# Control-to-Output Transfer Function Measurement of a Boost Converter Using Synchronized Multifrequency Excitation

Sophia Roesel, Daniel Breidenstein, Thomas Duerbaum

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Sophia Roesel, Daniel Breidenstein, Thomas Duerbaum. Control-to-Output Transfer Function Measurement of a Boost Converter Using Synchronized Multifrequency Excitation. The 26th European Conference on Power Electronics and Applications, GDR SEEDS France & EPE Association, Mar 2025, Paris, France. 10.34746/epe2025-0030. hal-05075716

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Distributed under a Creative Commons Attribution 4.0 International License
Control-to-Output Transfer Function Measurement of a Boost
Converter Using Synchronized Multifrequency Excitation

Sophia Roesel

Friedrich-Alexander-Universität
Erlangen-Nürnberg

Department EEI, Institute of Optoelectronics
Erlangen, Germany

E-Mail: sophia.roesel@fau.de

Daniel Breidenstein

Friedrich-Alexander-Universität
Erlangen-Nürnberg

Department EEI, Institute of Optoelectronics
Erlangen, Germany

E-Mail: daniel.breidenstein@fau.de

Thomas Duerbaum

Friedrich-Alexander-Universität
Erlangen-Nürnberg

Department EEI, Institute of Optoelectronics
Erlangen, Germany

E-Mail: thomas.duerbaum@fau.de

*Index Terms*—System identification, Converter Control, Frequency-domain Analysis, Aliasing Effects, Pulse Width Modulation (PWM)

*Abstract*—An optimal controller design requires knowledge about the control-to-output transfer function of the converter. Instead of a sine sweep, this paper presents a faster method by a synchronized multifrequency excitation, including the compensation of aliasing effects. The measurement results show excellent agreements with the common sine method.

# I. INTRODUCTION

Switched mode power supplies (SMPS) have to meet a high standard regarding efficiency, electromagnetic compatibility, and dynamic performance. With respect to the topic of this paper, the last point is of special interest. Independent of line and load variations, the controller has to ensure a constant output voltage of the converter. For designing the control loop, the small signal control-to-output transfer function needs to be obtained in a first step. Typically, for measuring the transfer function with a gain-phase analyzer, a sweep of sine signals with different perturbation frequencies is injected into the system and the response is measured. This measurement procedure takes a lot of time since it has to be repeated for a set of sinusoidal waveforms. To reduce the measurement time, literature

discusses other possibilities of receiving the control-to-output transfer function for control loop design or adaptive control [1, 2]. For digital controlled systems [1–3] receive the transfer function with cross-correlation. [4] and [5] disturb an analog controlled system with a digital, multifrequency signal and measure the loop gain. This paper focuses on measuring the control-to-output transfer function of voltage mode controlled topologies. It uses exemplarily a boost converter. The controller relies on an analog approach, leading to the generation of the pulse width of the gate signal with natural sampling. Therefore, this paper proposes a method of disturbing the analog controller with a multifrequency maximum binary length sequence (MLBS) signal to receive the control-to-output transfer function. Synchronizing the clock of the analog controller with the microcontroller leads to an improvement in the accuracy of the obtained results. When perturbing with a multifrequency signal, the problem of aliasing effects occurs in the process of the PWM-signal generation. The dealing with compensation of the aliasing effects is explained. Overall, the measurements with the two methods obtain the control-to-output transfer function with very good agreement.

Before the method of receiving the transfer function with a multifrequency perturbation is explained in Section III, Section II revises the commonly used method of perturbing the system with one sinusoidal waveform
at a time. The results obtained with this approach serve as a reference for the measurements with the multifrequency perturbation. Section III additionally deals with the compensation of the aliasing effects.

## II. BOOST CONVERTER AND SINE SWEEP MEASUREMENT

The gain-phase analyzer disturbs the system at one perturbation frequency $f_m$ and measures the response of the system at the reference and test channels. Equation (1) delivers the complex value of the transfer function at $f_m$. Magnitude and phase can be extracted from Eq. (1).

$$ \underline{G}(j2\pi f_m) = \frac{\hat{u}_{\text{Test}}(j2\pi f_m)}{\hat{u}_{\text{Reference}}(j2\pi f_m)} \quad (1) $$

The procedure has to be repeated for multiple perturbation frequencies to obtain the transfer function. Due to the non-linear properties of switching converters, necessary considerations are discussed in the following.

### A. Transfer function measurement of SMPS and Comp-Signal as Reference

[6] describes the approach of measuring the loop gain of a SMPS. This method can be adapted to measure only the control-to-output transfer function. Fig. 1 shows the setup for measuring the plant, which is, in case of this paper, a boost converter. Regardless of studying the loop gain or the plant, the measurement has to be executed in closed loop control to keep the SMPS at its operation point [6]. Therefore, a resistor $R_{inj}$ is added to the

feedback path to inject the disturbance. The output of the controller-OPV is chosen as the injection point, but other options are also possible [7], but not examined in this paper. The sine, which disturbs the system, is then injected across the resistor $R_{inj}$ by means of an injection transformer $T_{inj}$. A careful consideration of the amplitude of the perturbation sine has to be made [8]. A too small amplitude renders the measurement inaccurately because of noise. This occurs at low frequencies due to the controller gain and at high frequencies because of the attenuation of the plant. On the other hand, at a too high amplitude non-linearities can occur, and the operation point changes. After evaluating an accurate disturbance amplitude, the perturbation of the output voltage $u_{out}$ (o) is measured and compared to the voltage at the positive input of the PWM-comparator $u_{comp}$ (c), in order to measure the control-to-output transfer function $\underline{G}_{c\to o}$:

$$ \underline{G}_{c\to o}(j2\pi f_m) = \frac{\hat{u}_{\text{out}}(j2\pi f_m)}{\hat{u}_{\text{comp}}(j2\pi f_m)} \quad (2) $$

To receive the control-to-output transfer function of the plant, a simple controller with integrating behavior can be used. A very low corner frequency can thereby reduce noise at the lowest frequencies of interest. The knowledge of the control-to-output transfer function $\underline{G}_{c\to o}$ allows for the optimization of the controller in the next step.

### B. PWM-Signal as Reference

Although $\underline{G}_{c\to o}$ represents the desired control-to-output transfer function for designing the controller, a

<figure><img src="image_2.png" /><figcaption>Figure 1. Single frequency measurement setup</figcaption></figure>
second measurement with the PWM-signal (p) at $u_{PWM}$ as a reference is useful for further explanations on the occurrence of aliasing effects in case of a multifrequency perturbation. The PWM-to-output transfer function results then from:

$$ \underline{G}_{p \to o} (j2\pi f_m) = \frac{\hat{u}_{out} (j2\pi f_m)}{\hat{u}_{PWM} (j2\pi f_m)} \quad (3) $$

With the control-to-PWM function $\underline{G}_{c \to p}$, the control-to-output transfer function $\underline{G}_{c \to o}$ can be received by:

$$ \underline{G}_{c \to o} (j2\pi f_m) = \underline{G}_{p \to o} (j2\pi f_m) \cdot \underline{G}_{c \to p} \quad (4) $$

For a single frequency small signal perturbation, $\underline{G}_{c \to p}$ corresponds to a constant factor given by Eq. (5).

$$ \underline{G}_{c \to p} = \hat{u}_{PWM} \cdot f_s \cdot \frac{\Delta t_{ramp}}{\Delta u_{ramp}} \quad (5) $$

$\underline{G}_{c \to p}$ depends on the amplitude $\hat{u}_{PWM}$ of the PWM-signal, the switching frequency $f_s$ as well as the slope of the sawtooth, both visualized in Fig. 2.

<figure><img src="image_3.png" /><figcaption>Figure 2. PWM generation with natural sampling</figcaption></figure>

### C. Sine Sweep Measurement Results

To verify the multifrequency method, measurements are performed on the boost converter shown in Fig. 3. The PCB (printed circuit board) consists of the boost power stage and the analog controller. As already mentioned, the controller has an integrating behavior with a corner frequency at a very low frequency in order to facilitate the measurement of the plant. If the multifrequency signal disturbs the system, the microcontroller is necessary for perturbation. In the setup shown in Fig. 1 the microcontroller also provides the clock for the analog controller for synchronized perturbation in Section III. Furthermore, the measurement setup includes measurement points for $u_{out}$, $u_{PWM}$ and $u_{comp}$, and the injection point for perturbing the system.

<figure><img src="image_4.png" /><figcaption>Figure 3. Analog controlled boost converter with injection and measurement points</figcaption></figure>

Table I summarizes the specification and parameters of the boost converter.

<table><caption>Table I
PARAMETERS OF THE BOOST CONVERTER</caption><thead><tr><td>Parameter</td><td>Value</td></tr></thead><tbody><tr><td>$U_{in}$</td><td>5 V</td></tr><tr><td>$U_{out}$</td><td>12 V</td></tr><tr><td>$f_s$</td><td>100 kHz</td></tr><tr><td>$R_L$</td><td>50 Ω</td></tr><tr><td>$L$</td><td>220 μH</td></tr><tr><td>$C_{out}$</td><td>66 μF</td></tr><tr><td>$\hat{u}_{PWM}$</td><td>7.6 V</td></tr><tr><td>$\frac{\Delta u_{ramp}}{\Delta t_{ramp}}$</td><td>0.25 V μs<sup>-1</sup></td></tr></tbody></table>

Fig. 4 shows the resulting magnitude and phase of the control-to-output transfer function, received by measuring the disturbance of the output voltage $u_{out}$ in comparison to the reference $u_{comp}$ on one hand and on the other hand in comparison to $u_{PWM}$. The measurements are performed with a HP 4194A gain-phase analyzer and the measurements are shown and valid until half of the switching frequency ($f_s/2$). They serve as reference for a comparison with the multifrequency perturbation.

Due to the nature of the measurement procedure, obtaining the entire transfer function takes a long time. It is additionally stretched by the settling time to reach the steady state for each frequency $f_m$ after injecting the perturbation. To avoid such long measurement times the multifrequency perturbation is presented in the following.
<figure><img src="image_5.png" /><figcaption>Figure 4. Measured control-to-output transfer function with the single frequency method at OSC-level = 20 mV</figcaption></figure>

### III. SYNCHRONIZED MULTIFREQUENCY MEASUREMENT

Now that the commonly used procedure for measuring the control-to-output transfer function of the plant is revised, a faster method is explained. This is done by exciting the system with a multifrequency binary sequence in order to receive the control-to-output transfer function. The corresponding measurement setup is illustrated in Fig. 5. The microcontroller provides the clock of the analog controller and outputs the binary

perturbation sequence, which are both synchronized. This synchronization is necessary to ensure that exactly one perturbation bit of the MLB sequence is injected during one switching period.

The binary excitation signal is injected over an isolated gate driver supplied by an accumulator ($U_{bat}$). The binary perturbation is then either zero or one for each switching period in the PWM-signal generated by the microcontroller. The amplitude of the perturbation $A_{perturb}$ is set by the voltage divider $R_{inj}$ and $R_{div}$ and the voltage of the accumulator $U_{bat}$:

$$A_{perturb} = U_{bat} \cdot \frac{R_{inj}}{R_{inj} + R_{div}} \quad (6)$$

#### A. Injection Signals

The literature discusses numerous multifrequency binary signals suitable for system identification, summarized in [9, 10]. The characterization of SMPS commonly uses the maximum-length binary sequence (MLBS) and discrete-interval binary sequence (DIBS) [4, 5, 11].

MLBS has a flat spectrum, resulting in the same disturbance amplitude at all multiples of the fundamental perturbation frequency. If this results in measurement problems at high frequencies because of a too small amplitude, a desired spectrum can be optimized using DIBS. Fig. 6 shows the simulated spectrum for both perturbation options. The DIB sequence is optimized in this case for higher amplitudes at higher frequencies.

<figure><img src="image_6.png" /><figcaption>Figure 5. Multifrequency measurement setup</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Figure 6. Spectrum of MLBS and DIBS</figcaption></figure>

Nevertheless, this paper considers only the excitation with MLBS. In short, MLBS is created with a m-bit shift register and an XOR-feedback. The length of the resulting sequence equals $N = 2^m - 1$. Fig. 7 visualizes the creation of the MLBS. A more detailed analysis is provided for example in [12].

<figure><img src="image_8.png" /><figcaption>Figure 7. MLBS generation</figcaption></figure>

The measurements in time domain for $u_{out}$, $u_{comp}$ and $u_{PWM}$ are recorded with the digitizer card PXIe-5122. This card has a sufficient vertical and horizontal resolution, which is necessary for accurate results. Applying AC-coupling further increases the resolution since only the response to the perturbation, and not the DC operation point, is relevant for the control-to-output transfer function. Fig. 8 shows the waveforms in time domain for $u_{out}$ and $u_{comp}$ for the whole perturbation period. The MLB sequence is 1023 Bits long and sampled with a rate of 100 MS s⁻¹ by the used ADC capture card. The value of $R_{div}$ equals 10 kΩ and the resistor $R_{inj}$ has a value of 50 Ω. The accumulator has a voltage $U_{bat}$ of 8 V. This leads, according to Eq. (6), to a perturbation amplitude of 40 mV, which results in a small signal perturbation of 1.5 %.

Due to the superimposed interactions of the controller on the COMP-signal the post-processing with commonly used cross-correlation [3, 12], is not feasible to extract the control-to-output transfer function of the plant. This is the reason because the spectrum of $u_{comp}$ is not flat in

<figure><img src="image_9.png" /><figcaption>Figure 8. Measured waveforms (AC-coupled)</figcaption></figure>

contrast to MLBS, which is a necessity in order to get the control-to-output transfer function via cross-correlation. Therefore, the spectrum method is used.

## B. Spectrum Method

After receiving the signals in time domain, the next step performs a Fast Fourier Transformation (FFT) on $u_{out}$ and either $u_{comp}$ or $u_{PWM}$ over one perturbation period. The amplitude and phase of the control-to-output transfer function $\underline{G}_{c \to o}$ result from:

$$|\underline{G}_{c \to o}(jk\omega_m)| = 20 \cdot \log_{10} \left( \frac{|\hat{u}_{out}(jk\omega_m)|}{|\hat{u}_{comp}(jk\omega_m)|} \right) \quad (7)$$

$$\angle (\underline{G}_{c \to o}(jk\omega_m)) = \angle \hat{u}_{out}(jk\omega_m) - \angle \hat{u}_{comp}(jk\omega_m) \quad (8)$$

To receive the transfer function $\underline{G}_{p \to o}$, $\hat{u}_{comp}$ has to be exchanged with $\hat{u}_{PWM}$ in Eq. (7) und Eq. (8).

The resulting control-to-output transfer function for $\underline{G}_{c \to o}$ as well as $\underline{G}_{p \to o}$ are shown in Fig. 9. The measurements from Section II serve as the reference.
<figure><img src="image_10.png" /><figcaption>Figure 9. Measured control-to-output transfer function with multi-frequency method without compensation for aliasing</figcaption></figure>

A comparison between the results received by the single frequency perturbation and multifrequency perturbation for the PWM-to-output transfer function $G_{p→o}$ shows good agreement. For $G_{c→o}$ a magnitude and phase error at higher frequencies can be observed, which is caused by aliasing effects during the PWM-generation.

### C. Aliasing in PWM-Generation

Considering the PWM-generation process, $u_{PWM}$ is only defined by its discrete duty cycle values $d[n]$. For a periodically disturbed PWM-signal, like the one illustrated in Fig. 10, the spectrum of $u_{PWM}$ can be expressed with Eq. (9) in case of a small signal perturbation around the unperturbed duty cycle $D$ (see [13]).

$$ \hat{u}_{PWM}(jk\omega_m) = \frac{2\hat{u}_{PWM}}{N} \sum_{n=0}^{N-1} \left( e^{-j\frac{2\pi}{N}k \cdot n} d[n] \right) e^{-j\frac{2\pi}{N}kD} \quad (9) $$

<figure><img src="image_11.png" /><figcaption>Figure 10. Periodic disturbance of a PWM-signal</figcaption></figure>

Note that it only depends on the DFT (Discrete Fourier Transformation) of the discrete duty cycle values of each switching interval as well as a phase correction with the unperturbed duty cycle $D$.

### D. Compensation of Aliasing

To cancel out the effect of aliasing, the discrete value of $u_{comp}$ at the switching instance ($d[n] \cdot T_s$) for each switching period has to be taken as a reference instead of the continuous $u_{comp}$-signal.

Fig. 11 displays two measured switching periods. This figure illustrates exemplarily how the discrete values for the first and second switching periods are obtained. This is done by extracting the value of $u_{comp}$ at the time $d[n] \cdot T_s$. The switching instances at $d[n] \cdot T_s$ and $T_s$ are obtainable from the high frequency oscillations occurring at the time domain waveforms of $u_{out}$. In addition to the discrete $u_{comp}$ voltage for each switching period, the unperturbed duty cycle $D$ is necessary. It equals the average value of all switching periods:

$$ D = \bar{d}[n] \quad (10) $$

<figure><img src="image_12.png" /><figcaption>Figure 11. Extraction of discrete $u_{comp}$ voltages</figcaption></figure>

The control-to-output transfer function $G_{c→o}$, with elimination of the aliasing effects, can now be calculated with Eq. (11) when considering the DFT of the discrete $u_{comp}$ voltages and the phase correction as the reference signal:

$$ \underline{G}_{c \to o}(jk\omega_m) = \frac{\hat{u}_{out}(jk\omega_m)}{\frac{2}{N} \sum_{n=0}^{N-1} \left( e^{-j\frac{2\pi}{N}k \cdot n} \cdot u_{comp}[n] \right)} \cdot e^{j\frac{2\pi}{N}kD} \quad (11) $$
The comparison between the multifrequency and single frequency method when compensating the aliasing effects shows good agreement in Fig. 12.

<figure><img src="image_13.png" /><figcaption>Figure 12. Measured control-to-output transfer function with multifrequency method with compensation for aliasing</figcaption></figure>

## IV. CONCLUSION

This paper describes and analyzes different approaches for getting the control-to-output transfer function of the plant. This information is necessary for optimizing the controller, which can be done in the next step. A boost converter served as a simple example for the measurement of the plant. First, the commonly used method of a single frequency sweep is explained and the results are taken as a reference. As a second option, the perturbation of the system with MLBS, is presented. The advantage of this multifrequency perturbation is a shorter measurement time compared to the traditional sine sweep of a gain-phase analyzer. The PWM-to-output transfer function $G_{\mathrm{P}\to\mathrm{o}}$ shows good agreement with the results of the single frequency perturbation. For the control-to-output transfer function $G_{\mathrm{C}\to\mathrm{o}}$, deviations at higher frequencies are visible. The reason for this are aliasing effects in the generation of the PWM-signal. After an explanation on how the aliasing effects can be compensated, the control-to-output transfer function, received by perturbing with MLBS, shows good agreement with the results from the gain-phase analyzer. Therefore, a faster way of receiving the control-to-output transfer function is presented.

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