Reduction of Output Voltage Ripples in Frequency Modulated Power Converter

Nowadays switch-mode power converters (SMPC) are essential to electric power conversion with high efficiency (linear regulators used widely in the past had very low efficiency and therefore nowadays they are in rather limited use) [1–3]. SMPC are widely used in power supplies of various electronic devices, active power factor correctors, etc. Despite their advantages (mainly high efficiency and specific power) [3, 4], they have also their drawbacks: high electromagnetic interference (EMI) both conducted and radiated [2, 5]. Filters, shielding, grounding etc are classical ways to mitigate EMI. Another successful method to reduce EMI is spread spectrum which is usually based on the use of switching frequency modulation (FM) [1, 2, 6]. Despite the fact that FM has its benefits in reducing EMI, it can increase output voltage ripples significantly [713]. In addition to natural high-frequency (HF) output voltage ripples, FM causes also low-frequency (LF) output voltage ripples which are more problematic than HF ones [8, 9]. Detailed examination of the effect of FM on the output voltage ripples in SMPC operating in continuous and discontinuous conduction modes (CCM and DCM respectively) was performed mainly in [7, 9–13]. Although the main causes of the increase in the output voltage ripples in CCM are found out in [7, 9, 10] and expressions to calculate the peak-to-peak output voltage ripples Vofmp-p in closed-loop FM SMPC are derived in [9], closed-form expression for Vofmp-p, which can appreciably simplify analysis of the effect of SMPC and modulating signal parameters on Vofmp-p, is not derived. Recommendations to reduce the ripples proposed in [9] consider mainly the LF ripples not Vofmp-p. So the main aim of the paper is quantitative analysis of the ripples followed by worked-out recommendations to reduce them.


Introduction
Nowadays switch-mode power converters (SMPC) are essential to electric power conversion with high efficiency (linear regulators used widely in the past had very low efficiency and therefore nowadays they are in rather limited use) [1][2][3].SMPC are widely used in power supplies of various electronic devices, active power factor correctors, etc.Despite their advantages (mainly high efficiency and specific power) [3,4], they have also their drawbacks: high electromagnetic interference (EMI) both conducted and radiated [2,5].Filters, shielding, grounding etc are classical ways to mitigate EMI.Another successful method to reduce EMI is spread spectrum which is usually based on the use of switching frequency modulation (FM) [1,2,6].
Despite the fact that FM has its benefits in reducing EMI, it can increase output voltage ripples significantly [7][8][9][10][11][12][13]. In addition to natural high-frequency (HF) output voltage ripples, FM causes also low-frequency (LF) output voltage ripples which are more problematic than HF ones [8,9].Detailed examination of the effect of FM on the output voltage ripples in SMPC operating in continuous and discontinuous conduction modes (CCM and DCM respectively) was performed mainly in [7,[9][10][11][12][13].Although the main causes of the increase in the output voltage ripples in CCM are found out in [7,9,10] and expressions to calculate the peak-to-peak output voltage ripples V ofmp-p in closed-loop FM SMPC are derived in [9], closed-form expression for V ofmp-p , which can appreciably simplify analysis of the effect of SMPC and modulating signal parameters on V ofmp-p , is not derived.Recommendations to reduce the ripples proposed in [9] consider mainly the LF ripples not V ofmp-p .So the main aim of the paper is quantitative analysis of the ripples followed by worked-out recommendations to reduce them.

Output voltage ripples of FM SMPC
For the examination closed-loop FM buck converter (Fig. 1) will be used.It is assumed that the converter operates in CCM.As it is concluded in [7][8][9][10][11][12][13] FM increases output voltage ripples.In general the ripples consist of HF switching ripples with switching frequency f sw and LF ripples with modulation frequency f m , as it can also be seen in Fig. 2 (b-d), where simulated in SIMULINK output ripples for the open-loop and closed-loop buck converter (using simulation circuit shown in Fig. 2   A general expression to calculate the peak-to-peak output voltage ripples in FM SMPC is derived in [9] as follows where ṽ LF (t) is the LF ripples; A HF (t) is the envelope of the HF ripples, which can be derived by substituting instantaneous switching frequency f sw (t) into the expression for the unmodulated SMPC output voltage ripples V p-p .As it is derived in [9], for instance, for FM buck SMPC with typical output filter capacitor in CCM where L is the power inductor inductance; r cout is the equivalent series resistance (ESR) of the output capacitor; D is the average duty ratio; V out is DC output voltage.
T h e i n s t a n t a n e o u s s w i t c h i n g f r e q u e n c y i s where Δf sw is the switching frequency deviation; m(t) is the modulating signal (e.g.sine, triangular, sawtooth, etc); f sw is the nominal switching frequency.
The main cause of the LF ripples for FM SMPC in CCM is nonzero difference |t d | between the switching delays [7,9,10,12], which are mainly due to power transistor switches, their drivers and logic circuits (e.g.flipflops) of pulse width modulated (PWM) integrated circuits.A general expression to calculate the LF ripples of closedloop FM SMPC in CCM considering nonzero |t d | is derived in [9] in operator form as follows [15] (where H PWM (s)is PWM gain; H div (s) and H c (s) are voltage divider and compensation circuit transfer functions).As an example the transfer functions for the closed-loop buck SMPC are shown in Fig. 3.The buck converter

where H co (s) is the control-to-output transfer function; the open loop gain T(s)=H div (s)H c (s)H co (s)H PWM (s)
To derive the steady-state LF ripples ṽ LF (t) in time domain, the inverse Laplace transform of the (4) should be applied.The LF ripples in the time domain for sinusoidal FM can be easily obtained from (4) as follows , (6) where As it can be seen from ( 4) and (7) V LFp-p are proportional to |t d | and ∆f sw .

Derivation of closed-form expression for V ofmp-p
Expression (1) derived in [9] does not clearly show the effect of FM and SMPC parameters on V ofmp-p , because it can be evaluated only numerically.Therefore closedform expression for V ofmp-p needs to be derived.In order to get the expression, (1) can be used, as it can also be deduced from Fig. 4. Since ∆f sw is usually lower than 0.25f sw [2], then (2) can be approximated using the first- order-Taylor-series-approximation as follows where δ=Δf sw /f sw ; unmodulated buck SMPC output voltage peak-to-peak ripples V p-p are as follows: Since for sinusoidal FM m(t)=sin(2πf m t), then closedform expression for V ofmp-p can be simply derived It can be seen that difference between unmodulated and FM buck SMPC output peak-to-peak ripples V ofmp-p -V p-p is proportional to Δf sw and V in .Similarly closed-form expressions for V ofmp-p can also be derived for other m(t).
In order to prove that derived expression is useful experimental verification and SIMULINK simulations were performed for closed-loop buck SMPC with sinusoidal FM.
The comparison results for different f m are shown in Table 1.As it can be seen the results are in a good agreement proving that ( 10) is precise enough to calculate the peak-topeak output voltage ripples in buck FM SMPC.

Effect of D and R load on V ofmp-p
Since D and R load (or output power P out ) in real SMPC are continuously changing parameters, it is of importance to investigate how they affect V ofmp-p of FM buck SMPC.By analyzing (10) it is concluded that V ofmp-p is almost independent on P out (as it can also be seen in Fig. 5).
In fact unmodulated buck SMPC output ripples are also independent on R load in CCM (as it can be deduced from ( 9)).After analyzing the (10), it is concluded that V ofmp-p depends on D, as it can also be seen in Fig. 6.Moreover D max at which V ofmp-p is maximum is equal to 0.5, as it is also for unmodulated buck SMPC V p-p .(Parameters are the same as in Fig. 5)

Recommendations to reduce V ofmp-p
The analysis provided allows us to propose some recommendations to reduce the ripples:  t d should be as small as possible; for this purpose the techniques proposed in [9] can be used;  Choice of f m and Δf sw : (a) from V ofmp-p point of view: the lower Δf sw is the lower V ofmp-p is; f m should not be chosen in the vicinity of the resonance frequency f max of the buck converter output filter (see Fig. 3); (b) from EMI attenuation point of view: the higher Δf sw /f m is the better EMI attenuation can be achieved [1]; so when selecting the modulation parameters, trade-off between EMI attenuation and V ofmp-p should be considered.

Conclusions
Increase in peak-to-peak output voltage ripples due to FM in buck SMPC in CCM can be simply analyzed using the closed-form expression derived.The expression that takes into account also non-zero t d , can give us a possibility also examine the effect of D and the output power on the ripples.It is concluded that V ofmp-p is almost independent on the output power, but depends on D. Moreover the worstcase-scenario D=0.5 at which the peak-to-peak ripples are maximum, is the same as for unmodulated buck converter.
The expressions derived as well as SIMULINK simulations and experiments for closed-loop buck SMPC show that V ofmp-p can be effectively neutralized when t d is small as possible.When selecting the modulating signal parameters, the trade-off between EMI attenuation and V ofmp-p should be considered.The results obtained in the analysis and the recommendations proposed can be used when designing high-quality FM SMPC.

Table 1 .
Comparison of the theoretical, simulated and experimental results for closed-loop buck FM SPC with Δf sw