Power Semiconductor Devices  UCF Department of EECStomwu/course/eel6208/notes/22 Power...
Embed Size (px)
Transcript of Power Semiconductor Devices  UCF Department of EECStomwu/course/eel6208/notes/22 Power...

Power Semiconductor Devices

UCF Power Diode

UCF Ideal Power Diode

UCF PNPN Diode
Breakover voltage
Holding current

UCF DIAC (or PNPNP Diode)
Behave like two PNPN diodes connecting back to back
Can conduct in either direction

UCF Silicon Controlled Rectifier (SCR) or Thyristor

UCF Ideal SCR (1)

UCF Ideal SCR (2)

UCF TRIAC
like two SCRs connect back to back sharing a common gate

UCF Ideal TRIAC (1)

UCF Ideal TRIAC (2)

UCF Power MOSFET (1)

UCF Power MOSFET (2)

UCF IGBT (1)
Insulated Gate Bipolar Transistor

UCF IGBT (2)

AVAILABLE SEMICONDUCTOR SWITCHING DEVICES
In this section, the emphasis will be on the iv switching characteristics of devices and their corresponding power ratings and possible applications. Selecting the most appropriate device for a given application is not an easy task, requiring knowledge about the device's characteristics and unique features, innovation, and engineering design experience. Unlike lowpower (signal) devices, power devices are more complicated in structure, driver design, and operational iv characteristics. This knowledge is very important for enabling power electronics engineers to design circuits that will make these devices close to ideal. In this section, we will briefly discuss two broad families of power devices:
Bipolar and Unipolar Devices
1. Power diodes
2. Bipolar junction transistors
3. Insulated gate bipolar transistors (IGBTs)
4. Metal oxide semiconductor fieldeffect transistors (MOSFETs)

38 Chapter 2 Review of Switching Concepts and Power Semiconductor Devices
ThyristorBased Devices 1. Siliconcontrolled rectifiers (SCRs)
2. Gate tumoff {GTO) thyristors
3. Triode ac switches (triacs)
4. Static induction transistors (SITs) and thyristors (SITHs)
5. MOScontrolled thyristors (MCTs)
2.5.1 Bipolar and Unipolar Devices
The Power Diode
The power diode is a twoterminal device composed of a pn junction and whose tumon state cannot be controlled (uncontrolled switch). The diode tumon and tumoff is decided by the external circuitry: A positive voltage imposed across it will tum it on and a negative current through it turns it off.
The symbol and the practical and ideal iv characteristic curves of the power diode are shown in Fig. 2.8(a), (b), and (c), respectively. In the conduction state, the forward voltage drop, ~, is typically 1 V or less. The diode current increases exponentially with the voltage across it; i.e., a small increase in ~ produces a large increase in IF (see Problem 2.5). In the reversebias region, the device is in the off state and only a reverse saturation current, Is, exists in the diode (also known as leakage current). The breakdown voltage, V8R, is the maximum inverse voltage the diode is capable of blocking. V8R is a dioderated parameter with values up to a few kilovolts, and in normal operation the reverse voltage should not reach V8 R. Zener diodes are special diodes in which the breakdown voltage is approximately 612 V, controlled by the doping process.
In power circuits, power diodes have two important features:
1. Powerhandling capabilities, including forward current carrying and reverse voltage blocking
2. Reverse recovery time {trr) at tumoff
The parameter trr is very significant because the speed of turning off the diode could be large enough to affect the operation of the circuit. At tumon, the delay time is normally insignificant compared to the transient time in power electronic circuits.
+
Broadly speaking, two types of power diodes are available:
1. The bipolar diode, which is based on the pn semiconductor junction. Depending on the applications, a bipolar diode can be either the standard linefrequency
On
Off
M W ~ Figure 2.8 Power diode. (a) Circuit symbol. (b) Practical and (c) ideal switching characteristics.
type oc 61: 50 nsaodS ing of3 tv
2. The Scball a lowerilr bipolar dial carriers., 6 Finally.*= teristics fewerc:a:e rily geDCII cuit, it is
To study tbcJ of Fig. 2.9(a). wli Fig. 2.9(b). Forsi normally en~ either a BIT oral current I0. At I = input voltage Viii since tumon .1! ward conductical diode voltage &I transient voltaF why during the is in the steady a since significallll
As shownia its forward vablc, reaches a negalit whichthec~
broken down j forward volfaF i stant and is dell:ll ample, since dlr: parasitic induclll turning off by~
....
Figure 2.9 Typil at t = t0 . (b) DiD

I
2.5 Available Semiconductor Switching Devices 39
type or the fastrecovery highfrequency type, with its trr varying between 50 ns and 50 JJS. Typical voltage drop is 0.71.3 V, with reverse voltage blocking of3 kV and forward current of3.5 kA.
2. The Schottky diode, which is based on the metalsemiconductor junction. It has a lower forward voltage drop than the bipolar (about 0.5 V or less). Unlike the bipolar diode, whose current conduction depends on the minority and majority carriers, the Schottky diode current depends mainly on the majority carriers. Finally, the iv characteristics of the Schottky diode are similar to the iv characteristics of the bipolar diode. Unlike bipolar diodes, Schottky diodes generate fewer excess minority carriers than majority carriers; hence its current is primarily generated due to the drift of majority carriers. Due to its largeleakage circuit, it is normally used in lowvoltage, highcurrent de power supplies.
To study the reverse recovery characteristics of the diode, we consider the circuit ofFig. 2.9(a), which has a typical diode current waveform during tumoff as shown in Fig. 2.9(b). For simplicity we assume the switch is ideal. Such a circuit arrangement is normally encountered in switchmode detode converters with the switch replaced by either a BJT or a MOSFET. Initially we assume the diode is conducting with forward current /
0. At t = t0, the switch is turned on, forcing the diode to tum off due to the de
input voltage fin The tumon characteristics of the diode are simpler to deal with since tumon only involves charging the diode depletion capacitor. The diode's forward conduction begins when its depletion capacitor has been charged. At tumon, the diode voltage drop is larger than the normal forward drop during conduction. This transient voltage exists due to the large value of diode resistance at tumon. This is why during the diode's tumon time the power dissipation is much larger than when it is in the steady conduction state. The diode tumoff characteristics are more complex since significant stored charges exist in the body of the pn junction and at the junction.
As shown in Fig. 2.9(b), during tumoff, the diode current linearly decreases from its forward value, /
0, at t = t0 to zero at t = t1 and then continues to go negative until it
reaches a negative peak value at t = t2, known as the reverse recovery current, /RR at which the current starts to rise exponentially to zero at t = t3 The time intervals can be broken down as follows: Between t0 and t1, the diode current is positive and the diode forward voltage is small. Hence, we assume the rate of change of diode current is constant and is determined by the total circuit inductance in series with the diode. In our example, since the switch is ideal, di0 / dt and /RR are limited by the diode's and lead's parasitic inductances. At t = t1, the current becomes zero and the diode should begin turning offby supporting reverse voltage. But because of the excess minority carriers.in
r=ro
Nonideal /V' tumoff
(a)
Io
io
Io 1.
to tu I I I ;~~~ lrr
(b)
Figure 2.9 Typical diode switching characteristics. (a) Switching circuit with S closed at t = t0 . (b) Diode current.

40 Chapter 2 Review of Switching Concepts and Power Semiconductor Devices
the pn junction that need to be removed before the diode's reverse voltage begins to rise, the diode remains in the conduction state for longer time, i.e., until t = t2
The delay from t 1 to t2 is due to the minority carriers in the depletion region, whereas the delay from t2 to t3 is caused by the charge stored in the bulk of the semiconductor material. At t = t3, all charge carriers are removed, causing the device to be fully switched off. The time it takes from the moment the diode current becomes negative until it becomes zero is the reverse recovery time, tmas shown in Fig. 2.9(b).
Between t2 and t3, the junction behaves like a capacitor whose voltage goes from zero to the reverse voltage via a charging current in this interval. The total charge carriers that cause a negative diode current flow when it is turned off constitute the reverse recovery charge, Qm and can be expressed in terms of /RR and trr.
The time between t2 and t3 may be very short compared to tm resulting in high di I dt. The ratio between (t3  t2) and trr is a parameter that defines what is known as diode snappiness. The smaller this ratio, the quicker the diode recovers its reverse blocking voltage, resulting in what is known as a fastrecovery or hardrecovery diode. Meanwhile, a diode with a high ratio of (t3  t2) to trr takes a relatively long time to bring its forward current to zero from its negative peak value. These diodes are known as softrecovery diodes. The fastrecovery diodes have high di I dt and normally experience oscillation at turnoff. Standard or generalpurpose diodes have soft recovery time and are used in lowspeed applications where the frequency is less than a few kHz.
In general, an attempt to reduce either trr or JRR will result in an increase in the other. The forward recovery voltage limits the efficiency because of device stresses and higher switching losses, and trr limits the frequency of operation. Power electronics engineers should keep in mind that the transient voltage at tumon and the transient current at tumoff might affect the external circuitry and cause unwanted stresses. External snubber circuits are added to suppress these transient values.
EXAMPLE2.4
Consider the switching circuit shown in Fig. 2.10 by modeling the circuit parasitic inductance as a lumped discrete value, L5 Assume the switch was open for a long time before being turned on at t = t0 Assume the same diode switching characteristics of Fig. 2.9( b), except that it is a fastrecovery diode with t3  t2 ""0. Derive the expressions for /RR and the peak switch current in terms of the diode reverse recovery time.
SOLUTION While the diode is in the conduction state, its forward current is / 0 When the switch is closed at t = t0, the diode voltage remains zero and its current is given by
and i5
fort 2: /0 is given by
v. is{t) = Lm(t to) to ~ I~ t2
s
Io
Figure 2.10 Diode switching circuit with parasitic inductor.
.\1 t = t. tbediodcc ~
:\t this poilll die ..
renl /IT Siacc . l,.r,.,lL, .. .._.a IIJUSl be addedAIIolla".....a..TCOI~ ... t
The pea ... less m.a. cqllll .. CXli]IOCICIICdaolk6 n:cO\"Cn tim:.de COll"e:nas. ....
THBipelar.._
The scbemalic li lBJT) are saa terminal ~;a: , ~;tcbing spcllt:t drawback is 6c
lJnlikedlcSI BITs ba"'e twoti blocking voiiiF 011 state. BJT ...
suggest dull
Becausesmaller fJ.r.....1 collectoremilll::r largely by tbc a known asao .. uansistor is.. Facurrent I 8 , tbc:a, tr.msistor ~
~ .. 1f111311~. tbdiall ~=ebias~ pb._~""DOO caalll } !i._ is dcfinoll

".
2.5 Available Semiconductor Switching Devices 41
:\t 1 = 11 the diode current becomes zero and is becomes I 0 . Hence, the interval 11  t0 is given by
vin IRR = lrr
Ls
The peak switch current occurs at t = 12 when iD = IRR and is given by
v I = _!!!.t +I s,peak L rr 0
s
:\t this point the diode is turned off and the peak inductor current is higher than the load current I 0 . Since the load is highly inductive, its value cannot increase suddenly by the amount Jintrr/ Ls without creating high reverse voltage across the diode. As a result, a snubber circuit must be added across the diode to dissipate excess stored energy in the inductor.
Another important point is that when L5
becomes very small, a very large reverse recovery current occurs that could damage the diode and cause large switching losses.
The peak value of the reverse current, IRR is a very important parameter and it can be less than, equal to, or larger than the forward current I 0, depending on the external circuitry connected to the diode, and the diode parasitic inductance. The fastrecovery diodes have low recovery time, nonnally less than 50 ns, and are used in applications such as highfrequency dede converters, where the speed of recovery is critical.
The Bipolar Junction Transistors (BJT)
The schematic symbol and iv characteristics for the bipolar junction transistors (BJT) are shown in Fig. 2.ll(a), (b), and (c), respectively. It is a twojunction, threeterminal device with the minority carriers being the main conducting charges. The switching speed of the BJT is much faster than that of thyristortype devices. A major drawback is the second breakdown problem.2
Unlike the SCR, the BJT is turned on by constantly applying a base signal. Power BJTs have two different properties from the lowpower BJT and logic transistor: large blocking voltage in the off state and high forward currentcarrying capabilities in the on state. BJT power ratings reach up to 1200 V and 500 A. These high rating values suggest that the power BJT's driving circuits are more complicated.
Because the BJT is a currentdriven device, the larger the base current, the smaller f3rorcel and the deeper the transistor is driven into saturation. In saturation, the collectoremitter voltage is almost constant and the collector current is determined largely by the external circuit to the switch. It is sometimes useful to define what is known as an overdrive factor, which gives a measure to how deep in saturation the transistor is. For example, if the transistor is at the edge of saturation with given base current 18 , then with an overdrive factor of 10 the base current becomes 1018 and the transistor becomes deeper in saturation.
:::ormally, the first breakdown voltage refers to the avalanche breakdown caused by the increase in the reverse bias voltage, which can be nondestructive. The second breakdown voltage is a destructive phenomenon caused by localized overheating spots in the device. 3
f3rorccd is defined as the ratio I c I /8 when the transistor is operating in the saturation mode.
I f ~ l

42 Chapter 2 Review of Switching Concepts and Power Semiconductor Devices
ie
+ On state
veE Off state
4~~.veE
(a) (b)
ie
Saturation (on state)
,;V ,il_ 
Active region
r Increasing base current
(c)
Figure 2.11 BJT switching characteristics. (a) npn transistor. (b) Ideal iv characteristics. (c) Practical iv characteristics.
Since the base thickness is inversely proportional to the current gain {3, Darlingtonconnected BIT pairs have been developed in which the collectors of two devices are joined and the base of the first is connected to the emitter of the second, as shown in Fig. 2.12. This arrangement results in an overall gain that approximately equals the product of the individual {3's of the two transistors. Transistor Q1 serves as an auxiliary transistor, which provides the base current necessary to tum on Q2 Because there is a high current gain, a smaller base current to Q2 is needed to drive the power Darlington pair. Darlington power transistors are widely used in UPSs and various ac and de motor drives up to hundreds of kilowatts and tens of kilohertz. A modern Darlington pair has ratings up to 1.2 k V with current up to 800 A and operating frequency up to several kilohertz.
Triple Darlingtons are also available, in which the current gain becomes proportional to the product of the three individual currents gains of the transistors. To turn off the Darlington switch, all base currents must become zero, resulting in slower switching speed compared to a single transistor. Also, the overall collectoremitter saturation voltage, VeE sat is higher than for a single transistor, as will be illustrated in Exercise 2.5. '

2.5 Available Semiconductor Switching Devices 43
Figure 2.12 Darlingtonconnected BJT.
There are three regions of operation: saturation, active, and cutoff. As a power S'o\itch, the BJT must operate either in the saturation region (on state) or in the cutoff region (off state). The third state is when the transistor is in the linear region and is :lSed as an analog amplifier.
To investigate the tumon and tumoff processes, we consider a simple inverter .:ircuit shown in Fig. 2.13(a) with its switching waveforms as shown in Fig. 2.13(b). Tlk: voltage v1 is the base driving voltage with positive polarity, V1, to push positive .current into the base, I 81 = ( V1  V8E) I R8 , and a negative polarity, V2, to quickly dis.:barge the base current, 182 = (V2 + VBE) I R 8 . At time t = t0, V1 is applied with positive de voltage, +V1 Because it takes time to charge the internal depletion capacillor to tum the junction on at V8E = 0.7 V, a delay time, td, elapses before the colleclL"'f" current starts flowing. After the junction is turned on, the collector starts flowing exponentially through R8 and the emitterbase junction capacitor. During this period, 1lhe minority carriers are being stored in the transistor base region. The collector curl:'Clt increases until it reaches its maximum saturated value, /on determined by
_ ~n VcE,sat /on R
The time it takes for the collector current to rise from 10% to 90% of its maxiDllm value, /on is called the rise time. For simplicity, Fig. 2.13(b) shows the rise time itm Ic = 0 to /00 The total switching on time is given by ton = td + tr To tum off die transistor, a negative (or zero) base voltage is normally applied, resulting in a base arrent 182 being pulled out of the base as shown in Fig. 2.13(b). The collector current Jbes not start decreasing until sometime later after the stored saturation charge in the 'htie bas been removed. This time is called the storage time, ts; it is normally longer 6aJ rbe delay time, td, and usually determines the limiting range of the switching ~ If the base voltage is not negative (i.e., in the absence of 182), the entire base 'ewtmt must be removed through the process of recombination.
To tum on the BJT, a large current must be pushed to the base. This base current 1&11& be large enough to saturate the transistor. In the saturation region, both baseCIIIlllfier and basecollector junctions are forward biased. This is why the BJT is known a a currentdriven device. When it is operated in the saturation region, 18 > lei {3, wilcre 13 is the de current gain. In this region a new de current gain 13 is defined to IDiiicate the depth of the transistor saturation. The saturation collectoremitter voltage P'en as V CE sat and f3rorced is defined as
Ic f3rorced =I
B

44 Chapter 2 Review of Switching Concepts and Power Semiconductor Devices
v~rH
VzrI I I I
to
fBI 111 ~~
Isz
I I I I I
I I 1 r
ic I I I I I I I I
Ion ++1 I I I I I I I I I I I
(b)
Figure 2.13 Switching characteristics for the BJT. (a) Circuit. (b) Switching waveforms.
where Ic and 18 are the collector and base currents in saturation, respectively, and f3rorced < {3. The smaller f3rorced the deeper the transistor is driven into saturation. T~ ically, f3rorced can be as low as 1. Ideally, vCE,sat = 0, but in practice this value varies between 0.1 and 0.6 V, depending on how deep in saturation the device is driven. The new ratio of collector to emitter current is much smaller than the case when the transistor is operated in the active mode. At the edge of saturation, f3rorced = {3.
The total power dissipation in the transistor is obtained by adding the input supplied by the collector current and the input power supplied by the base ~ .. ~~o.,.thence, the total power dissipation is defined as follows:
Pdiss = VcEic+ VBEIB
EXERCISE 2.4
Consider the transistor circuit shown in Fig. E2.4. Assume the transistor is operating in the saturation region with VCEsat = 0.5 V, V8E = 0.75 V, VD = 0.7 V, and D = 0.5. Sketch i8, veE and iv. Determine the o~erall efficiency of the circuit (neglect the power supplied through the base).
lit .....

2.5 Available Semiconductor Switching Devices 45
t .. T, Figure E2.4
1J = 80.6%
To determine the voltage, current, and power operational limits, normally a plot of iv characteristics is given, as shown in Fig. 2.14. It gives the region in which the transistor can operate within its limits, the region is known as the safe operation area (SOA). It represents the permissible range of current, voltage, and power of the device in operation. The locus of switch voltage versus switch current during tumon and tumoff must lie within the SO A.
(a) Show that the current gain of the triple Darlington transistors shown in Fig. E2.5 is given by
(b) Assume transistor Q1 has collectoremitter saturation voltage VcI,sat Show that VcJ,sat for transistor Q3 is given by
VC3,sat = VCEI,sat+2VBE
Assume identical V8 E for the three transistors.
Current limit
SOA
Max power
(Pcmax)
Second breakdown limit
/Voltage limit
veE. max
+L. veE
Figure 2.14 Safe Operation Area (SOA) for a BJT.

46 Chapter 2 Review of Switching Concepts and Power Semiconductor Devices
ic
Figure E2.5
EXERCISE 2.6 Consider the simple BJT switch shown in Fig. E2.6. Determine f3rorced for R8 = 1 k.O. R8 = 10 k!l, R8 = 20 kO. Use V8E = 0.7 V, VCE,sat = 0.3 V and assume an ideal diode.
JOY D
Figure E2.6
ANSWER 1.25, 12.5, 25
Initially, the BJT was developed to be used in linear audio output amplifiers. Soon BJT devices were used in switchmode and highfrequency converters for aerospace applications to reduce the size and weight of magnetic components and filter capacitors. In applications where selftumoff devices are needed, such as de choppers and inverters, BJTs quickly replaced thyristors.
The Power MOSFET
In this section, an overview of power MOSFET semiconductor switching devices will be given. A detailed discussion of the physical structure, fabrication, and physical behavior of the device and its packaging is beyond the scope of this chapter. The emphasis here will be on the device's regions of operation and its terminal iv switching characteristics.
Unlike the bipolar junction transistor, the metal oxide semiconductor fieldeffect transistor (MOSFET) device belongs to the unipolar device family, since it uses only the majority carriers in conduction. The development of metal oxide semiconductor technology for microelectronic circuits opened the way for the power MOSFET device in 1975. Selecting the most appropriate device for a given application is not an easy task, requiring knowledge about the device characteristics, and unique features, as well as innovation and engineering design experience. Unlike lowpower (signal) devices, power devices are more complicated in structure, driver design, and operational iv characteristics. This knowledge is very important in enabling a power electronics engineer to design circuits that will make these devices close to ideal. .... I.A ..

Drain (D)
JD

48 Chapter 2 Review of Switching Concepts and Power Semiconductor Devices
Gate Source
Drain
(a)
Gate
Drain
(b)
Figure 2.17 (a) Vertical crosssectional view of a power MOSFET. (b) Simplified representation.
MOSFET Structure Unlike the lateralchannel MOSET devices used in man IC technologies, in which the gate, source, and drain terminals are located on th' same surface of the silicon wafer, power MOSFETs use a vertical channel struc ture to increase the device's power rating. In the vertical channel structure, the source and drain are on opposite sides of the silicon wafer. Figure 2.17(a) shows avertical crosssectional view of a power MOSFET. Figure 2.17(b) shows a simplified representation. There are several discrete types of the verticalstructure power MOSFET available commercially today, such as the VMOSFET, UMOSFET, DMOSFET, and SMOSFET. The pn junction between the phase region (also, referred to as the body or bulk region) and the ndrift region provides the . forward voltage blocking capabilities. The source metal contact is connected directly to the phase region through a break in the n+ source region to allow for a fixed potential to the phase region during normal device operation. When the gate and source terminals are set at the same potential ( V GS = 0), no channel is established in the phase region (i.e., the channel region remains unmodulated). The lower doping in the ndrift region is needed to achieve higher drain voltage blocking capabilities. For the drainsource current, iD, to flow, a conductive path must be established between the n+ and n regions through the phase diffusion region.
.,t;.

2.5 Available Semiconductor Switching Devices 49
OnState Resistance When the MOSFET is in the on state (triode region), the channel of the device behaves like a constant resistance, RDs(on) that is linearly proportional to the change between vDS and iD, as given by the following relation:
dVDSi RDS(on) = 7JT;;
V GS=constant
The total conduction (onstate) power loss for a given MOSFET with forward current ID and onresistance RDS(on) is given by
p on,diss = IbRDS(on)
The value of RDS(on) can be significant and varies between tens of milliohms and a few ohms for lowvoltage and highvoltage MOSFET, respectively. The onstate resistance is an important data sheet parameter, since it determines the forward voltage drop across the device and its total power losses.
Unlike the currentcontrolled bipolar device, which requires base current to allow the current to flow in the collector, the power MOSFET is a voltagecontrolled unipolar device and requires only a small amount of input (gate) current. As a result, it requires less drive power than the BJT. However, it is a nonlatching current like that of the BJT; i.e., a gatesource voltage must be maintained. Moreover, since only majority carriers contribute to the current flow, MOSFETs surpass all other devices in switching speed, with speeds exceeding a few megahertz. Comparing the BJT and the MOSFET, the BJT has higher powerhandling capabilities and lower switching speed, while the MOSFET device has lower powerhandling capabilities and relatively fast switching speed. The ~OSFET device has a higher onstate resistance than the bipolar transistor. Another difference is that the BJT parameters are more sensitive to junction temperature compared to the MOSFET parameters. Unlike the BJT, MOSFET devices don't suffer from second breakdown voltages, and sharing current in parallel devices is possible.
Internal Body Diode The modem power MOSFET has an internal diode called a body diode connected between the source and the drain, as shown in Fig. 2.18(a). This diode provides a reverse direction for the drain current, allowing a bidirectional switch implementation. Even though the MOSFET 's body diode has adequate current and switching speed ratings, in some power electronic applications that require the use of ultrafast diodes, an external fastrecovery diode is added in an antiparallel fashion, with the body diode blocked by a slowrecovery diode, as shown in Fig. 2.18(b).
Internal Capacitors Another important parameter that affects the MOSFET's switching behavior is the parasitic capacitances between the device's three terminals, namely, the gatetosource (Cgs), gatetodrain (Cgd), and draintosource (Cds) capacitances, shown in Fig. 2.19(a). The values of these capacitances are nonlinear and a function of the device's structure, geometry, and bias voltages. During tumon, capacitors cgd and cgs must be charged through the gate; hence, the design of the gate control circuit must take into consideration the variation in these capacitances. The largest variation occurs in the gatetodrain capacitance as the draintogate voltage varies. The MOSFET parasitic capacitances are given in terms of the device's data sheet parameters ciss coss and crss as follows,
cgd = crss
cgs = ciss crss
cds = coss crss

2.5 Available Semiconductor Switching Devices 51
charge and discharge the gatetosource and gatetodrain parasitic capacitances to tum the device on and off, respectively.
In power electronics, the aim is to use power switching devices to operate at higher and higher frequencies. Hence, the size and weight associated with the output transformer, inductors, and filter capacitors will decrease. As a result, MOSFETs are used extensively in power supply designs that require high switching frequencies, including switching and resonantmode power supplies and brushless de motor drives. Because of the device's large conduction losses, its power rating is limited to a few kilowatts. Because of its many advantages over BJT devices, modern MOSFET devices have received high market acceptance.
Regions of Operation Most MOSFET devices used in power electronics applications are of thenchannel, enhancement type, like that shown in Fig. 2.16(a). For the ~OSFET to carry drain current, a channel between the drain and the source must be created. This occurs when the gatetosource voltage exceeds the device threshold voltage, VTh For vcs > VTh the device can be either in the triode region, which is also called "constant resistance" region, or in the saturation region, depending on the value of vDs For a given Vcs with a small vDS (vDS < Vcs VTh) the device operates in the triode region (saturation region in the BJT), and with a large vDs (vDs > Vcs  VTh), the device enters the saturation region (active region in the BJT). For Vcs < VTh, the device turns off, with the drain current almost equal to zero. Under both regions of operation, the gate current is almost zero. This is why the MOSFET is known as a voltagedriven device and, therefore, requires a simple gate control circuit.
The characteristic curves in Fig. 2.16(b) show that there are three distinct regions of operation, labeled as triode region, saturation region, and cutoff region. When used as a switching device, only the triode and cutoff regions are used; when it is used as an amplifier, the MOSFET must operate in the saturation region, which corresponds to the active region in the BJT.
The device operates in the cutoff region (off state) when v cs < vTh resulting in no induced channel. In order to operate the MOSFET in either the triode or saturation region, a channel must first be induced. This can be accomplished by applying a gatetosource voltage that exceeds vTh i.e.,
Vcs > VTh
Once the channel is induced, the MOSFET can operate either in the triode region (when the channel is continuous with no pinchoff, resulting in the drain current being proportional to the channel resistance) or in the saturation region (the channel pinches off, resulting in constant /D). The gatetodrain bias voltage (vcv) determines whether the induced channel undergoes pinchoff or not. This is subject to the following restrictions.
For the triode mode of operation, we have
VcD > VTh
and for the saturation region of operation, we have
vcD < VTh
Pinchoff occurs when vcD = VTh In terms of vDS the preceding inequalities may be expressed as follows:
1. For the triode region of operation,

52 Chapter 2 Review of Switching Concepts and Power Semiconductor Devices
2. For the saturation region of operation,
vDs > vGs VTh
3. For the cutoff region of operation
vGs < VTh
It can be shown that the drain current, iD, can be mathematically approximated follows:
where
iD = k[2(vGs VTh)vDs vbs1
iD = k(vGs VTh)2
1Ln = electron mobility Cox = oxide capacitance per unit area
L = length of the channel
W = width of the channel
(triode region)
(saturation region)
Typical values for these parameters are given in the PSPICE model discussed later. At the boundary between the saturation (active) and triode regions, we have
resulting in the following equation for iD:
. k 2 1D = VDS
The input transfer characteristic curve for iD vs. vGs when the device is operating in the saturation region is shown in Fig. 2.20.
The largesignal equivalent circuit model for an nchannel enhancementtype MOSFET operating in the saturation mode is shown in Fig. 2.21. The drain current is represented by a current source as a function of VTh and vGs
If we assume the channel is pinched off, the drainsource current will no longer be constant but rather will depend on the value of v DS as shown in Fig. 2.22. The increased
Figure 2.20 Input transfer characteristics for a MOSFET device operating in the saturation region.

2.5 Available Semiconductor Switching Devices 53
io D G
+ +
vos
s
Figure 2.21 Largesignal equivalent circuit model.
\'alue of v0 s results in reduced channel length, resulting in a phenomenon known as channel length modulation. If the v05 i0 lines are extended as shown in Fig. 2.22, they all intercept the v os axis at a single point labeled 1 I A, where A is a positive constant MOSFET parameter. The term (1 + Av0 s) is added to the i0 equation to account for the increase in i0 due to the channel length modulation. i0 is thus given by
i0 = k(vcs V11,)2(1 + Av0 s) (saturation region)
From the definition of r0
given, it is easy to show that the MOSFET output resistance can be expressed as follows:
If we assume the MOSFET is operating under smallsignal conditions, i.e., the ,ariation in Vcs on the i0 vs. Vcs characteristic curve is in the neighborhood of the de operating point Q at /0 and Vcs' as shown in Fig. 2.23, the i0 current source can be represented as the product of the slope gm and vcs' as shown in Fig. 2.24.
Figure 2.22 MOSFET characteristic curve including output resistance.

54 Chapter 2 Review of Switching Concepts and Power Semiconductor Devices
Io
Figure 2.23 Linearized i0 vs. vcs curve with operating de point (Q).
Ge.. e +
s
....tf0
ro
Figure 2.24 Smallsignal equivalent circuit including MOSFET output resistance.
Input Capacitance Because the MOSFET is a majoritycarrier transport device, it is inherently capable of highfrequency operation. Still, the MOSFET has two limitations:
1. High input gate capacitances
2. Transient/delay due to carrier transport through the drift region
As stated earlier, the input capacitance consists of two components: the gatetosource and gatetodrain capacitances. The input capacitances can be expressed in terms of the device junction capacitances by applying the Miller theorem to Fig. 2.25(a). Using the Miller theorem, the total input capacitance, Cin between the gate and source is given by
Cin = Cgs + (1 + gmro)Cgd
The frequency response of the MOSFET circuit is limited by the charging and discharging times of Cin Miller effect is inherent in any feedback transistor circuit with resistive load that exhibits a feedback capacitance from the input to the output. The objective is to reduce the feedback gatetodrain resistance. The output capacitance between the drain and source, Cds does not affect the tumon and tumoff MOSFET switching characteristics. Figure 2.26 shows how Cgd and Cgs vary under increased drainsource voltage, Vos
In power electronics applications, power MOSFETs are operated at high frequencies in order to reduce the size of the magnetic components. To reduce the switching

G
I I s
+
(b)
2.5 Available Semiconductor Switching Devices 55
.* D
(a)
D
F"~ure 2.25 (a) Smallsignal equivalent circuit including parasitic .:apacitances. (b) Applying the Miller theorem.
Up;Icitance
Voltage
Figure 2.26 Variation of Cgd and Cgs as a function of v0 s.
losses, power MOSFETs are maintained in either the on state (conduction state) or off state (forward blocking state).
Safe Operation Area The safe operation area (SOA) of a device provides the current and voltage limits the device must handle to avoid destructive failure. The typical SOA for a MOSFET device is shown in Fig. 2.27. The maximum current limit while the device is on is determined by the maximum power dissipation.
As the drainsource voltage starts increasing, the device starts leaving the on state and enters the saturation (linear) region. During the transition time the device exhibits

56 Chapter 2 Review of Switching Concepts and Power Semiconductor Devices
Current limit
SOA
Max power
(Pcmaxl
Figure 2.27 Safe operation area for a MOSFET.
Second breakdown limit
Voltage limit
vos,max
large voltage and current simultaneously. At higher drainsource voltage values that approach the avalanche breakdown, it is observed that a power MOSFET suffers from a second breakdown phenomenon. The second breakdown occurs when the MOSFET is in the blocking state (oft), and a further increase in v05 will cause a sudden drop in the blocking voltage. The source of this phenomenon in MOSFETs is the presence of a parasitic ntype bipolar transistor, as shown in Fig. 2.28. The inherent presence of the body diode in the MOSFET structure makes the device attractive to applications in which bidirectional current flow is needed in the power switches.
Temperature Effect Today's commercial MOSFET devices have excellent response for high operating temperatures. The effect of temperature is more prominent on the onstate resistance, as shown in Fig. 2.29. As the onstate resistance increases, the conduc
Drain
Gate _j
Source
Ros(on)
Temperature
Figure 2.28 MOSFET equivalent circuit including the parasitic BJT
Figure 2.29 The onstate resistance as a function of temperature.

2.5 Available Semiconductor Switching Devices 57
Collector
c c
~ G ct ~
E E
Emitter
(a) (b) (c)
F"tgure 2.30 (a) IGBT equivalent circuit, (b) simplified equivalent circuit, and (c) symbol.
tion losses also increase. This large vDS(on) limits the use of the MOSFET in high,oltage applications. The use of silicon carbide instead of silicon has reduced vDS(on) manifold.
As the device technology keeps improving in terms of switch speeds and powerhandling capabilities, it is expected that the MOSFET will continue to replace the BJT in all types of power electronic systems.
The Insulated Gate Bipolar Transistor (IGBT)
The detailed equivalent circuit model, the simplified twotransistor circuit model, and the schematic symbol for the insulated gate bipolar transistors (IGBT) are shown in Fig. 2.30(a), (b) and (c), respectively. Its iv characteristic is similar to the ~OSFET device and is not shown here. Since the IGBT architecture consists of a ~OSFET and a BJT as shown in Fig. 2.30(b), it is clear that the IGBT has the high input impedance of the MOSFET along with the high current gain and small onstate conduction voltage of the BJT. The device was commercially introduced in 1983 and combines the advantages of MOSFETs, BJTs, and thyristor devices: the high current density allowed in BJT devices and the lowpower gate drive needed in ~OSFET devices.
The device is turned off by zero gate voltage, which removes the conducting channel. However, a negative base cannot turn off the pnp transistor current. As a result the turnoff time is higher in the IGBT than in the bipolar transistor. However, like the GTO (to be discussed shortly), the IGBT has a tail current at turnoff due to the recombination of carriers from the base region. At turnon, a positive gate voltage is applied with respect to the emitter of the npn transistor, creating an nchannel in the MOS device that causes the pnp transistor to start conducting.
Its input capacitance is significantly smaller than that of the MOSFET device, and the device does not exhibit the secondbreakdown phenomenon. It is faster than the BJT and can operate up to 20 kHz in mediumpower applications. Currently, it is available at ratings as high as 1.2 kV and 400 A. The improvement in its fabrication is promising, and it is expected that it will replace the BJT in the majority of power electronics applications.

58 Chapter 2 Review of Switching Concepts and Power Semiconductor Devices
2.5.2 ThyristorBased Devices
The generic term thyristor refers to the family of power semiconductor devices made of three pn junctions (four layers of pnpn) that can be latched into the on state through an external gate signal that causes a regeneration mechanism in the device. In this section, we will discuss four main members of the thyristor family that are currently used in power electronic circuits: The siliconcontrolled rectifier (SCR), gate turnoff thyristor (GTO), triode ac switch (triac), static induction transistor (SIT), static induction thyristor (SITH), and MOScontrolled thyristor (MCT).
The SiliconControlled Rectifier
The siliconcontrolled rectifier (SCR) is the oldest power controllable device utilized in power electronic circuits, introduced in 1958. Unlike the diode, the SCR can block voltages bidirectionally and carry current unidirectionally. Until the 1970s, when power transistors were presented, the conventional thyristor had been used extensively in various industrial applications. The SCR is a threeterminal device composed of a foursemiconductor pn junction. Unlike the diode, the SCR has a third terminal called the "gate" used for control purposes.
The symbol and iv characteristics for the SCR are shown in Fig. 2.3l(a) and (b), respectively. The ideal switching characteristic curves are shown in Fig. 2.3l(c), where v AK and i A are the voltage across the anodecathode terminals and the current through the anode, respectively.
The latching current is always less than the minimum trigger current specified in the device's data sheet. The holding current is the minimum forward current the SCR can carry in the absence of a gate drive. The forward breakover voltage, V80, is the voltage across the anodecathode terminal that causes the SCR to tum on without the application of a gate current. Reverse avalanche (breakdown) occurs when v AK is negatively large.
The normal operation of the SCR occurs when its gate is used to control the tumon process by injecting a gate current iG to allow the forward current to flow; vAK is positive and can be turned off by applying a negative vAK across it.
It must be noted that once the SCR is turned on, the gate signal can be removed. For this reason, this device is also known as a latch device. The gate current must be applied for a very short time and normally can go up to I 00 rnA. Once the SCR is turned on, it has a 0.52 V forward voltage.
The physical structure of the SCR consists of three pn junctions, as shown in Fig. 2.32. The different doping levels shown are used to help sustain a large block voltage and speed the breakdown process. Under no external bias voltage, the majority carriers diffuse across the junctions and recombine with the minority carriers, resulting in zero net current.
The Off State Generally speaking, thyristor tumoff can be carried out by reversing the anodecathode voltage (i.e., through natural ac commutation), or it can be turned off through forced commutation by switching a previously negatively charged capacitor across the SCR or by the insertion of a series impedance to reduce the forward current below the device's holding current.
In order to tum off the SCR, v AK must be negative ( v AK < 0); of course, the triggering gate signal is immaterial (the presence of iG only increases electron movement across the junction J2). Under this condition, junctions J 1 and J3 are reverse biased and

Max reverse vol~age
Reverse ...amche region
2.5 Available Semiconductor Switching Devices 59
Anode (A)
Cathode (K)
(a)
Latching current ll
Reverse blocking region
Reverse voltage blocking
(b)
Forward current On +carrying (on)
(c)
Forward voltage blocking (oft)
Forward blocking region
__ 1_,; Holding current
Forward breakover voltage
F"~gure 2.31 SCR switching characteristics. (a) Symbol. (b) iv characteristics. (c) Ideal ~itching characteristics.
Anode
Pt
nj
f1 12
P2
Cathode Figure 2.32 Simplified physical structure of the SCR: no biasing.

';1.
;' 60 Chapter 2 Review of Switching Concepts and Power Semiconductor Devices
junction J2 is forward biased. Like two reversebias diodes, the only currents that through the device are the leakage currents, /s1 and Is3 , of junctions J 1 and J3, tively. Notice that the larger the depletion region driven to the low doping level of n1 compared to p 1, as shown in Fig. 2.33(a), the larger the reverse voltage and the the depletion regions of J 1 and J3 become. Junction J2 is forward biased with Tnr"""""" current,IF2, equal to (ls1 = Id. The junction J3 breaks down at lower voltages thanJ1 due to different doping levels, and the large reverse blocking voltage is sustained by junction J1
Typical tumoff times for the SCR range from a few microseconds to I 00 J.LS for low and high voltage ratings, respectively.
The On State To tum on the SCR in the conduction state, a positive anodecathode voltage must be applied (vAK > 0) and a gate current must be injected to initiate the anstate regeneration process. For vAK > 0 and iG = 0, junctions J 1 and J3 are in the forwardbias states and J2 is in the reversebias state. Since n 1 is less doped than the p 2 region, the depletion region grows mostly into the n1 region, as shown in Fig. 2.33(b).
Without a trigger current (iG = 0), the junction J 2 is in the forwardblocking condition, and the only current that flows in the device is the small leakage current, lsz through J2 since it is reverse biased. The forward current remains small until the critical forward break over voltage, V80, is exceeded. At this point, the potential energy of the barrier at junction J 2 increases, accelerating the electronhole pair generation until avalanche breakdown occurs, resulting in the thyristor being switched rapidly into the conduction state. This trigger mechanism should be avoided unless specified as safe by the manufacturer.
Gate triggering is achieved when a small pulse current, iG, is injected at the gate, introducing an avalanche condition across J 2 and forward currents, I Fl and IF3, that flow through J 1 and J3, respectively. In order to explain the thyristor gate firing mechanism, normally the SCR is replaced by the model of two interconnected complementary pnp and npn transistors Q1 and Qz, respectively, as shown in Fig. 2.34.
We consider only the forward blocking and forward conducting states, since in the reverse blocking state the only current that flows is the leakage current of Q1 and Q2
Anode
Depletion r__..._.., region
Cathode
(a)
Anode
 lt
Cathode
(b)
Figure 2.33 Depletion layer (a) under reverse bias ( v AK < 0) and (b) under forward bias (vAK > 0).

2.5 Available Semiconductor Switching Devices 61
p
iA
n
iC2
G ic p iK
n
Figure 2.34 Twotransistor analogy model for the SCR. (a) pn junction
K representation. (b) Transistor circuit (a) (b) representation.
To derive the terminal current relation for the SCR, we tum to the largesignal model of the transistor, called the EbersMoll (EM) model, which is used for both transistor junctions, the npn and the pnp. Figure 2.35(a) and (b) shows the EM models for the npn and pnp transistors, respectively. Here ap1
4 and an denote the forward a, which is close to unity, and aRI and aR2 denote the reverse a, which is normally very small (0.02). For simplicity we will assume aRI and aR2 are zero. The currents Io,Bc and Io.cs represent the diode leakage currents for the basecollector diode of Q2 and the collectorbase diode of Q1 when both transistors are in the active mode. These currents are also known as the collectortobase saturation current while the emitter is opencircuit, Ic802, for the npn transistor, and basetocollector current, I 8c01 , for the pnp transistor. The direction of Ic802 and I8 c01 are opposite to the direction of the leakage currents as shown in Fig. 2.35(a) and (b). Recall that, in the active region, the baseemitter junction is forward biased and the basecollector junction is reverse biased.
First, let us assume there's no gate triggering (ic = 0). Replacing the equivalent EM models of Fig. 2.35(a) and (b) into Fig. 2.35(c), we obtain the twotransistor equivalent circuit model. With simple algebraic manipulation, we can show that
. . IcsOI + Inc02 IK = lA = 1(al +a2)
With ic = 0, the only current that will flow is the leakage current (a1 and a2 are small). Normally a 1 + a2 ~ 1 to keep it off. If a 1 + a2 = 1, the SCR will enter a sustained breakdown, with the anode current limited only by the external circuitry.
To avoid entering the breakdown region, a gate signal is injected, resulting in the following forward current:
(2.1)
4Forward a (aF) is the same as the single transistor's a when operating in the active region, given by a = {31 (I +{3), where f3 is the transistor current gain, f3 = lei 18 .

62 Chapter 2 Review of Switching Concepts and Power Semiconductor Devices
IcB02~ anio.BE IBcOlj CXFJiD,EB iB2 iBI
82 81
aR2ID,BC""'O io,EBj aRI/D.CB""'O
82 + ~ i2=iK iB2 2
2
(c)
Figure 2.35 Complete EbersMoll model for (a) the npn transistor and (b) the pnp transistor. (c) Equivalent circuit model for Fig. 2.34(b).
For the generation process to start, we design the SCR for a 1 + a2 "' I.
EXERCISE 2.7
Show that for the generation process to start, the following relation must be satisfied:

2.5 Available Semiconductor Switching Devices 63
"lien the thyristor was invented, all the schemes for forcecommutating mercury:~a.aifiers of the 1930s soon became thyristorbased circuits with expanded applications ildude ac drives and UPSs. However, because of their cost and low efficiency, r6pl5lor circuits did not penetrate the adjustablespeeddrive application area. Today's waions range from single phasecontrolled rectifier circuits to static var compen~ in utility systems. Because of its limited frequency of operation, the application .t'lllr thyristor has reached saturation.
'lllle schematic symbol and the practical and ideal switching iv characteristics for the 111r111moffthyristor (GTO) are shown in Fig. 2.36(a), (b), and (c), respectively. The *Hc:e is as old as the SCR and was introduced commercially in 1962. Like the SCR, em be turned on with a positive gate signal, but unlike the SCR, applying a negative !lillie signal, as shown in Fig. 2.36(b), turns off the GTO. Once the GTO is turned on or 11111[ 1be gate signal can be removed. The device has a higher onstate voltage than the SCJt at comparable currents. The GTO is normally an off device; it has a very poor .off current gain and it exhibits a secondbreakdown problem at tumoff .
. ,.:_;
K
Ia)
iA
Reverse blocking state
(b)
(c)
Trigger ojj'by negative gate signal
On state
Off' state VAK
lttgore 2.36 GTO switching characteristics. (a) Symbol. (b) iv characteristics. lcl Ideal switching characteristics.

64 Chapter 2 Review of Switching Concepts and Power Semiconductor Devices
Because of its high switching power dissipation, the GTO's frequency of operation is limited to less than 1 kHz, and modern GTO devices are rated at 4.5 kV and at currents as high as 3 kA. The GTO is used in high current and voltage applications, such as voltagefed inverters and induction heating resonant converters.
The Triode AC Switch (Triac)
Like the GTO, the triode ac (triac) switching device was introduced immediately after the SCR. In fact, the triac is nothing but a pair of SCRs connected in reverseparallel on one integrated chip, as shown in Fig. 2.37. It is also known as a bidirectional SCR. The triac's equivalent circuit and the circuit schematic symbol are shown in Fig. 2.37(a) and (b), respectively. The device can be triggered in the positive and negative halfcycle of the ac voltage source by applying a positive or a negative gate signal, respectively. Today's triac ratings are up to 800 V at 40 A. The iv characteristics and the ideal switching characteristics are shown in Fig. 2.37(c) and (d). Use of the triac is considerably limited due to its low rates of rise of voltage and current. Applications include light dimming, heating control, and various home appliances.
The Diac
Finally, we should mention another power device known as the diac, which is essentially a gateless triac constructed to break down at low forward and reverse voltages. The diac is mainly used as a triggering device for the triac.
Static Induction Transistors and Thyristors
In 1987 a device known as the static induction transistor (SIT) was introduced. One year later, the static induction thyristor (SITH) was introduced. The symbols for the SIT and SITH are shown in Fig. 2.38(a) and (b), respectively.
The SIT is a highpower and highfrequency device. The device is almost identical to the JFET, but with its special gate construction it has a lower channel resistance compared to the JFET.
The SIT and SITH are normally on devices and have no reverse voltage blocking capabilities. The SITH device turns off in the same way as the GTO, by applying a negative gate current, but it has a higher conduction drop than the GTO. Finally, both devices are majoritycarrier devices with positive temperature coefficients, allowing device paralleling.
Among the SIT's major applications are audio and VHF/UHF amplifiers, microwaves, AM/FM transmitters, induction heating, and highvoltage, lowcurrent power supplies. It has a large forward voltage drop compared to the MOSFET; hence, it is not normally used in power electronic converter applications. The applications of the SITH include static var compensators and induction heating.
The MOSControlled Thyristor
The simplified equivalent circuit model and the schematic symbol for a ptype MOScontrolled thyristor (MCT) are shown in Fig. 2.39(a) and (b), respectively. Its ideal iv switching characteristic is similar to that of the GTO, as shown in Fig. 2.39(c). The device was commercially introduced in 1988. Like the GTO device, it has a high turnoff current gain.

SCR,
Gp~el
A (anode)
SCR2
K (cathode)
(a)
Off state
Reverse current
carrying
Off
".
2.5
iA
(c)
iA
On
(d)
Available Semiconductor Switching Devices
On
~ Off
On state
Forward current carrying
VAK
A
iA
G
K
(b)
Off state ~
MT1
+
VAK
MT2
Figure 2.37 Triac switching characteristics. (a) Equivalent representation using two SCRs. (b) Symbol. (c) iv characteristics. (d) Ideal switching characteristics.
65
The pMCT is turned on by applying a negative gate voltage (less than 5 V) with respect to the cathode, turning on the pFET and turning off the nFET, initiating the regenerative mechanism in the SCR connected npn and pnp transistors. Similarly, applying a positive gate signal with respect to the cathode initiates the turnoff. The nMCT has the same device structure, except that the pFET and nFET are

66 Chapter 2 Review of Switching Concepts and Power Semiconductor Devices
D (drain)
G (gate)
G (gate)
S (source)
(a)
A (anode)
G(ga~
K (cathode)
Reverse blocking state
(a)
On state
Off state
(c)
A (anode)
/ K (cathode)
(b)
0
G
G 0
Figure 2.38 {a) SIT symbol. (b) SITH symbol.
A
I~ + VAK
~ K
(b)
A
~ I~
K
(d)
Figure 2.39 MCT switching characteristics. (a) Equivalent circuit. (b) pMCT symbol. (c) Ideal switching characteristics. (d) nMCT symbol.
interchanged; hence, a positive and a negative gate signal turns the nMCT on and off, respectively. The schematic symbol for the nMCT is shown in Fig. 2.39(d).
The MCT's current and voltage ratings exceed 1 kV and 100 A and are continuously being improved. The device can be easily connected in series and in parallel combinations to boost power rating.

2.7 Future Trends in Power Devices 67
This device is serious competition for the IGBT. It has the same frequency of operation as the IGBT but with a smaller voltage drop and a higher operating temperature. Intensive efforts are under way to introduce a new improvement in the device, and it is expected to receive wider acceptance in medium and highpower applications.
Other Power Devices
Other devices of the thyristor family include the reverseconducting thyristor (RCT), which is nothing but a builtin antiparallel body diode connected across the SCR to allow cmTent to flow in the opposite direction, and the lightactivated SCR (LASCR), which is used in highvoltage and highcurrent applications such as HVDC systems. Their power ratings go up to hundreds of kilovolts and hundreds of kiloamperes, and they provide complete electrical isolation between the power and control circuits.
CO)IPARISON OF POWER DEVICES
Depending on the applications, the power range processed in power electronics is very widefrom hundreds ofmilliwatts to hundreds of megawatts. Therefore, it is very difficult to find a single switching device type to cover all power electronics applications. Today's available power devices have tremendous power and frequency rating ranges as well as diversity. Their forward current ratings range from a few amperes to a few kiloamperes, their blocking voltage rating ranges from a few volts to a few kilovolts, and their switching frequency ranges from a few hundred hertz to a few megahertz, as illustrated in Table 2.2. This table gives only relative comparison between available power semiconductor devices because there is no straightforward technique that gives a ranking for these devices. Devices are still being developed very rapidly with higher current, voltage, and switching frequency ratings. Figure 2.40 shows a plot of frequency versus power, illustrating these rating ranges for various available power devices.
FLIURE TRENDS IN POWER DEVICES
It is expected that improvements in powerhandling capabilities and increases in the frequency of operation of power devices will continue to drive the research and developments in semiconductor technology. From power MOSFETs to power MOSIGBTs
Table 2.2 Comparison of Power Semiconductor Devices
Device Year made Rated Rated Rated Rated Forward type available voltage current frequency power voltage
Thyristor 1957 6 kV 3.5 kA 500Hz IOOsMW 1.52.5 v (SCR)
Triac 1958 lkV IOOA 500Hz IOOs kW 1.52V
GTO 1962 4.5 kV 3kA 2kHz !OsMW 34 v BJT 1960s 1.2 kV 800A lOkHz IMW 1.53 v
(Darlington) MOSFET 1976 500V 50 A IMHz IOOkW 34 v IGBT 1983 1.2kV 400A 20kHz IOOs kW 34 v SIT 1987 4kV 600A 100kHz IOskW I020V
SITH 1975 4kV 600A 10kHz IOskW 24 v MCT 1988 3 kV 2kV 20100 kHz !OsMW 12 v

68 Chapter 2 Review of Switching Concepts and Power Semiconductor Devices
p As power rating
GTO
MCT
SITH e
BJT
IGBT
Frequency (Hz)
As frequency increases, power
decreases
f
Figure 2.40 Frequency versus power rating ranges for various power devices.
to power MOScontrolled thyristors, the power rating has consistently increased by a factor of 5 from one type to another. Major research activities will focus on obtaining new device structures based on the MOSBJT technology integration to rapidly increase power ratings. It is expected that the power MOSBJT technology will capture more than 90% of the total power transistor market.
The continuing development of power semiconductor technology has resulted in power systems with driver circuits, logic and control, device protection, and switching devices designed and fabricated on a single chip. Such power IC modules are called "smart power" devices. For example, some oftoday's power supplies are available as ICs for use in lowpower applications. There is no doubt that the development of smart power devices will continue in the near future, addressing more power electronics applications.
2.8 SNUBBER CIRCUITS
To relieve switches from overstress during switching, switching aid circuits, known as snubber circuits, are normally added to the power switching device. The objectives of snubber circuits may be summarized as (1) reducing the switching power losses in the main power device in the power electronic circuit, (2) avoiding second breakdowns, and (3) controlling the device's dv I dt or di I dt in order to avoid latching in pnpn devices. There are a wide range of tumon and tumoff snubber circuits available in today's power electronic circuits. These include dissipative and nondissipative passive snubber circuits, and nondissipative active snubber circuits. In dissipative snubber circuits a capacitor is used to slow the device's voltage rise during tumoff, or an inductor to slow the device's current rise during tumon. Figure 2.41(a) and (b) shows popular tumoff and tumon snubber circuits, respectively. In Fig. 2.4l(a), a capacitor is used to reduce the voltage rise dv5wl dt across the switch during turnoff. In Fig. 2.41 (b), a snubber inductor, Ls, is used to slow down the rise of the inductor current

Problems 69
L.,. D
(a) (b)
F"tgt~re 2.41 Passive snubber circuits: (a) tumoff and (b) tumon :!llubber circuits.
With turnon
llld turnoff ~ubbers
Figure 2.42 isw versus vsw switching loci.
di5"/ dt (the inductor current equals the switch current, i5w). Figure 2.42 shows the
switching loci for a practical switch (transistor) with and without snubber circuits. For a detailed discussion on all types of snubber circuits and their design methods, refer to the references at the end of the textbook.
(b) Determine the average output voltage. (c) Determine the average output power delivered to the load.
is ideal and operating at a duty ratio of (d) Determine the average output power supplied by the de source. (e) Determine the efficiency of the circuit.

DCDC Converter

Buck and Boost Converter Topologies
Implementation Using a Switch and a Diode Implementation Using Two Switches
Buck Vo/Vin = D (duty ratio)
Implementation Using a Switch and a Diode Implementation Using Two Switches
Boost Vo/Vin = 1/(1D)
For bidirectional Buck/Boost applications, only topologies using two switches can be used.

Combined Bidirectional Topology
Buck Boost
Boost Buck
11
1
DV
V
low
high
11
low
high
VD
V
Consistent since D2=1D1.
Bidirectional

Example: Excitation from Low Voltage Side
When D1=0.6 and Vlow =40 V, we have Vhigh = 40/(10.6)=100 V.

Example: Excitation from High Voltage Side

136 Chapter 4 Nonisolated SwitchMode dede Converters
EXERCISE 4.2
Repeat Example 4.1 by assuming the switch has a 1.8 V voltage drop across it when
ANSWER 0.64, 0.75, 93.6%
4.2.1 The Buck Converter
Topology and Basic Operation
Figure 4.9(a) and (b) shows the circuit configuration for a buck converter with a and twoswitch implementation. Figure 4.9(c) shows the transistordiode , .. ,1"'"''uwo tion. This topology is known as a buck converter because it steps down the average voltage below the input voltage.
Throughout this chapter to obtain the steadystate characteristic equations, we assume that power switching devices and the converter components are lossless. over, the exact steadystate analysis of these converters requires solving sec:ona

4.2 Continuous Conduction Mode 137
output voltage v0(t) consists of the desired de and the undesired ac components. Prac
tically, the output ripple due to switching is very small (less than 1%) compared to the level of the de output voltage. As a result, we will assume the output ripple voltage is small and can be neglected when evaluating converter voltage gains, i.e., v
0 = ~ In
other words, the ripplefree output voltage assumption is made since the output time constant for the filter capacitor and the output resistor, RC, is very large. Moreover, the analysis will be based on the converter operating in the steadystate condition, i.e., the converter currents and voltages have reached their steadystate values. These assumptions can be summarized and represented mathematically as follows:
1. Since we assume lossless components and ideal switching devices, the average input power, Pin , and the average output power, P 0 , are equal:
(4.4)
2. Since we assume steadystate operation, the inductor current and the capacitor voltage are periodic over one switching cycle, i.e.,
iL(to) = iL(to + T)
vc(to) = vc(tO + T) where t0 is the initial switching time and Tis the switching period.
(4.5a)
(4.5b)
3. Since we assume ideal capacitors and inductors, the average inductor voltage and the average capacitor current are zero:
1 IT+ 10 fc =  ic(t) dt = 0 Tt
0
(4.6)
(4.7)
In fact, Eq. (4.7) is a representation of Faraday's law, which states that voltage time during charging equals voltage time during discharging. This is also known as the voltsecond principle. These two relations suggest that the total energy stored in the capacitor or the inductor over one switching cycle is zero. Finally, throughout the analysis in this chapter, the typical switching waveform for the power devices given in Fig 4.9(d) will be used to represent the switching action of the power switch. For simplicity we set the initial switching time to zero, t0 = 0.
Again, Dis known as the duty ratio or duty cycle, defined in Eq. (4.2). The power transistor is turned on for a period of DT and turned off for the remaining time ( 1  D) T. Depending on whether the switch is turned on or off, the inductor current will be either charging through fin or discharging through the diode, respectively. As a result, there are two modes of operation. We first consider mode 1, when the switch is on, shown in Fig. 4.10(a).
As shown in the figure, when the switch is on, the input voltage, fin, forces the diode into the reverse bias region. To determine the voltage conversion ratio, the average input and output currents, and the output voltage, we use the inductor current as a state variable in the following equation:
Vin = VL + Vo d'
=L~+V dt 0
(4.8)

138 Chapter 4 Nonisolated SwitchMode dede Converters
+
c R
(a)
+
D R Vo
(b)
Figure 4.10 Equivalent circuit modes for the buck converter. (a) Mode I: The power on. (b) Mode 2: The power switch is off.
Equation ( 4.8) can be rearranged as follows:
diL 1 dt = z(Vin Vo)
Integrating Eq. (4.9) from t = 0 tot with JL(O) as the initial condition, we obtain
iL(t) = !(Vin V0 )t+JL(0) L
Equation (4.10) suggests that the inductor current charges linearly with a slope ( Vin  V0 )1 L , where I L ( 0) is the initial inductor current value at t = 0, when switch is first turned on. This equation applies as long as the switch is on. the equivalent circuit model changes when the power switch is turned off at t = resulting in the equivalent circuit of mode 2 shown in Fig. 4.IO(b), during which diode is conducting.
As shown in Fig. 4.10(b), in order for the inductor current to maintain its conttlnlll" ity, the diode is forced to conduct by becoming forward biased so that the diode up" the current in the direction shown. The diode is known asjlyback or tre.~vtne.~LliiP because of the manner in which it is forced to tum on. The resultant equation that scribes mode 2 operation is
diL 1 =V dt L o
Integrating both sides ofEq. (4.11) fort~ DT with iL(DT) as an initial condition, we obtain
where IL (DT) is the initial inductor current when the switch is first turned off.

4.2 Continuous Conduction Mode 139
Equation (4.12) suggests that the inductor current starts discharging at t = DT with the slope ofV/ L , as shown in Fig. 4.11 (a). In steadystate operation we have
(4.13)
Evaluating Eq. (4.10) at t = DT and Eq. (4.12) at t = T and using Eq. (4.13), we obtain the following two relations for IL(O) and IL(DT):
IL(DT) = !(vin V0 )DT+ IL(O) L
IL(O) = _Vo(lD)T+ IL(DT) L
(4.14a)
(4.14b)
The steadystate current and voltage waveforms are shown in Fig. 4.11. I Lmax and I Lmin are the inductor current values at the instants the switch is turned off and on, respectively.
Voltage Conversion
Next we use the preceding relations to derive expressions for the voltage conversion, and average input and output currents. From Eqs. ( 4.14) we obtain
iL
o I
vo =D vin
hmax=h(D~)~=1  _ /Lmin = IL(O) 1 ~ t
0 1 DT 1 T I
(a) I
V;,~~~l~+L~~~~~~F~ t I I I (b) I
IL~~~~7i \~ ~~;/f=it=t ~ t
I I I I (c) 1 I I I I
ioi N ~ +~~~~~~+~~~ t
: (d) : I I I 1c+ I I I
lLmaxlo ~ : )
lLminloV DT ~ ~ t
(e)
( 4.15)
Figure 4.11 Steadystate wavefonns for the buck converter: (a) inductor current, (b) inductor voltage, (c) input current, (d) diode current, and (e) capacitor current.

140 Chapter 4 Nonisolated SwitchMode dede Converters
Hence, the maximum output voltage gain is 1. We should point out that Eq. ( 4.1 be obtained easily by using the voltsecond principle across the inductor, given as follows:
Mode 1
(interval D T)
Mode2
(interval (1 D)T)
(Inductor voltage)( time) + (Inductor voltage)(time) = 0
where VL equals ( vin Vo) and Vo during time intervals DT and respectively.
We can make two observations on the buck voltage gain equation V0
= D v. First, since all the converter components (L, C, D, Q) are ideal, they don't any power, resulting in 100% voltage efficiency. Second, the average input and voltage ratio has a linear control characteristic curve, as shown in Fig. 4.12. By ing the value of the duty cycle, D, we can control the average output voltage to desired level.
Average Input and Output Currents
The input current, iin, as illustrated in Fig. 4.1l(c), with an average value of /in, given by
fin= !fTiin(t) dt To
Since iin = iL in mode 1, we substitute for iL(t) from Eq. (4.10), and by evaluating integral between t = 0 and t = D T, we obtain
1 lin=
2L(Vin V0 )D
2T+IL(O)D
Using Eq. (4.14b), we obtain
where !Lmax and /Lmin represent IL(DT) and IL(O), respectively. Similarly, by inspection, the average output current is given by
J = J = ]Lmin + hmax _ Vo 0 L 2  R
0.5
if+++ D Figure 4.12 Ideal output control characteristic 0.5 curve for the buck converter.

4.2 Continuous Conduction Mode 141
From Eqs. (4.14) and (4.20) we can solve for the maximum and minimum inductor currents, to obtain
= (1+(1D)TJ ]Lmax DVin R 2L
_ (1 (1D)TJ ]Lmin DVin R 2L
Substituting these equations in Eqs. ( 4.19) and ( 4.20), we obtain
Hence, the current gain is given by
I = DVin o R
D2Vin Jin=~
(4.21)
(4.22)
(4.23)
This relation can be obtained by equating the average input and output power, to yield
From Eqs. (4.15) and (4.23), it is clear that the current and voltage relations for the converter are equivalent to a de transformer model with a ratio of D, as shown in Fig. 4.13. The sinusoidal curve and straight line drawn across the transformer windings indicate that the transformer is capable of transferring ac and de, respectively.
Critical Inductor Value
It is clear that for I Lmin ;;t:. 0 , the converter will operate in the continuous conduction mode (cern). To find the minimum inductor value that is needed to keep the converter in the cern, we set JLmin to zero and solve for L:
I :D
+
_ (1 (1D)TJ _ ]Lmin  D vin R 2L  0
/()
(1D) Lcrit = 2 TR (4.24)
+
Figure 4.13 Equivalent circuit representation for the buck converter, referred to as a dede transformer.

142 Chapter 4 Nonisolated SwitchMode dede Converters
where Lcrit is the critical inductance minimum value for a given D, T, and R converter enters the discontinuous conduction mode ( dcm) of operation.
Output Voltage Ripple
Since we have assumed that the output voltage has no ripple, the entire ac current from the inductor passes through the parallel capacitor, and only de rent is delivered to the load resistor. In practice, the value of the output is an important design parameter since it influences the overall size of the de converter and how much of the switching frequency ripple is being Having said that, it is design practice to choose a larger output capacitor in to limit the ac ripple across V
0 Theoretically speaking, if C ~ oo, the
acts like a short circuit to the ac ripple, resulting in zero output voltage we assume Cis finite, then there exists a voltage ripple superimposed on the age output voltage. In order to derive an expression for the capacitor ripple age, we first obtain an expression for the capacitor current, which is given by following relation:
As a result, the initial capacitor current at t = 0 is given by
and at t = DT,
Jc(DT) = JL(DT) 10
= + (/Lmax; /Lmin)
The resultant capacitor current and voltage are shown in Fig. 4.14.
DT T
DT l+DT 2
T
Figure 4.14 Capacitor current and voltage waveforms.

4.2 Continuous Conduction Mode 143
The instantaneous capacitor current can be expressed in terms of /).I from Eqs. ( 4.21) and ( 4.22), as shown in the following equations:
i {t) = ILmax ILmin! _ ILmax ILmin = /).I! _/).I e DT 2 DT 2 0 $ t < DT (4.25a)
I I . IL IL i (!} = _ Lmax Lmm{tDT) + max mm e (I D)T 2
= /).I (tDT)+/).I (I D)T 2 0$ t < DT (4.25b)
where /).I= ( Vin(I D)TD)/ L.
From the capacitor voltagecurrent relation, ie = C(dv/ dt), the capacitor voltage, vel(!), can be expressed by the following integral fort 2::0:
Vel (t) = .!.Jt ie dt + Ve(O) c 0
where Ve(O) is the initial capacitor voltage at t = 0. Substituting for ie(t) from Eq. (4.25a), we obtain the following equation:
Vel(!)= bf: (~~~~I) dt+ Ve(O) Evaluating this integral yields
1 /).I t2 /).I v l(t) =  t+ v (0) e CDT2 2C e 0$ t < DT
Similarly, for t 2:: D T, the capacitor voltage is given by
vc2(t) = .!.Jt ie dt + Ve(DT) C DT
(4.26)
where Ve(DT) is the initial capacitor voltage when the switch is turned off at t = DT. From Eq. (4.27) vc2(t) is given by
v (t) = /).I (tDT)2
+ /).I(tDT)+ V (DT) DT$t < T e
2 C(l D)T 2 2C e (4.27)
Since the capacitor voltage is in the steady state, we have v e 1 (t = D T) =
ve2(t = DT) and vel (0) = ve2(T), resulting in the following boundary conditions for the capacitor voltage:
Since the average capacitor voltage is V0
, then we have in general
V" ~ ~[ J,"' vc~ (t) dt + [ v "'(t) d~ Substituting for vel and ve2 from Eqs. (4.26) and (4.27), we obtain
V = /).I (l2D)T+ V (0) o 12C e (4.28)

144 Chapter 4 Nonisolated SwitchMode dede Converters
Substitute for aJ = (DVin(lD)T)/L in Eq. (4.28) to yield
V (O) = DV [1 (1D)(l 2D)T2] c m 12CL (4.29}
Hence, the capacitor initial values at t = 0 and t = D T are equal, as expected since the capacitor current is symmetrical. Since the peak capacitor voltage occurs when the inductor current is zero, we have the capacitor minimum voltage occurring at t = DT/2, which is obtained from Eq. (4.26):
V = 1 ai (DT)2
aJ(DT) + V (O) c,min C2DT T  2C T c
(4.30a)
= _aJ DT+ V (0) 8C c
and the maximum capacitor voltage occurring at t = (1 + D)T/2 as obtained from Eq. (4.27):
V =  aJ ((1 +D) T D r)2 + M ((1 + D) T Dr)+ V (D T) c,max 2C(lD)T 2 2C 2 c
(4.30b)
= :~(1D) + Vc(DT) Substituting for Vc(O) from Eq. (4.29) and using aJ = (DVin(lD)T)IL, it can be shown that vc, min and vc, max are expressed as follows:
V . =V[1(lD)(2 D)T2] c,mm o 24CL
v = v [' + (lD2)r2] c,max o 24CL
Hence, the variation in the capacitor peak voltage is given by
and from Eqs. (4.31) we obtain
av =~(1D) c 8LCj2
(4.31a)
(4.31b)
Sometimes it is useful to express the ratio of the ripple to the output voltage,
ave_ 1D Vo  8LCJ2
(4.32)
This term is known as the output voltage ripple and represents the regulation. As expected, when the filtering capacitor and the frequency increase, the voltage ripple decreases.
Using Capacitor Charge to Evaluate !i Vc
Another useful way to evaluate the expression for a vc without having to obtain the exact expression for vc(t) is to use the total charge, Q, deposited on the capacitor current interval. Figure 4.14 shows the waveform for ic and vc with areas of positive

4.2 Continuous Conduction Mode 145
charge(+) and negative charge(). Because of waveform symmetry, t = DT/2 and t = (1 + D) T 12 represent the ic zero crossing times when the capacitor voltage is minimum, Vc, min , and maximum, Vc, max , respectively. Hence, the capacitor voltage ripple is Ll Vc. The total charge stored in the capacitor between t = D T 12 and t = ( 1 +D) T 12 is obtained from the following equation:
dQ = cdvc dt dt
So the total charge Q between capacitor current ic zero crossings (DT/2 s t < (1 + D)T/2) is given by
(4.33)
However, since the total charge is related to the current according to the relation
then we have
._dQ l
dt
LlQ =  i dt =Area under the curve 1 I(I+D)T/2
T/2 DT/2
= ~( 1 + DT P..r)~Lll 2 2 2 2
= ~I~Lli 222
From Eqs. (4.33) and (4.34), we obtain
LlVc = g~LlJ
Substituting for Lll = (D Vin(l D)T)/ L, we obtain
LlVc 1 D 2 =T
Vo 8LC
_ 1D 8LCj2
(4.34)
(4.35)
Consider a buck converter with the following circuit parameters: Vin = 20 V, V0
= 15 V, and lo = 5 A, for f= 50 kHz. Determine: (a) D, (b) Lcrit, (c) maximum and minimum inductor currents for L = lOOLcrit, (d) average input and output power, and (e) capacitor voltage ripple for C = 0.47 ~tF.
(a) D = 0.75
(b) Using R = 3 .Q and T = 20 JLS, the critical inductor value is given by
(1D) Lcrit =  2 TR = 7.5 ~tH

146 Chapter 4 Nonisolated SwitchMode dede Converters
(c) For L = lOOLcrit = 750 JLH = 0.75 mH, we have
I . = D V. (.!. _ ( 1 D) T) Lmm m R 2L
= (0.75)(20)G3.33 X I03)
]Lmin = 4.95 A
fLmax = (0.75)(20)G + 3.33 X I03)
]Lmax = 5.05 A
(d) Since it is an ideal converter, the average output and input powers are given by
(e) The capacitor voltage ripple is given by
LlVo 1D Vo  8LCj2
(1 0.75)
8(0.75 mH)(0.47 JLF)(50 x 103)2
dV0 v = 0.035 = 3.5% 0
EXAMPLE4.3
Design a buck converter with the following specifications: Ll V0 / V0 = 0.5%, Vin = 20 V, P0 = 12 W, f= 30kHz, and D = 0.4.
SOLUTION In order to design this converter, we need to calculate the values for L, C, and R. The output voltage is given by
Hence, the output current is
The output resistance is
8 R = = 5.33 n
1.5
The critical inductance for cern is given by
1D Lent =2TR
= ( 1
20.4 )(30: 103 } 33 = 53.3 JLH

4.2 Continuous Conduction Mode 147
Let us select L = 600 JLH. Based on this value, the maximum and minimum inductor currents are given by
1 = DV. (.!. + (1D)T) Lmax m R 2L
The ripple voltage is given by
Solving for C,
= (0.4)(20{5.~3 + 0.0167) = 1.63 A
lLmin = (0.4)(20)(5 .~3  0.0167) = 1.37 A
L1Vo 1D  = =0.005
Vo 8LCj2
C= 1D (8Lj2)0.005
C = 27.78 JLF
Redesign Example 4.2 to achieve an output ripple voltage not to exceed I% and an inductor current ripple not to exceed I 0% at the average load current.
0.15 mH, 8.33 JLF, 3 fl
Determine the diode and transistor average and rms current values for Exercise 4.3.
1.25 A, 3.75 A, 2.5 A, 4.33 A
Show that the expression for the peak capacitor voltage at t = (I + D)T /2 is as given by Eq. (4.31b).
The Boost Converter
Basic Topology and Voltage Gain
Other possible switch and transistordiode arrangements are shown in Fig. 4.15(a) and (b), respectively. This topology is known as a boost converter since the output voltage is higher than the input, as will be shown in this section.

148 Chapter 4 Nonisolated SwitchMode dede Converters
c
(a)
ic
c
(b)
io
R
+
+
v,
Figure 4.15 Boost converter. (a) Twoswitch implementation. (b) Transistordiode implementation.
Similar to the case for the buck converter, we assume all the converter components are ideal and the transistor switching waveform is as shown in Fig. 4.9(d)_ When the switch is turned on, the equivalent circuit of mode 1 is shown in Fig. 4.16(a). This is a charging interval, and the voltage across the inductor is Vin and iL(t) is given by
iL(t) = ~ Vint + !L(O) 0::; t < DT
where JL(O) is the initial inductor current value at t = 0 .When the switch is turned off at t = DT, the resultant equivalent mode 2 circuit is shown in Fig. 4.16(b).
The inductor voltage is Vin V0 , and iL(t) is given by
s
(a)
D
c
(b)
+
R v"
DT::; t < T
Figure 4.16 Equivalent circuit modes for the boost converter. (a) Mode 1: The switch is on. (b) Mode 2: The switch is off.

4.2 Continuous Conduction Mode 149
Evaluating Eqs. ( 4.36) and ( 4.3 7) at t = D T and t = T, respectively, and using the fact that IL(T) = IL(O), we obtain
(4.38a)
(4.38b)
From Eqs. (4.38a) and (4.38b), the resulting voltage conversion is given by
vo 1 V;n 1D
(4.39)
Hence, the voltage gain is always greater than 1. Also from Eqs. (4.38), the inductor ripple current is given by
1 = V DT L m
Substituting for V;n from Eq. ( 4.39), we obtain
!1! = ~ VOD(l D)T L
Key current and voltage waveforms are given in Fig. 4.17.
Average Input and Output Currents
(4.40a)
(4.40b)
The input current is the same as the inductor current as shown in Fig. 4.17(a). Hence, the average input current by inspection is given by
f. = /Lmax + /Lmin m 2
(4.41)
The average output current is the same as the average diode current and is given by
(4.42)
Since we assume an ideal converter, the average input and output powers must be equal. Using Eqs. (4.41) and (4.42), we get
resulting in
f;n Vo 10 V;n 1D
(4.43)
As with the buck converter, the inputoutput current and voltage ratios are equivalent to a de transformer with a transformer mode ratio equal to 1 I ( 1 D) , as shown in Fig. 4.18.

150 Chapter 4 Nonisolated SwitchMode dede Converters
iL
IL(DT)~~ 1[(0)~
o :or :r
VL I I I
(Vo Vin )
(a)
(b)
I I I
I I I I I
I I
f[(DT) t I : +~+__________ ~: ~"~ t
(c)
hmax lo
10 1'
(d)
Figure 4.17 Current and voltage waveforms for the boost converter.
Figure 4.18 Equivalent transformer circuit representation for the boost converter.
Using Eqs. (4.38) and (4.42), we can solve for the maximum and minimum inductor current values:
( 1 DT)
IL(O) = ILmin = vin R(lD)2 2L (4.44a)
I (DT) I  V ( 1 + DT) L  Lmax in R(lD)2 2L
(4.44b)
~

4.2 Continuous Conduction Mode 151
For positive values of JLmax and /Lmin, the converter will operate in the continuous conduction mode. To solve for the minimum critical inductor value that will keep the converter in the cern, we set /Lmin to zero:
/Lmin = 0
Under this boundary condition, the critical inductor value is given by
(4.45)
Output Ripple Voltage
It is clear from Fig. 4.17 that when the diode is reverse biased, the capacitor current is the same as the load current. Since we assume the load current is purely de, the capacitor current is given by
ic = Jo
ic=iLJo
0 '5:.1 < DT
DT'5:.t < T
The capacitor current waveform is shown in Fig. 4.17(d) and redrawn in Fig. 4.19 along with the capacitor voltage waveform. Mathematical expressions for ic can be obtained directly from this figure.
The current iAt) is expressed mathematically as
ic(t) = !!.I (tDT)+I (DT) (1D)T c
0
/() f...1 I I I I I I I I I I
DT'5:.t'5:.T
Vc(O) t 
Vr, ========t =========L_l L\vc= l"gT Vc(DT) I I I I I I I I I I I I I I
0 DT T
Figure 4.19 Capacitor current and voltage waveforms for the boost converter, assuming
/Lmin > /o
(4.46)

152 Chapter 4 Nonisolated SwitchMode dede Converters
where Ic(DT) is the initial ic(t) at t = DT. The capacitor voltage for 0:::; t < DT given by
vc(t) = .!.ft I0
dt + Vc(O) c 0
I = ~t+ Vc(O)
where Vc(O) is the initial capacitor voltage at t = 0. At t = D T we have
Since the average capacitor voltage is V0
, we can solve for Vc(O) and Vc(DT) as
IODT Vc(O) = Vo + 2C