is good.

At mains frequencies, Iron cored inductors are used to obtain a high induc-tance for smoothing purposes. Often called chokes, they usually have an airgap in the magnetic circuit to avoid saturation by the dc current. However,more turns are then needed to achieve a given inductance as the reluctance ofthe iron circuit is greater. Consequently chokes for 50Hz use are large andheavy. As with capacitor smoothing, at high frequencies this method ofsmoothing is quite attractive. The regulation of the circuit is made worse bythe dc resistance of the inductor.

3.2 Combinations of inductors and capacitors

Consider the usual case of series-inductor parallel-capacitor:


Using the Fourier analysis method, the amplitude of the voltage component at 2f,from before, is

43� V̂

By circuit analysis, the peak voltage at this frequency across the load is

where ZL is the parallel combination of RL and capacitive reactance XC. Both XC

and XL are evaluated at 2f.

The reactance of the capacitor is usually small compared to the load resistance,which is again small compared to the reactance of the inductor. Therefore themagnitude of the peak ripple voltage is approximately

A common choice in dc supplies is to allow the ripple to be high in the induc-tor (small inductor) and use a large capacitor to give a smooth dc outputvoltage. This makes the input waveform closely sinusoidal and near unitypower factor. Further inductor-capacitor stages may also be added to giveeven greater reduction in ripple, in the manner of a filter. However, the dcregulation will worsen due to the additional resistance in the inductors.

3.3 Summary

At 50Hz, capacitor smoothing is usually cheaper and lighter than inductorsmoothing, but waveform distortion of the supply current is a problem.Combining capacitor and inductor smoothing is very attractive. In switch-modepower supplies, which operate at 20 kHz or greater, combined inductor-capacitorsmoothing is standard.


4. Three-phase Rectifiers

4.1 Three-phase Rectifier Circuits

4.1.1 Single-way circuits

A half-wave circuit is used for each phase.

The load current changes from D1 to D3 when Vb > Va at 30O for Vb. The currenttransfers later from D3 to D5, and then from D5 back to D1. The average voltageis found by integration,


4.1.2 The three phase bridge

This offers improved performance, notably twice the output voltage for the sameinput voltages.

When phase a is most positive, D1 conducts. Current flows through the loadand returns to the most negative phase, initially phase b, then 30O later phasec (as the most negative), and so on. The whole sequence of combinationsfollows in 30O sections: ab,ac,bc,ba,ca,cb,.......


The average voltage is 3 6V�

If the load voltage VO is sketched the waveform is quite smooth, even withoutany further smoothing.

The lowest frequency harmonic component is at six times the supply frequencyand the three-phase bridge is commonly known as a six-pulse converter to avoidany possible confusion.

The output voltage is twice that of the single way three phase circuit, asexpected, since the bridge can be thought of as two single way circuits withtheir outputs combined. Thus the bridge circuit gives the highest dc outputfor a given voltage rating of the diodes.

P.R. Palmer October 2008


1. diodes and rectifier circuits - university of cambridgeprp/3b3/3b3lec1-4.pdf · diodes and...

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