Resonant Pulse Converters
Chapter 11 (or 8)
Prepared by Dr. Mohammad Abdul Mannan Assistant P f A i t t Professor, Department of EEE D t t f American International University - Bangladesh

Disadvantages of PWM Controlled Switching Devices in Converters
The disadvantages of PWM controlled switching devices in converters are as follows: 1. The devices are turned “on” or “off” at the load current with a high di/dt value, 2. The it h 2 Th switches are subjected t a hi h lt bj t d to high-voltage stress, t 3. The switching power loss of a device increases linearly with the switching frequency. 4. The electromagnetic interference (EMI) is produce due to high di/dt and dv/dt in the converter wave forms. The above mentioned disadvantages can be eliminated or minimized if the switching devices are turned “on” and “off” when th voltage across a d i and/or it current b h the lt device d/ its t becomes zero.

The inverter in which the voltage and current are forced to pass through zero crossing by using an LC-resonant circuit is circuit, called calling a resonant pulse inverter.

Classification of Resonant Pulse Inverters
The resonant converters can be classified into eight types: 1. Series-resonant inverters, 2. Parallel-resonant inverters, 3. Class E resonant converters, 3 Cl t t 4. Class E resonant rectifiers, 5. Zero-voltage-switching (ZVS) resonant inverters, 6. Zero-current-switching (ZCS) resonant inverters, 7. Two-quadrant zero-voltage-switching (ZVS) resonant inverters, and 8. Resonant dc-link inverters.

Series Resonant Circuit
A series resonant circuit is shown in Fig 11-11 Fig. 11 11. The input voltage is a square wave whose peak fundamental component is Vi(pk) = 4Vs/π, and its rms value is Vi = 4Vs/√2π. Using the voltage divider rule in frequency domain, the voltage gain is given by: RV ( jω) i Vo( jω) = 2 ωo =1/ LC LC =1/ωo R + jωL − (1/ jωC) Vo 1 G ( jω ) = ( jω ) = Qs = ωo L / R u = (ω /ωo) V 1+ j(ωL / R) + j /(ωRC) i G( j ω ) = 1 1+ jQs[u − (1/ u)]

If ω =ωo then u =1 Thus G( jω) =1 max

Parallel Resonant Circuit
A parallel resonant circuit is shown in Fig 11-15 Fig. 11 15. The input current is a square wave whose peak fundamental component is Ii(pk) = 4Is/π, and its rms value is Ii = 4Is/√2π. Using the current divider rule in frequency domain, the voltage gain is given by: V R Z ( jω ) = o ( jω ) = I 1+ j(R /ωL) + j(ωRC) i 2 ωo =1/ LC LC =1/ωo Qs = ωo RC = R /ωo L = R C / L u = (ω /ωo)
Z ( jω ) = 1 1+ jQ p[u − (1/ u)]

If ω =ωo then u =1 Thus Z ( jω) =1 max

Series Resonant Inverter
The commutating components and switching device are placed g p g p in series with the load to form an underdamped circuit. The current through the switching devices falls to zero due to the natural characteristics of the circuit circuit. It is said to be self-commutated for thyristor. It produces an approximately sinusoidal waveforms at high output frequency (from 200 Hz to 100 kHz) and i commonly is used in relatively fixed output voltage applications (e.g. induction heating, sonar transmitter, fluorescent lighting, or g, , g g, ultrasonic generators).

Classification of Series Resonant Inverters
Depending on the connections of the switching devices and load, series resonant inverter may be classified into two g categories: 1. Series resonant inverters with unidirectional switches, and 2. Series resonant inverters with bidirectional switches.

Series Resonant Inverters with Unidirectional Switches
Fig. 11-1(a) Fig 11 1(a) switch the circuit diagram of a simple series resonant inverter using two unidirectional thyristor switches. The series resonant circuit formed by L C and load (assumed L, C, resistive) must be underdamped. That is, R2 < 4 L C In order to discuss of operation of this circuit, at first it is assumed that the capacitor initially is charged –Vo volts. The operation of this circuit can be divided into three modes modes.

Mode 1: This mode begins when T1 is fired. The current flows through T1 and the load. The current starts to increase and capacitor starts t di h it t t to discharge. Th current b The t becomes maximum i when the voltage across capacitor becomes zero. When capacitor starts to charge up, the current stars to falls. The current reach to zero when capacitor charged up to source voltage. Since current falls to zero T1 is turned off off.

Mathematical Expression of Mode 1: The instantaneous load current for this mode is given by: di L 1 + Ri + 1 ∫ i dt + v (t = 0) =Vs (11.2) 1 C 1 c1 dt With initial conditions i1(t = 0) = 0 and vc1(t = 0) = - Vc. Since the circuit is underdamped, the solution of Eq. (11.2) yields: i (t) = A e−(R / 2L)t sin ωrt (11.3) 1 1 Where ωr is the resonant frequency and ω = 1 − R2 (11.4) r LC 4L2 The constant, A1, in Eq. (11.3) can be evaluated from the initial condition: di 1 = Vs +Vc = A i (t) = Vs +Vc e−αt sinω t (11.5) r 1 1 ωr L ωr L dt t =0

α = R (11.6)
2L

The time tm when the current i1(t) in Eq. (11.5) becomes maximum can be found from the condition ⎡ω ⎤ di 1 −αtm −αtm tm = ω tan−1⎢⎢ αr ⎥⎥ (11.7) 1 =ω e cosωrtm −αe sinωrtm = 0 r ⎢ ⎥ r ⎣ ⎦ dt t The Th capacitor voltage can b f it lt be found f d from v (t) = 1 ∫0i (t)d −Vc dt c1 C 1 (11.8) vc (t ) = −(Vs −Vc )e−αt (α sin ωrt +ωr cosωrt) /ωr +Vs This mode is valid for 0 ≤ t ≤ t1m(=π/ωr) and ends when i1(t) becomes zero at t1m. At the end of this mode, i1(t = t1m) = 0 and −α /ωr απ v (t = t ) =V = −(Vs +Vc )e +Vs = −(Vs +Vc )e− z +Vs (11.9) c1 1m c1 z = απ ωr

Mode 2: During this mode, T1 and T2 are off. Redefining the time origin, t = 0, at the beginning of this mode, this mode is valid f 0 ≤ t ≤ t2m. lid for i (t ) = 0, v (t ) =V ; 2 c2 c1 v (t = t ) =V =V c2 2m c2 c1

Mode 3: This mode begins when T2 is fired. The current flows through T2 and the load in reverse direction. The Th current starts t i t t t to increase and capacitor starts t di h d it t t to discharge. The current becomes maximum when the voltage across capacitor becomes zero. p When capacitor starts to charge up, the current stars to falls. The current reach to zero when capacitor charged up to -Vc. Since current falls to zero T2 is turned off off.

Mathematical Expression of Mode 3: The instantaneous load current for this mode is given by: di L 3 + Ri + 1 ∫ i dt + vc (t = 0) = 0 (11.10) 3 C 3 dt With initial conditions i3(t = 0) = 0 and vc3(t = 0) = - Vc1. The solution of Eq. (11.10) yields: V i (t) = c1 e−αt sinωrt (11.11) 3 ωr L t The capacitor voltage can be found from v (t) = 1 ∫0i (t)dt −V c3 c1 C 3 v (t ) = −V e−αt (α sin ωrt +ωr cosωrt ) /ωr (11.12) c3 c1 This mode is valid for 0 ≤ t ≤ t3m(=π/ωr) and ends when i3(t) ( () becomes zero at t3m. At the end of this mode, i3(t = t3m) = 0 and −απ /ωr v (t = t ) =V =Vc = −V e = −V e− z (11.13) c3 3m c3 c1 c1

Solving Eqs. (11.9) and (11.13), we obtained: V =Vc +Vs (11.16) c1 Eq. (11.16) indicates that under steady state conditions, the peak E (11 16) i di h d d di i h k values of positive current and of negative current through the load are the same. To avoid a short-circuit, the load current i1(t) must be zero and T1 must be turn off before T2 is fired. Therefore, the available off-time t2m (=toff), known as the dead zone, must be greater than the turn-off time of thyristors, tq.

π − π =t >t ωo ωr off q f o ≤ f max = 1 2(tq +π /ωr )

ωo is the frequency of the output voltage in rad/s.

The resonant inverter circuit in Fig. 11-1(a) is very simple. It has the following disadvantages: (i) the power flow from the dc supply i di l is discontinuous and (ii) th d supply will h ti d the dc l ill have a hi h high peak current and would contain harmonics. The first disadvantage can be removed by closely coupled g y y p inductors as shown in Fig. 11-2. When T1 is fired and current i1(t) begins rise, the voltage across L1 will be positive with polarity as shown The induced voltage on shown. L2 will now add to the voltage of C in reverse biasing T2; and T2 will be turned off. The result is that firing of one thyristor will turn off the other, even before the load current reaches to zero.

The second disadvantage can be overcome in a half-bridge configuration as shown in Fig. 11-3, where L1 = L2 and C1 = C2. The Th power i d is drawn f from th d source d i b th h lf the dc during both half-cycles l of output voltage. One half- of the load current is supplied by capacitor C1 or C2 and the other half by the dc source. A full-bridge inverter, which allows higher output power, is shown in Fig. 11-4. The resonant frequency and available dead-zone depend on the load and for this reason; resonant inverters are most suitable for fixed-load application. pp

Series Resonant Inverters with Bidirectional Switches
Disadvantages the resonant inverters with unidirectional switches: 1. The power devices have to be turned on in every halfcycle of output voltage. l f t t lt 2. Limits the inverter frequency and the amount of energy transfer from the source to the load. 3. The thyristors are subjected to high peak reverse voltage.

The above mentioned disadvantages can be removed by using bidirectional switch as shown in Fig. 11-7.

When T1 is fired, a resonant pulse of current flows and T1 is selfcommutated at t = t1. However resonant oscillation continuous through diode D1 untill the current falls again to zero at the end of a cycle. The waveforms for currents and capacitor voltage are as shown in Fig. 11-7(b). s s ow g. 7(b). If the conduction time of the diode is greater than the turn off time of the thyristor there is no need of a dead zone and the thyristor, output frequency, fo, is the same as the resonant frequency, fr, ω fo = fr = r ; f o ≤ f max = 1 2π 2tq

The diode D1 should be connected as close as possible to the thyristor to reduce any stray inductance in the loop formed by T1 and D1. Any inductance in the diode path would reduce the net reverse vo ge c oss e e voltage across the terminal of T1 and T1 may not turn off. o d y o u o . To overcome this problem, a reverse conducting thyristor (RCT) is normally used. An RCT is made by integration of a thyristor and a fast recovery diode into a single silicon chip and RCTs are fast-recovery chip, ideal for series resonant inverters.

The circuit diagram for the halfbridge version in Fig 11-8(a) Fig.

The inverters can be operated in two different modes: (i) Nonoverlapping: In this mode, the firing of a thyristor is delayed until the load current oscillation through a diode has been completed as in Fig. 11-8(b). (ii) Overlapping: In this mode, a thyristor is fired, while the Ove pp g: s ode, y s o s ed, w e e current in the diode of the other part still conducting, as shown in Fig. 11-8(c). Although overlapping operation increases the output frequency frequency, the output power is increased.

The full-bridge configuration is shown in Fig. 11-9. The maximum frequency of resonant inverters are limited due to the turnoff or commutation requirements of thyristors, typically 12 to 20 µs, whereas transistor, which require only a microsecond or less, can replace the thyristors. The inverter can operate at the resonant frequency. frequency A transistorized half-bridge inverter is shown in Fig. 11-10 with transformer-connected load. Transistor Q2 can be turned on almost instantaneously after transistor Q1 is turned off.

Quasi-Square Voltage Control for Series Resonant Inverter A b id topology as shown i Fi 11 12( ) can b applied to bridge l h in Fig. 11-12(a) be li d achieve the output voltage control. From this circuit a quasisquare wave is obtain. q The switching frequency fs is kept constant at the resonance frequency fo. By switching two devices simultaneously a quasi-square wave as shown in Fig. 11-12(b) can be obtained.

The rms fundamental input voltage is given by:
4V V = s cosα (11.36) i 2π

By varying α from 0 to π/2 at a constant frequency, the voltage Vi can be controlled from 4Vs/(π√2) to 0. ( )

Parallel Resonant Inverter
A parallel is supplied from a current source so that the circuit offers high impedance to the switching current. A parallel resonant inverter is shown in Fig. 11-16(a). Inductor Le acts as a current source and C is the resonant element. Lm is the mutual inductance of the transformer and acts as resonating inductor. A constant current is switched alternatively into the resonant y circuit by transistors Q1 and Q2 .

The instantaneous output voltage for dvo vo 1 C + + ∫ vodt = I s dt R Lm this circuit is given by: g y With initial condition vo(t=0) = I vo(t) = s e−αt sinωrt (11.40) 0 and iL(t=0) = 0. This solution ( ) ωrC of above equation is: ωr = 1 − 1 (11.41) α= 1 LmC 4R2C 2 2RC The voltage vo(t) in Eq. (11.40) becomes maximum at tm given by: 1 tan−1⎡⎢ωr ⎤⎥ (11.7) tm = ω ⎢α ⎥ ⎢ ⎥ r ⎣ ⎦ Approximately maximum time at tm is given by: π tm = ω r

Quasi-Square Current Control for Parallel Resonant Inverter A bridge topology as shown in Fig. 11-12(a) can be applied to g p gy g ( ) pp achieve the output voltage control. From this circuit a quasisquare wave of current is obtain. The switching frequency fs is kept constant at the resonance frequency fo. By switching two devices simultaneously a quasi-square wave as shown in Fig. 11.18(b) can be obtained.

The rms fundamental input current is given by: 4I 4I I = s cosα (11.45) i 2π By varying α from 0 to π/2 at a constant frequency, the voltage Ii can be controlled from 4Is/(π√2) to 0. This concept can be extended to HVDC applications in which the ac voltage is converted to the dc voltage and then converted back to ac. The transmission is normally done at a constant dc current Idc. A single-phase version is shown in Fig. 11-18(c).