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This work studies the smallsignal direct and crosscoupling audiosusceptibilities of the independentinput seriesoutput boost converter with peak currentmode control. The converter functions in the continuousconduction mode and comprises nconnected identical boost modules whose inputs are fed from separate voltage sources and outputs connected in series. Expressions for the module direct (self) and crosscoupling audiosusceptibilities are derived in symbolic form. The expressions explicitly show (n) as a variable and take into account the sampling action of the current loops. In addition, audiosusceptibilty frequency responses following the closure of the voltage feedback loops are generated and the influence of increasing n is discussed. Detailed simulations using PSIM are performed to support the analysis.
audiosusceptibility, modular boost dcdc converters, independentinput seriesoutput, peak currentmode control, smallsignal modeling
Over the past few decades, the modular connection of dcdc converters has shown effectiveness in reducing current and voltage stresses on the participating semiconductor devices and also in increasing the conversion system reliability. A modular converter allows power to be processed by a number of converter modules connected in different combinations to fulfil certain input/output requirements. The four wellknown conventional arrangements of modular converters are: the parallelinput/paralleloutput (PIPO), the parallelinput/seriesoutput (PISO), the seriesinput/paralleloutput (SIPO), and the seriesinput/seriesoutput (SISO); the works [1, 2] are comprehensive references on the subject. Other arrangements, however, have appeared in the literature like the seriesparallelinput/seriesoutput [3], the seriesinput/seriesparalleloutput [4], and the independentinput arrangements, namely the independentinput/seriesoutput (IISO) [515], and the independentinput/paralleloutput (IIPO) [1619].
Several control methodologies have been proposed for these modular converters, among them is the peak currentmode control (PCMC) technique due to its renowned merits such as fast response, stable current sharing and precise output voltage regulation [1]. PCMC has been used for the control of PIPO [2023], PISO [2426], SIPO [2729], SISO [30, 31], IISO [14], and IIPO [16] converters. Initial examination of the dynamics of these systems is dominantly done with the help of linearised smallsignal (SS) models. Better insight into the converter performance can be obtained if SS modeling is supported by transfer function expressions showing how the system behaves as the number of modules is varied.
The SS linetooutput voltage transfer function (audiosusceptibility) of a dcdc converter cell is an important dynamic performance parameter. It describes the converter’s ability to attenuate SS input voltage disturbances. The audio susceptibilities of several PCMC singlestage dcdc converters have been formally studied [3236]. Also many papers addressed the audiosusceptibilities of the conventional PCMC singleinput modular converters [20, 24, 26, 28]. Nevertheless, susceptibilities of PCMC modular converters supplied from nindependent sources and how these susceptibilities react to varying the number of modules under open and closedloop conditions have never been treated in the literature. In such converters like the IISO and the IIPO there are two audiosusceptibilities to consider: 1) The direct (self) audiosusceptibility which defines the output voltage response of a certain module due to a disturbance in its input voltage; and 2) the crosscoupling audio susceptibility which gives the output voltage response of one module due to a disturbance in the source voltage of another.
The objective of this work is to study analytically the SS direct and crosscoupling audiosusceptibilities of an IISO PCMC boost converter intended for dc power supplies applications. Converters using the IISO structure have the distinct feature of being supplied from independent voltage sources; with higher output voltages obtained by the series connection of the outputs of a number of singlecell converters. The pioneer work on IISO dcdc converters dates back to the 1990’s [5] where a two module IISO boost configuration was used to interface a smallsize windphotovoltaic system to the utility circuit. Since then many articles have been written on IISO converters for different applications related to renewable energy systems, distributed power systems and dedicated dc power supplies, but none of them have employed PCMC for the control of the constituent modules until recently [14] where the rampcompensated PCMC is used to control an IISO boost converter. The work [14], however, has only focused on the controltooutput voltage SS responses produced numerically using Simulink software. Other performance parameters like audiosusceptibility and crosscoupling effect have not been considered.
The contributions of this paper are: 1) the SS direct and crosscoupling audiosusceptibilities of the PCMC IISO boost converter are analytically studied. Symbolic expressions for these transfers are derived after establishing a smallsignal model based on statespace equations. The expressions explicitly show the number of used cells (n) and take into account the sampling action of each of the converter’s current loops. These expressions help the user to readily plot SS responses for different n without needing a circuit simulator; 2) the study also addresses the effect of closing the voltage feedback loops on the audiosusceptibilities when classical controllers are employed. To validate the analysis, two, three, and fourmodule converters are implemented using PSIM and frequency responses are obtained using the “ac sweep” tool which allows the user to obtain the frequency response while the converter is in its switchedmode form.
The IISO converter considered in this study is depicted in Figure 1. Several identical PCMC boost modules are operated in the continuouscurrent conduction mode and joined in series at the output side to supply a common load. A separate voltage source feeds each of the independently controlled modules. The PCMC of each module in Figure 1 can be briefly reviewed as follows: Converter switching cycle is started by a constantfrequency clock. Transistor’s duty ratio (D) is set when the inductor current, after being sensed by (R_{i}), becomes equal to a value that depends on control voltage (V_{c}), produced by the compensated error amplifier (EA) circuit, and an external ramp slope (S_{e}). This ramp is necessary to stabilize the current loop if D is greater than 0.5 [32]. Also in Figure 1, the attenuation factor of the module output voltage is denoted by K_{v} while F_{v} represents the module voltageloop compensator transfer function.
The following sections of this paper are: Section 2 presents the converter SS model. Section 3 studies the direct and crosscoupling audiosusceptibilities with closed current loops and voltage loops left open. Section 4 addresses the effect of closing the voltage feedback loops on the audiosusceptibilities. Finally, Section 5 contains the conclusion.
Figure 1. IISO converter schematic
2.1 Modeling the power stage
To develop the SS model of the multimodule PCMC converter, a two module converter is considered first. Referring to Figure 1, the input and output voltages of each module is denoted by V_{g} and V_{o} respectively; the module inductor current is indicated by I_{L}, and the transistor duty ratio of each module is symbolised by D. Using statespace averaging and linearization [37], the powerstage SS model of a twomodule IISO boost converter assuming ideal components can be characterized as in Ref. [14] by:
$\dot{\hat{x}}=\frac{d}{d t}\left[\begin{array}{c}\hat{\imath}_{L 1} \\ \hat{v}_{o 1} \\ \hat{\imath}_{L 2} \\ \hat{v}_{o 2}\end{array}\right]=[A]\left[\begin{array}{c}\hat{\imath}_{L 1} \\ \hat{v}_{o 1} \\ \hat{\imath}_{L 2} \\ \hat{v}_{o 2}\end{array}\right]+[B]\left[\begin{array}{c}\hat{d}_1 \\ \hat{d}_2 \\ \hat{v}_{g 1} \\ \hat{v}_{g 2} \\ \hat{\imath}_o\end{array}\right]$ (1a)
where
$[A]=\left[\begin{array}{cccc}0 & \frac{D_1^{\prime}}{L_1} & 0 & 0 \\ \frac{D_1^{\prime}}{C_1} & \frac{1}{R C_1} & 0 & \frac{1}{R C_1} \\ 0 & 0 & 0 & \frac{D_2^{\prime}}{L_2} \\ 0 & \frac{1}{R C_2} & \frac{D_2^{\prime}}{C_2} & \frac{1}{R C_2}\end{array}\right]$ (1b)
$[B]=\left[\begin{array}{ccccc}\frac{V_{g 1}}{D_1^{\prime} L_1} & 0 & \frac{1}{L_1} & 0 & 0 \\ \frac{2 V_{g 1}}{D_1^{\prime 2} R C_1} & 0 & 0 & 0 & \frac{1}{C_1} \\ 0 & \frac{V_{g 2}}{D_2^{\prime} L_2} & 0 & \frac{1}{L_2} & 0 \\ 0 & \frac{2 V_{g 2}}{D_2^{\prime 2} R C_2} & 0 & 0 & \frac{1}{C_2}\end{array}\right]$ (1c)
where the hat symbol (^) is used for SS changes; and $D^{\prime}=1D$.
2.2 Modeling the PCMC stage
The PCMC stage of each module is modeled in a similar fashion to that of the single boost dcdc converter [32, 33]. The SS model for a 2module converter is illustrated in Figure 2(a). Each module is comprised of: The current sensing resistance R_{i}; the PWM modulator gain F_{m}; the current loop sampling gain H_{e}; and feedforward gains K_{f}and K_{r}formed as a result of closing the current loop of each module. Variations in the inductor voltage during the ON and OFF intervals of the transistor are respectively denoted as $\hat{v}_{o n}$ and $\hat{v}_{o f f}$. Model parameters are given in Table 1. The module duty ratio with only the current loop closed is:
$\hat{d}=F_m\left(\hat{v}_cR_i H_e \hat{\imath}_L+K_f \hat{v}_{o n}+K_r \hat{v}_{o f f}\right)$ (2)
After the application of Laplace transforms to (1) and the substitution for duty ratio ($\hat{d}$) from (2), we get the following when the modules are identical:
$\left[\begin{array}{c}s \hat{v}_{L 1} \\ s \hat{v}_{o 1} \\ s \hat{l}_{L 2} \\ s \hat{v}_{o 2}\end{array}\right]=\left[\begin{array}{cccc}A_1 & A_2 & 0 & 0 \\ A_3 & A_4 & 0 & A_5 \\ 0 & 0 & A_1 & A_2 \\ 0 & A_5 & A_3 & A_4\end{array}\right]\left[\begin{array}{c}\hat{l}_{L 1} \\ \hat{v}_{o 1} \\ \hat{l}_{L 2} \\ \hat{v}_{o 2}\end{array}\right]+\left[\begin{array}{ccccc}B_1 & 0 & B_2 & 0 & 0 \\ B_3 & 0 & B_4 & 0 & B_5 \\ 0 & B_1 & 0 & B_2 & 0 \\ 0 & B_3 & 0 & B_4 & B_5\end{array}\right]\left[\begin{array}{c}\hat{v}_{c 1} \\ \hat{v}_{c 2} \\ \hat{v}_{g 1} \\ \hat{v}_{g 2} \\ \hat{\imath}_o\end{array}\right]$ (3)
The A and B elements are tabulated in Table 2, and parameters F_{m}, H_{e}, K_{r} and K_{f} are as given in Table 1.
The set of Eq. (3) represents the converter SS model with only the current loops closed.
(a) Block diagram
(b) Structure of Type II compensator
Figure 2. Twomodule smallsignal model
Table 1. Parameters of figure 2(a)
$F_m=\frac{1}{\left(S_n+S_e\right) T}=\frac{L}{M_c R_i V_g T} \quad \text { where } M_c=1+s_e / s_n 
$H_e \cong\left(1+\frac{s}{\omega_n Q_z}+\frac{s^2}{\omega_n^2}\right)$ where $Q_z=2 / \pi$ and $\omega_n=\pi / T$ 
$K_f=\frac{D T R_i(10.5 D)}{L}+\frac{D^2 T^2 R_i(32 D)}{12 L} s$ and $K_r=\frac{D^{\prime 2} T R_i}{2 L}$ 
$\hat{v}_{\text {on }}=\hat{v}_g ; \hat{v}_{\text {off }}=\hat{v}_o\hat{v}_g$ 
$F_v=\frac{k_i\left(1+s / \omega_z\right)}{s\left(1+s / \omega_p\right)}$ 
Table 2. Summary of the expressions of Eq. (3)
A_{1} 
$\frac{V_{o, \text { module }}\,\,\,\,\,\, R_i H_e F_m}{L}$ 
B_{1} 
$\frac{V_{o, \text { module }}\,\,\,\,\,\, F_m}{L}$ 
A_{2} 
$\frac{D^{\prime}+V_{o, \text { module }}\,\,\,\,\,\, F_m K_r}{L}$ 
B_{2} 
$\frac{1+V_{o, \text { module }}\,\,\,\,\,\, F_m\left(K_fK_r\right)}{L}$ 
A_{3} 
$\frac{D^{\prime}+I_{L, \text { module }}\,\,\,\,\,\, R_i H_e F_m}{C}$ 
B_{3} 
$\frac{I_{L, \text { module }}\,\,\,\,\,\, F_m}{C}$ 
A_{4} 
$\frac{1}{C}\left(\frac{1}{R}+I_{L, \text { module }} F_m K_r\right)$ 
B_{4} 
$\frac{I_{L, \text { module }}\,\,\,\,\,\, F_m\left(K_fK_r\right)}{C}$ 
A_{5} 
$\frac{1}{R C}$ 
B_{5} 
$\frac{1}{C}$ 
2.3 Modeling the voltage feedback loop stage
Referring to Figure 2(a), the voltage feedback loop of each module consists of an attenuator K_{v} and a classical (type II) compensator represented by F_{v}. The compensator structure is depicted in Figure 2(b). With voltage and current feedback loops closed, Eq. (3) is updated by substituting for the control voltages v_{C}_{1} and v_{C}_{2} which can be expressed as:
$\hat{v}_{c 1}=\left(\hat{v}_{\text {ref } 1}K_{v 1} \hat{v}_{o 1}\right) F_{v 1}$ (4a)
$\hat{v}_{c 2}=\left(\hat{v}_{r e f 2}K_{v 2} \hat{v}_{o 2}\right) F_{v 2}$ (4b)
The SS model for the 2module converter is represented as:
$\left[\begin{array}{c}s \hat{l}_{L 1} \\ s \hat{v}_{o 1} \\ s \hat{\imath}_{L 2} \\ s \hat{v}_{o 2}\end{array}\right]=\left[\begin{array}{cccc}A_1 & A_2B_1 K_{v 1} F_{v 1} & 0 & 0 \\ A_3 & A_4B_3 K_{v 1} F_{v 1} & 0 & A_5 \\ 0 & 0 & A_1 & A_2B_1 K_{v 2} F_{v 2} \\ 0 & A_5 & A_3 & A_4B_3 K_{v 2} F_{v 2}\end{array}\right]\left[\begin{array}{c}\hat{\imath}_{L 1} \\ \hat{v}_{o 1} \\ \hat{l}_{L 2} \\ \hat{v}_{o 2}\end{array}\right]+\left[\begin{array}{ccccc}B_1 F_{v 1} & 0 & B_2 & 0 & 0 \\ B_3 F_{v 1} & 0 & B_4 & 0 & B_5 \\ 0 & B_1 F_{v 2} & 0 & B_2 & 0 \\ 0 & B_3 F_{v 2} & 0 & B_4 & B_5\end{array}\right]\left[\begin{array}{c}\hat{v}_{r e f 1} \\ \hat{v}_{r e f 2} \\ \hat{v}_{g 1} \\ \hat{v}_{g 2} \\ \hat{l}_o\end{array}\right]$ (5)
3.1 Module direct audiosusceptibility
The direct (self) audiosusceptibility of a certain module is the output voltage response of that module due to a disturbance in its input voltage. The case of a twomodule converter (n = 2) will be used as a starting point to reach a general SS expression for the audiosusceptibilities with nconnected modules. Referring to (3), the module direct audiosusceptibility when n = 2 can be expressed as:
$\frac{\hat{v}_{o 1}}{\hat{v}_{g 1}}=\frac{\hat{v}_{o 2}}{\hat{v}_{g 2}}=\frac{\Delta_1\left[s^2\left(A_1+A_4\right) s+A_1 A_4A_2 A_3\right]}{\Delta_2\left[s^2\left(A_1+A_4+A_5\right) s+A_1\left(A_4+A_5\right)A_2 A_3\right]}$ (6)
$\Delta_1=B_4\left(sA_1+A_3 B_2 / B_4\right)$ (7a)
$\Delta_2=s^2\left(A_1+A_4A_5\right) s+A_1\left(A_4A_5\right)A_2 A_3$ (7b)
A general expression for the direct audiosusceptibility of a converter with nconnected modules can be concluded when we find the susceptibilities of the three and fourmodule converters which can be reached by following the steps used above with the twomodule case.
When n = 3, the direct audiosusceptibility can be derived as:
$\frac{\hat{v}_{o 1}}{\hat{v}_{g 1}}=\frac{\hat{v}_{o 2}}{\hat{v}_{g 2}}=\frac{\hat{v}_{o 3}}{\hat{v}_{g 3}}=\frac{\Delta_1\left[s^2\left(A_1+A_4+A_5\right) s+A_1\left(A_4+A_5\right)A_2 A_3\right]}{\Delta_2\left[s^2\left(A_1+A_4+2 A_5\right) s+A_1\left(A_4+2 A_5\right)A_2 A_3\right]}$ (8)
And for n = 4
$\frac{\hat{v}_{o 1}}{\hat{v}_{g 1}}=\frac{\hat{v}_{o 2}}{\hat{v}_{g 2}}=\frac{\hat{v}_{o 3}}{\hat{v}_{g 3}}=\frac{\hat{v}_{o 4}}{\hat{v}_{g 4}}=\frac{\Delta_1\left[s^2\left(A_1+A_4+2 A_5\right) s+A_1\left(A_4+2 A_5\right)A_2 A_3\right]}{\Delta_2\left[s^2\left(A_1+A_4+3 A_5\right) s+A_1\left(A_4+3 A_5\right)A_2 A_3\right]}$ (9)
In general, for the nconnected modules shown in Figure 1, the module direct audiosusceptibilty is:
$\frac{\hat{v}_{\text {on }}}{\hat{v}_{g n}}=\frac{\Delta_1}{\Delta_2} \times \frac{\left[s^2\left(A_1+A_4+(n2) A_5\right) s+A_1\left(A_4+(n2) A_5\right)A_2 A_3\right]}{\left[s^2\left(A_1+A_4+(n1) A_5\right) s+A_1\left(A_4+(n1) A_5\right)A_2 A_3\right]}$ (10)
The expression is 5^{th} order for the numerator and 6^{th} for the denominator when we substitute for the A and B terms given in Table 2. After performing these substitutions and rearranging terms the resultant expression is programmed into Matlab with the following parameters for each module:
V_{g} = (48/number of modules) V; D = 0.6; R = 30 Ω;
T = 10 µs; L = 115 µH; C = 40 µF; R_{i} = 0.1 Ω
The direct audiosusceptibility can be evaluated for different slope ratio (M_{c}) values by using Eq. (10).
Figure 3 shows the direct audiosusceptibility responses as we vary n with slope ratio M_{c}= 1.5 while Figure 4 depicts the responses when M_{c} = 2.9. The sloperatio values are selected to study the underdamped and criticallydamped behaviour. Figures 3 and 4 also present PSIM “ac sweep” results to validate the analytical model predictions. Matlab polezero locations corresponding to these cases are given in Table 3. In the module audiosusceptibility response, at low frequencies, one can notice that there are two real lefthalf splane poles sandwiching a real zero. At ½ the switching frequency (f_{s}/2), the response is influenced by a complex pair of poles in the lefthalf plane as a result of the sampleandhold (S/H) effect of PCMC, and hence the peaking in Figure 3 when M_{c} = 1.5. This double pole however splits up into two single real poles with M_{c} = 2.9. The S/H effect also creates a complex pair of lefthalfplane zeros in the module audiosusceptibility response at frequencies higher than f_{s}/2. Figures 3 and 4 show that increasing the number of modules increases the direct audiosusceptibility, but does not have a noticeable effect on the peaking at f_{s}/2; and critical damping of the responses is achieved for all cases with the same slope ratio.
(a) Plots from analytical model
(b) PSIM simulation results
Figure 3. Direct audiosusceptibility with current loops closed and variable number of modules with (M_{c} = 1.5)
(a) Plots from analytical model
(b) PSIM simulation results
Figure 4. Direct audiosusceptibility with current loops closed and variable number of modules with (M_{c} = 2.9)
Table 3. Polezero locations in (rad/sec) of direct audiosusceptibility with variable n

M_{C }= 1.5 
M_{C} = 2.9 


n = 2 
n = 3 
n = 4 
n = 2 
n = 3 
n = 4 
Zeros 
1.0e+05 * 0.1189 + 4.6336i 0.1189  4.6336i 0.3940 + 3.1066i 0.3940  3.1066i 0.0265 + 0.0000i 
1.0e+05 * 0.0792 + 4.6218i 0.0792  4.6218i 0.3940 + 3.1015i 0.3940  3.1015i 0.0350 + 0.0000i 
1.0e+05 * 0.0594 + 4.6157i 0.0594  4.6157i 0.3938 + 3.0965i 0.3938  3.0965i 0.0435 + 0.0000i 
1.0e+05 * 0.1189 + 4.8924i 0.1189  4.8924i 3.6775 + 0.0000i 2.6353 + 0.0000i 0.0288 + 0.0000i 
1.0e+05 * 0.0792 + 4.7963i 0.0792  4.7963i 3.7480 + 0.0000i 2.5643 + 0.0000i 0.0376 + 0.0000i 
1.0e+05 * 0.0594 + 4.7474i 0.0594  4.7474i 3.8112 + 0.0000i 2.5005 + 0.0000i 0.0465 + 0.0000i 
Poles 
1.0e+05 * 0.3941 + 3.1066i 0.3941  3.1066i 0.3940 + 3.1066i 0.3940  3.1066i 0.0349 + 0.0000i 0.0181 + 0.0000i 
1.0e+05 * 0.3940 + 3.1015i 0.3940  3.1015i 0.3939 + 3.1015i 0.3939  3.1015i 0.0434 + 0.0000i 0.0266 + 0.0000i 
1.0e+05 * 0.3939 + 3.0965i 0.3939  3.0965i 0.3938 + 3.0965i 0.3938  3.0965i 0.0520 + 0.0000i 0.0351 + 0.0000i 
1.0e+05 * 3.6779 3.6771 2.6358 2.6348 0.0373 0.0203 
1.0e+05 * 3.7485 3.7475 2.5650 2.5636 0.0461 0.0290 
1.0e+05 * 3.8117 3.8106 2.5013 2.4996 0.0552 0.0379 
3.2 Module crosscoupling audiosusceptibility
The crosscoupling audiosusceptibility is the output voltage response of one module due to a disturbance in the source voltage of another. Following the same procedure used above for deriving the direct audiosusceptibilty, the crosscoupling audio susceptibility can be derived as:
$\left(\frac{\hat{v}_{o n}}{\hat{v}_{g n}}\right)_{\text {cross }}=\frac{\Delta_1}{\Delta_2} \times \frac{A_5\left(sA_1\right)}{\left[s^2\left(A_1+A_4+(n1) A_5\right) s+A_1\left(A_4+(n1) A_5\right)A_2 A_3\right]}$ (11)
where, Δ_{1} and Δ_{2} are as given by Eq. (7); and A_{1}toA_{5} are defined in Table 2.
(a) Plots from analytical model
(b) PSIM simulation results
Figure 5. Crosscoupling audiosusceptibility with current loops closed and variable number of modules (M_{c} = 1.5)
Figure 5 shows the predicted crosscoupling audiosusceptibility responses as we vary n with slope ratio M_{c}= 1.5 while Figure 6 depicts the responses when M_{c} = 2.9. Corresponding Matlab polezero locations appear in Table 4; the denominator of this transfer function is of course the same as that of the direct audiosusceptibility, but the numerator is one order lower. From Figures 5 and 6, it can be noticed that: 1) the crosscoupling audiosusceptibility is less than the direct audiosusceptibility; 2) At low frequencies the audiosusceptibility is reduced with the addition of modules, but as perturbation frequency rises, the susceptibility will increase when modules are added.
(a) Plots from analytical model
(b) PSIM simulation results
Figure 6. Crosscoupling audiosusceptibility with current loops closed and variable number of modules (M_{c} = 2.9)
Table 4. Polezero locations in (rad/sec) for crosscoupling audiosusceptibilty with variable n

M_{C } = 1.5 
M_{C }= 2.9 


n = 2 
n = 3 
n = 4 
n = 2 
n = 3 
n = 4 
Zeros 
1.0e+05 * 0.1189 + 4.6336i 0.1189 – 4.6336i 0.3948 + 3.1167i 0.3948 – 3.1167i 
1.0e+05 * 0.0792 + 4.6218i 0.0792 – 4.6218i 0.3948 + 3.1167i 0.3948 – 3.1167i 
1.0e+05 * 0.0594 + 4.6157i 0.0594 – 4.6157i 0.3948 + 3.1167i 0.3948 – 3.1167i 
1.0e+05 * 0.1189 + 4.8924i 0.1189 – 4.8924i 3.4824 + 0.0000i 2.8341 + 0.0000i 
1.0e+05 * 0.0792 + 4.7963i 0.0792 – 4.7963i 3.4824 + 0.0000i 2.8341 + 0.0000i 
1.0e+05 * 0.0594 + 4.7474i 0.0594 – 4.7474i 3.4824 + 0.0000i 2.8341 + 0.0000i 
Poles 
1.0e+05 * 0.3941 + 3.1066i 0.3941 – 3.1066i 0.3940 + 3.1066i 0.3940 – 3.1066i 0.0349 + 0.0000i 0.0181 + 0.0000i 
1.0e+05 * 0.3940 + 3.1015i 0.3940 – 3.1015i 0.3939 + 3.1015i 0.3939 – 3.1015i 0.0434 + 0.0000i 0.0266 + 0.0000i 
1.0e+05 * 0.3939 + 3.0965i 0.3939 – 3.0965i 0.3938 + 3.0965i 0.3938 – 3.0965i 0.0520 + 0.0000i 0.0351 + 0.0000i 
1.0e+05 * 3.6779 3.6771 2.6358 2.6348 0.0373 0.0203 
1.0e+05 * 3.7485 3.7475 2.5650 2.5636 0.0461 0.0290 
1.0e+05 * 3.8117 3.8106 2.5013 2.4996 0.0552 0.0379 
In order to find the module direct and crosscoupling audiosusceptibilities when the current and voltage feedback loops are closed, the general expressions given by Eqns. (10) and (11) can be used but after updating four terms according to Eq. (5). These terms are A_{2}, A_{4}, B_{1} and B_{3} which include the output voltage attenuation factor K_{v} and the compensated error amplifier transfer function F_{v}.
Designing the module voltage feedback loop compensator is based on its controltooutput voltage response when no peaking is present (i.e. when M_{c} = 2.9). The controltooutput voltage transfer function is obtained following the same approach used for deriving the audiosusceptibilities and can be expressed as:
$\frac{\hat{v}_{\text {on }}}{\hat{v}_{c n}}=\frac{\Delta_3}{\Delta_2} \times \frac{\left[s^2\left(A_1+A_4+(n2) A_5\right) s+A_1\left(A_4+(n2) A_5\right)A_2 A_3\right]}{\left[s^2\left(A_1+A_4+(n1) A_5\right) s+A_1\left(A_4+(n1) A_5\right)A_2 A_3\right]}$ (12)
where, Δ_{2} is given by (7b) and
$\Delta_3=B_3\left(sA_1+A_3 B_1 / B_3\right)$ (13)
where the A and B terms are in Table 2.
The compensator should be designed for the case when n is maximum (i.e. n = 4) to ensure that the system is stable when lower number of modules is employed [14]. Usual compensator design procedure used with singlecell converters [38, 39] can be applied for each module. The chosen voltageloop crossover frequency and phase margin is 800 Hz and 60° respectively.
(a) Plots from analytical model
(b) PSIM simulation results
Figure 7. Direct (self) audiosusceptibility with current and voltage feedback loops closed, variable n, and F_{v }= [6974 (1+s/1919) / s(1+s/13170)]
(a) Plots from analytical model
(b) PSIM simulation results
Figure 8. Crosscoupling audiosusceptibility with current and voltage feedback loops closed, variable n, and F_{v }= [6974 (1+s/1919) / s(1+s/13170)]
Figures 7 and 8 show the direct and crosscoupling audiosusceptibilities with current and voltage feedback loops closed and variable number of modules. It can be seen that closing the voltage feedback loops benefit the region below the crossover frequency only. Lower audiosusceptibilities can be obtained if the module voltage loop has a higher crossover frequency, but in the IISO boost converter this crossover frequency is limited by the ndependent righthalfplane zero which appears in the module $\hat{v}_o / \hat{v}_c$ transfer function at the angular frequency $\omega=R(1D)^2 / n L$.
Figure 7 also shows that increasing n will slightly reduce the direct audiosusceptibility at frequencies below 500 Hz; otherwise, any increase in the number of modules will increase the direct audiosusceptibility. As for the crosscoupling audiosusceptibility, Figure 8 shows that adding modules reduces the susceptibility at frequencies below 900 Hz (by ≈ 7 dB when n is changed from 2 to 4); however, at higher perturbation frequencies this susceptibility increases when modules are added. Due to system’s complexity, the relationship between audiosusceptibilities and the number of modules cannot be quantified by a simple formula, but with a computer program based on the expressions proposed in this work the designer can straightforwardly find the effect of different parameters on the converter audiosusceptibilities.
Symbolic smallsignal expressions are presented for both the direct and crosscoupling audiosusceptibilities of the peak currentmode controlled independentinput seriesoutput (IISO) boost dcdc converter. The derived expressions contain the sampleandhold effect of the current loops. Using these expressions, the frequency responses under closed current and voltage feedback loop conditions are generated for different number of modules (n). The analytical model results correlate well with the ones obtained from PSIM simulations up to half the switching frequency region.
With the inner (current) and outer (voltage) feedback loops closed, the following can be stated about the module audiosusceptibility: 1) for a certain number of modules the crosscoupling susceptibility is always less than the direct susceptibility; 2) direct susceptibility is slightly reduced when n is increased but only over a small frequency range below the voltageloop crossover frequency; else, this susceptibility increases with the addition of modules; and 3) crosscoupling susceptibility decreases as n is increased, but only at frequencies below voltageloop crossover frequency, otherwise, this audiosusceptibility will increase when modules are added.
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