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C HAPTER 2
Sinusoidal Approximations
n this chapter, the properties of the series, parallel, and other resonant converters are investigated using the sinusoidal approximation. Harmonics of the switching frequency are neglected, and the tank waveforms are assumed to be purely sinusoidal. This allows simple equivalent circuits to be derived for the bridge inverter, tank, rectifier, and output filter portions of the converter, whose operation can be understood and solved using standard linear ac analysis. This intuitive approach is quite accurate for operation in the continuous conduction mode with a high-Q response, but becomes less accurate when the tank is operated with a low Q-factor or for operation in or near the discontinuous conduction mode. The important result of this approach is that the dc voltage conversion ratio of a continuous conduction mode resonant converter is given approximately by the ac transfer function of the tank circuit, evaluated at the switching frequency. The tank is loaded by the effective output resistance, nearly equal to the output voltage divided by the output current. It is thus quite easy to determine how the tank components and circuit connections affect the converter behavior. The influence of tank component losses, transformer nonidealities, etc., on the output voltage and converter efficiency can also be found. It is found that the series resonant converter operates with a step-down voltage conversion ratio. With a 1:1 transformer turns ratio, the dc output voltage is ideally equal to the dc input voltage when the transistor switching frequency is equal to the tank resonant frequency. The output voltage is reduced as the switching frequency is increased or decreased away from resonance. On the other hand, the parallel resonant converter is capable of both step-up and step- down of voltage levels, depending on the switching frequency and the effective tank Q-factor. Switching loss mechanisms are also considered in this chapter. “Zero voltage switching” is a property that can be obtained in resonant converters whenever the tank presents a lagging (inductive) load to the switch network. This occurs for operation above resonance in the series
I
Principles of Resonant Power Conversion
resonant converter, and it can lead to elimination of the switching loss which arises from the switch output capacitances. Likewise, “zero current switching” can be obtained when the tank presents a leading (capacitive) load to the switch network, as in the series resonant converter operation below resonance. This property allows natural commutation of thyristors, and elimination of switching loss mechanisms associated with package and other parasitic inductances.
2. 1. First Order Network Models
Consider the class of resonant converters which contain a controlled switch network N (^) S and drive a linear resonant tank network N (^) T. The latter in turn is connected to an uncontrolled rectifier N (^) R , filter N (^) F and load R, which is illustrated in Fig. 2.1. Many well-known converters can be represented in this form, including the series, parallel, LCC, et al.
controlled switch network
low pass filter
resonant tank network
Õ
power input
N S
Õ I Õ
R
uncontrolled load rectifier
N T N R N F
i (^) R
vR
V
Õ
i (t)
V
iS
vS
g
g
Fig. 2.1. A class of resonant converters which consist of cascaded switch, tank, rectifier, and filter networks. A series resonant converter example is shown.
In the most common modes of operation, the controlled switch network produces a square wave voltage output vS(t) whose frequency f (^) S is close to the tank network resonant frequency f 0. In response, the tank network rings with approximately sinusoidal waveforms of frequency f (^) S. The tank output waveform vR or iR is then rectified by network N (^) R and filtered by network N (^) F , to produce the dc load voltage V and current I. By changing the switching frequency f (^) S (closer to or farther from resonance f 0 ), the magnitude of the tank ringing response can be modified, and hence the dc output voltage can be controlled.
Principles of Resonant Power Conversion
The fundamental component is
v (^) S1 (t) = 4 Vg π sin (2π fSt) (2-2)
which has a peak amplitude of (4/π) times the dc input voltage Vg, and is in phase with the original square wave v (^) S(t). Hence, the switch network output terminal is modeled as a sinusoidal voltage generator, v (^) S1 (t). It is interesting to model the converter dc input. This requires computation of the dc component Ig of the switch input current ig(t). The switch input current ig(t) is equal to the output current iS(t) when the switches are in position 1, and its inverse -iS(t) when the switches are in position 2. Under the conditions described above, the tank rings sinusoidally and i (^) S(t) is well approximated by a sinusoid of some peak amplitude I (^) S1 and phase ϕS:
iS(t) ≅ IS1 sin (2πfSt – ϕS) (^) (2-3)
The input current waveform is shown in Fig. 2.5.
The dc component, or average value, of the input current can be found by averaging ig(t) over one half switching period:
i (^) g = T^2 S
i (^) g(t)dt 0
T (^) S /
≅ T^2
S
IS1 sin (2πfSt - ϕS ) dt 0
T (^) S /
=^2 π I (^) S1 cos ϕS (2-4) Thus, the dc component of the converter input current depends directly on the peak amplitude of the tank input current IS1 and on the cosine of its phase shift ϕS. An equivalent circuit for the switch is given in Fig. 2.6. This circuit models the basic energy conversion properties of the switch: the dc power supplied by the voltage source Vg is converted into ac power at the switch output. Note that the dc power at the source is Vg times the dc component of ig (t), and the ac power at the switch is the average of vS(t) times i (^) S (t). Furthermore, if the harmonics of v (^) S(t) are negligible, then the switch output voltage can be represented by its fundamental, a sinusoid vS1(t) of peak amplitude 4Vg/π.
i (t)S
ω (^) St
ϕS
i (t)g
Fig. 2.5. Switch terminal current waveforms i (^) g(t) and iS(t).
Chapter 2. Sinusoidal Approximations
v (t) = sin(2πf t)
Õ
__ π^2 I (^) S1cos( ϕ )S +– S1 S
i (^) S1(t) ≅ Ι (^) S1sin(2πf tS − ϕ )S
_____4V
π
g
ig
vg
Fig. 2.6. An equivalent circuit for the switch network which models the fundamental component of the output voltage waveform and the dc component of the input current waveform.
Uncontrolled rectifier with capacitive filter network
In the series resonant converter, the output rectifier is driven by the (nearly sinusoidal) tank output current iR(t) and a large capacitor CF is placed at the dc output, so that the output voltage V contains negligible harmonics of the switching frequency f (^) S , as shown in Fig. 2.8. The diode rectifiers switch when iR(t) passes through zero, as shown in Fig. 2.7, and the rectifier input voltage vR(t) is essentially a square wave, equal to +V when iR(t) is positive and -V when iR(t) is negative. Note that vR(t) is in phase with i (^) R(t).
If the tank output current iR(t) is a sinusoid with peak amplitude IR1 and phase shift ϕR: i (^) R(t) = I (^) R1 sin (2πfSt - ϕR) (^) (2-5)
then the rectifier input voltage may be expressed in the Fourier series
vR(t) = 4V π ∑ n=1,3,5,...
n sin (2πnfS^ t –^ ϕR)^ (2-6)
where ϕR is the phase shift of iR(t). This voltage waveform is impressed on the output terminal of the resonant tank network. Again, if the tank network responds primarily to the fundamental component of fS of v (^) R (t), and has negligible response at the harmonic frequencies nf (^) S , n =
i (t)R
ϕR
+V vR(t)
–V
IR
ω (^) St
Fig. 2.7 Rectifier input waveforms iR (t) and v (^) R (t).
R
V
I
Õ
low pass load filter
uncontrolled rectifier
Õ
||iR(t)||
Õ
i (^) R(t)
v (^) R(t)
D 5 D 6 D 7 D 8
Fig 2.8 Uncontrolled rectifier with capacitive filter network, as in the series resonant converter. The diodes switch when iR(t) passes through zero.
Chapter 2. Sinusoidal Approximations
Resonant tank network
We have postulated that the effects of harmonics can be neglected, and we have consequently shown that the bridge behaves like a fundamental voltage source vS1 (t) and that the rectifier behaves like a resistor of value R (^) e. We can now solve the resonant tank network by standard linear analysis.
As shown in Fig. 2-10, the tank circuit is a linear network with the voltage transfer function: v (^) R1 (s) v (^) S1 (s) = H(s)^ (2-11)
Hence, the ratio of the peak magnitudes of vR1 (t) and vS1(t) is given by: peak magnitude of vR1(t) peak magnitude of vS1(t) = || H(s) ||s=j 2π^ fs^ (2-12)
In addition, iR is given by:
iR(s) = v^ R1R^ (s) e
= H(s)R e
v (^) S1 (s) (2-13)
So the peak magnitude of iR is:
I R1 = ||^ H(s)^ R||^ S = j2 πfS
e
⋅ (peak magnitude of v (^) S1 (t)) (2-14)
Solution of converter voltage conversion ratio V/Vg
An equivalent circuit of a complete resonant converter is depicted in Fig. 2.11. The complete voltage conversion ratio of the resonant converter can now be found:
| vR1||
)
(s) || s =
} V
|| i R ||
V
V (^) g =^ (^ R)^ (^
π )^ ⋅^
⋅ ( (^) R (^) e)
( (^) I ) ⋅ ( ) ⋅^
|| i R||
( (^) || ||)
} } }
v R
I
j2πfs )^
( ) π
⋅ (
|| v S1 ||
V (^) g )
}
⋅ (^) ( ||^ He ⋅
(
||v (^) S1||
Simplification by use of Eq. (2-10) yields the final result:
vS
i (^) R
+– tank network
vR
Z
iS
R (^) e
H( )S
i
Fig. 2.10. The linear tank network, excited by an effective sinusoidal input source and driving an effective resistive load.
Principles of Resonant Power Conversion
V
Vg =^ ||^ H(s)^ ||^ s=j 2π f^ s^ (2-16)
Eq. (2-16) is the desired result. It states that the dc conversion ratio of the resonant converter is approximately the same as the ac transfer function of the resonant tank circuit, evaluated at the switching frequency f (^) S. This intuitive result can be applied to converters with many different types of tank circuits. However, it should be re-emphasized that Eq. (2-16) is valid only if the response of the tank circuit to the harmonics of vS(t) is negligible compared to the fundamental response, an assumption which is not always valid. In addition, we have assumed that the switch network is controlled to produce a square wave and that the rectifier network drives a capacitive-type network. Finally, the transfer function H(s) is evaluated assuming that the load, Re , is effectively resistive, and it is given by Eq. (2-10).
Õ R
π ||iR1^ ||
H( )S
tank network
Z (^) i
i (^) R
vR
R
vS1 =^ 4V_____π sin(2πf t)S R = ___π^82 R
vS
iS
Õ
__^2
π i (^) = ||i^ S1|| cos( ϕ )S
controlled rectifier with dc input switch ac tank network capacitive filter dc output
g g
vg
i (^) g
e
e
Fig. 2.11. Steady-state equivalent circuit which models the dc and fundamental components of resonant converter waveforms.
Converter efficiency
The effects of tank component losses can also be easily estimated using the model of Fig. 2.11. The converter input power is
P in = V g Ig = Vg π^2 ||iS|| cos ϕS (2-17)
Note that ||iS|| cos ϕS is equal to the real part of iS(s). In addition, iS(s) is equal to the switch output voltage vS1(s) divided by the tank input impedance Zi(s):
iS(s) = vZ^ S1 (s) i(s)
= Y (^) i(s) v (^) S1 (s) (2-18)
Hence, the real part of iS(s) is:
Principles of Resonant Power Conversion
where ω 0 = 1 LC = 2π f (^0)
R 0 = (^) CL
Q e =R 0 /R e
The magnitude of H(j2πf (^) S ), which coincides with the converter dc conversion ratio M = V/Vg, is
M = || H(j2πfS) || = 1 1 + Qe^2 F^1 – F 2
where F = f^ S / f 0
ωC ωL
ω 0 R 0
|| Z (jω) ||
Z (s) = sL + (^) sC^1 + R
ω 0 L = (^) ω^1 0 C
= R 0
so ω 0 = 1 LC R 0 = (^) CL and Z (jω 0 ) = jR 0 - jR0 + R = R
at = 0 :
Q = RR^0
R
ωL
R (^0) ω 0
|| H(jω) || (^) = R 0 Q R R
R
ωR C
H(s) = (^) Z (s)R
i
i
i
e
e (^) e e
e
e
e
e e e
e
e
i ω ω
Fig. 2.13. Construction of the Bode diagrams of Zi(s) and H(s) for the series resonant converter.
The Bode diagrams of Z (^) i (s) and H(s) are constructed in Fig. 2.13. Equation (2-25) is plotted in Figs. 2.14 and 2.15 for various values of Qe and of switching frequency, and the approximate results are compared with the exact results of chapter 3. It can be seen that the dc conversion ratio is unity when the converter is excited at resonance, regardless of load. This is true because the series L-C circuit behaves as a short circuit at resonance: the impedances of the
Chapter 2. Sinusoidal Approximations
tank inductor and capacitor are equal in magnitude but opposite in phase, and their sum is zero. The voltages vS and vR are therefore the same. It can also be seen that a decrease in the load resistance R, which increases the effective quality factor Qe, causes a more peaked response in the vicinity of resonance.
0.5 0.6 0.7 0.8 0.9 1.
exact M, Q= approx M, Q= exact M, Q= approx M, Q= exact M, Q=0. approx M, Q=0.
F
M = V/Vg
Fig. 2.14. Comparison of exact and approximate series resonant converter characteristics, below resonance.
1 2 3 4 5
exact M, Q=0. approx M, Q=0. exact M, Q= approx M, Q= exact M, Q= approx M, Q=
F
M=V/Vg
Fig. 2.15. Comparison of approximate and exact series resonant converter characteristics, above resonance.
Over what range of switching frequencies is Eq. (2-25) accurate? The response of the tank to the fundamental component of vS(t) must be sufficiently greater than the response to the
Chapter 2. Sinusoidal Approximations
A sketch of tank network efficiency vs. switching frequency is given in Fig. 2.17. To the extent that the sinusoidal approximation is valid, the efficiency at low frequency is asymptotic to the value Re / (R (^) S
- Re). The tank esr RS is effectively in series with the load, and leads to a power loss in direct proportion to the load current. Hence, to obtain high efficiency, one should choose R (^) S<<Re. As the switching frequency is increased, the inductor core loss begins to degrade the efficiency. This occurs when the denominator of Eq. (2-27) begins to increase. To avoid this, the switching frequency should be restricted to
fS << (^2) π^1 L RP (RP || (RS + Re)) (2-28)
Alternatively, given a maximum desired switching frequency, Eq. (2-28) can be used to determine a lower bound on R (^) P.
1
η R R +RS e
R +R +RP S e
____________R^ e f (^) S R (^) P (R || (R +R ))P S e
√ 2 πL
___RP
2 πL
e
Fig. 2.17. Tank network efficiency vs. switching frequency.
Thus, sinusoidal approximations give an effective means of estimating the losses and efficiency degradations which arise owing to tank component nonidealities.
vR
L C
R
RS
R P
v Z S1 i e
H( )S
Fig. 2.16 Equivalent circuit used to model tank component losses.
Principles of Resonant Power Conversion
2. 3. Subharmonic Modes of the Series Resonant Converter
If the n th^ harmonic of the switch output waveform vS(t) is close to the resonant tank frequency, nfS ~ f 0 , and if the tank effective quality factor Qe is sufficiently large, then as shown in Fig. 2.18, the tank responds primarily to harmonic n. All other components of the tank waveforms can then be neglected, and it is a good approximation to replace vS(t) with its nth^ harmonic component:
vS(t) ≅ v (^) Sn (t) = n4 Vgπ TS sin (nωSt) (2-29)
This differs from Eq. (2-2) because the amplitude is reduced by a factor of 1/n, and the frequency is nfS rather than fS.
f
switch output voltage spectrum
f
f
fS 3 fS 5 fS
fS 3 fS 5 fS
fS 3 fS 5 fS
resonant tank response
tank current i spectrum
s
Fig 2.18 When the tank responds primarily to the third harmonic of the switching frequency, then frequency components other than the third harmonic may be neglected.
Principles of Resonant Power Conversion
+–
Õ
controlled switch network
power input
N S
I
Õ
R
low load pass filter
resonant tank network
N T
uncontrolled rectifier
N R N F
Õ
iR
vR
V
Õ
i (^) S
i (t)
v vS
g
g
Fig. 2.20. Block diagram of the parallel resonant converter.
As in the series resonant converter, the switch network is controlled to produce a square wave v (^) S (t). If the tank network responds primarily to the fundamental component of vS (t), then arguments identical to those of section 2.1 can be used to model the output fundamental components and input dc components of the switch waveforms. The resulting equivalent circuit is identical to Fig. 2.6. The uncontrolled rectifier with inductive filter network can be described using the dual of the arguments of Section 2.1. In the parallel resonant converter, the output rectifiers are driven by the (nearly sinusoidal) tank capacitor voltage v (^) R (t), and the diode rectifiers switch when vR(t) passes through zero as in Fig. 2.21. The rectifier input current i (^) R(t) is therefore a square wave of amplitude I, and it is in phase with the tank capacitor voltage v (^) R(t). The fundamental component of iR(t) is
i (^) R1 (t) = 4 I π sin (2πfSt - ϕR) (^) (2-32) Hence, the rectifier again presents an effective resistive load to the tank circuit, equal to
Re = vi R1(t) R1(t)^
= π4 IV^ R1 (2-33) The ac components of the rectified tank capacitor voltage | vR(t) | are removed by the output low pass filter. In steady state, the output voltage V is equal to the dc component of | vR(t) |:
V = T^2
S
VR1 | sin (2πf St - ϕR) | dt
0
T (^) S 2 (2-34)
So the load voltage V and the tank capacitor voltage amplitude are directly related in steady state. Substitution of Eq. (2-27) and resistive load characteristics V = IR into Eq. (2-26) yields:
i (t)R
ϕR
vR(t) +I
–I
VR
ω (^) St
Fig. 2.21. Parallel resonant converter waveforms v (^) R (t) and i (^) R (t).
Chapter 2. Sinusoidal Approximations
Re = π 82 R = 1.2337 R (^) (2-35)
An equivalent circuit for the uncontrolled rectifier with inductive filter network is given in Fig. 2.22. This model is similar to the one used for the series resonant converter, Fig. 2.9, except that the roles of the rectifier input voltage v (^) R and current iR are interchanged, and the effective resistance R (^) e has a different value. The model for the complete converter is given in Fig. 2.23. Solution of Fig. 2.23 yields the converter dc conversion ratio: M = (^) Vg V = 8 π^2
|| H(s) ||s=j2πfs (2-36)
where H(s) is the tank transfer function H(s) = ZsL^0 (s) (2-37)
and Z^0 (s) = sL^ ||^ sC^1 ||^ Re (2-38)
4V____
π
g = sin(2πf t)
tank network
H(s)
Õ +–
vR(t)
Õ
iR1 (t)
vS1^ +– R
iS
Z (^) R = π__^2 R 8
I
Õ
V = π^2 VR
S v (t) = VR^ R1^ sin(2^ πf t -S^ ϕ^ R)
controlled rectifier with dc input switch ac tank network inductive filter dc output
i (^) g
vg +- i^ e
= (^) π||i ||cos(ϕ ) i (^) g (^2) S1 S vS
Fig. 2.23. Equivalent circuit for the parallel resonant converter, which models the fundamental components of the tank waveforms, and the dc components of the input current and output voltage.
Õ
I
Õ
+–
vR(t)
iR1 (t)
R = __π
2 8 R
v (t) = VR R1 sin(2πf t -S ϕ )R
e
V =^ π^2 VR
Fig. 2.22. An equivalent circuit for the rectifier with inductive output filter, which models the fundamental component of the ac side voltage, and the dc component of the dc side voltage, in the parallel resonant converter.
Chapter 2. Sinusoidal Approximations
M = 8
π^2 ||^
1 + (^) Qs eω 0
|| (^) s=j2πf (^) s
π 2
(1 - F 2 ) 2 + ( QF
e
Equation (2-40) is compared with the exact converter solution in Fig. 2.25.
0.50 1.00 1.50 2.00 2.50 3.
M
F
Q (^) e =
Q (^) e =
Q (^) e =
Q (^) e =0. Q (^) e =0.
Fig. 2.25. Comparison of exact parallel resonant converter characteristics (solid lines) vs. the approximate solution, Eq. (2-40) (shaded lines).
Principles of Resonant Power Conversion
2. 5. Switching at Zero Current or Zero Voltage
Operation of the Full Bridge Below Resonance
An advantage of resonant converters is their reduced switching loss. When the series resonant converter is operated below resonance, a phenomenon known as “zero current switching” can occur, in which the transistors naturally switch off at zero current. With a simple circuit modification, the transistor turn-on transition can also be caused to occur at zero current. Let us consider the operation of the full bridge switch in more detail. A full bridge circuit, realized using power MOSFET’s and antiparallel diodes, is shown in Fig. 2.26. The switch output voltage v (^) S (t), and its fundamental component v (^) S1 (t), as well as the approximately sinusoidal tank current waveform i (^) S(t), are plotted in Fig. 2.27. At frequencies less than the tank resonant frequency, the input impedance of the tank network Z (^) i (s) is dominated by the tank capacitor impedance. Hence, the tank presents an effective capacitive load to the bridge, and switch current i (^) S(t) leads the switch voltage fundamental component v (^) S1 (t), as shown in Fig. 2.27. In consequence, the zero crossing of the current waveform iS(t) occurs before the zero crossing of the voltage v (^) S(t). For the half cycle 0 ≤ t ≤ T (^) S /2, v (^) S = +V (^) g. For 0 ≤ t ≤ tβ, the current iS(t) is positive and transistors Q 1 and Q 4 conduct. Then the diodes D 1 and D 4 conduct when
D 1 D 3
D 4
full bridge remainder of converter (series resonant example)
i (t)S
Q 1
Q 2 D 2
i (^) Q
v (^) DS Vg^ +–
Fig. 2.26. Full bridge circuit implemented with power MOSFETs and antiparallel diodes in a series resonant converter circuit.
___^ T^ S
v (t)S
v (^) S1(t) Vg
-Vg
T S
___T^ S
t
t
I (^) S1 i (^) S(t)
"hard" turn-on of Q , Q 1 4
"soft" turn-off of Q , Q 1 4
"hard" turn-on of Q , Q 2 3
"soft" turn-off of Q , Q 2 3
Q
Q
Q
Q
D
D
D
D
1 1 (^4 )
2 3
2 3
tβ (^) ___T (^) S 2 + tβ
Fig. 2.27. Tank waveforms for the series resonant converter, operatedbelow resonance. “Zero- current switching” aids the transistor turn off transitions.
