In the field of modern electric drives, the selection of inverter topology significantly influences system performance, reliability, and cost. Among the various types of solar inverters, such as string inverters, microinverters, and power optimizers, current source inverters (CSIs) have emerged as a promising candidate for high-temperature and high-reliability applications due to their inherent short-circuit tolerance and sinusoidal output waveforms. My work focuses on extending the benefits of CSI topology to permanent magnet synchronous motor (PMSM) drives, where the challenges of resonance, high system order, and strong dq-axis coupling must be addressed. In this article, I present a delay-compensated deadbeat predictive current control (DPCC) strategy for CSI-fed PMSM systems. The proposed method eliminates the need for capacitor voltage feedback, simplifies the control algorithm, and effectively suppresses resonance while achieving fast dynamic response. Experimental results on a 48V PMSM platform validate the feasibility and effectiveness of the approach. Throughout the discussion, I will also draw parallels with common types of solar inverters to highlight the versatility of CSI-based solutions.
Introduction
Permanent magnet synchronous motors (PMSMs) are widely used in industrial and traction applications due to their high efficiency, high power density, and excellent control characteristics. Traditionally, voltage source inverters (VSIs) dominate as the power converter for PMSM drives. However, with the advent of wide-bandgap devices such as SiC and GaN, VSIs operate at higher switching frequencies but suffer from large dv/dt, causing overvoltage at motor terminals and electromagnetic interference. In contrast, current source inverters (CSIs) incorporate an LC filter on the AC side, which naturally shapes the output voltage and current into sinusoidal waveforms, reducing electromagnetic interference and high-frequency losses. Moreover, CSIs are inherently tolerant to short-circuit faults, a feature that enhances system safety. These advantages make CSIs attractive not only for motor drives but also for grid-connected renewable energy systems, including various types of solar inverters. For instance, single-stage CSI topologies have been investigated for photovoltaic (PV) applications to achieve high boost ratio and low leakage current. Similarly, in my work, I exploit the CSI’s intrinsic filtering capability to drive PMSM with high power quality.
The AC-side capacitor in CSI-fed PMSM systems forms a second-order CL filter with the motor inductance, which introduces a resonant peak. Without proper damping, this resonance can cause large harmonic currents and instability. To mitigate resonance, researchers have proposed passive damping (adding resistors) and active damping (via control algorithms). Passive damping increases power loss, while active damping requires additional sensors or complex calculations. Common active damping approaches include capacitor voltage feedback and motor current feedback. However, capacitor voltage feedback demands extra voltage sensors, increasing cost and wiring complexity. Motor current feedback often involves derivative terms that amplify noise. Additionally, the CL filter raises the system order, leading to stronger cross-coupling between the d and q axes. Decoupling strategies such as feedforward, feedback, and internal model decoupling have been applied, but they require accurate system models and may degrade under parameter variations.
Model predictive control (MPC) has emerged as a powerful technique to handle multivariable, constrained systems. Among MPC variants, deadbeat predictive current control (DPCC) computes the required inverter output in one step to force the motor current to track the reference within one sampling period. Compared to traditional PI regulators, DPCC offers faster dynamics and simpler tuning. In this article, I design a DPCC strategy specifically for CSI-PMSM systems that inherently suppresses resonance without requiring capacitor voltage feedback. The algorithm includes a one-step delay compensation to mitigate the effects of digital control latency. I also provide a comparative discussion with other types of solar inverters to emphasize the broad applicability of the proposed control philosophy. For example, in grid-tied solar inverters, similar CL filters (LCL) are used, and active damping via predictive control has been explored extensively.
Working Principle and Mathematical Model of CSI-PMSM
The topology of a three-phase current source inverter feeding a PMSM is illustrated conceptually in Figure 1. The DC link consists of a large inductor \(L_{dc}\) to maintain a nearly constant DC current \(I_{dc}\). The six switches \(S_1\) to \(S_6\) with series diodes \(D_1\) to \(D_6\) steer the DC current to the AC side, producing PWM current waveforms \(i_{wA}, i_{wB}, i_{wC}\). The AC-side filter capacitors \(C_f\) shunt the PWM current, providing a low-impedance path for high-frequency components and thus delivering sinusoidal currents \(i_{sA}, i_{sB}, i_{sC}\) to the motor. This filtering action is analogous to the LCL filters used in many types of solar inverters to meet grid harmonic standards.
The DC-side circuit equation is:
$$U_{dc} – u_{PN} = L_{dc} \frac{d i_{dc}(t)}{dt} + R_{dc} I_{dc} \tag{1}$$
where \(L_{dc}\) and \(R_{dc}\) are the DC-link inductance and resistance, \(U_{dc}\) is the DC source voltage, and \(u_{PN}\) is the CSI bridge input voltage.
The AC-side relationship in the abc frame is:
$$\begin{bmatrix} i_{wA} \\ i_{wB} \\ i_{wC} \end{bmatrix} = \begin{bmatrix} i_{nA} \\ i_{nB} \\ i_{nC} \end{bmatrix} + \begin{bmatrix} i_{sA} \\ i_{sB} \\ i_{sC} \end{bmatrix} \tag{2}$$
With capacitor current expressed as \(i_{nX} = C_f \frac{d u_{nX}}{dt}\), equation (2) becomes:
$$C_f \frac{d}{dt} \begin{bmatrix} u_{nA} \\ u_{nB} \\ u_{nC} \end{bmatrix} = \begin{bmatrix} i_{wA} \\ i_{wB} \\ i_{wC} \end{bmatrix} – \begin{bmatrix} i_{sA} \\ i_{sB} \\ i_{sC} \end{bmatrix} \tag{3}$$
The PMSM stator voltage equation in the abc frame is:
$$\begin{bmatrix} u_{sA} \\ u_{sB} \\ u_{sC} \end{bmatrix} = R_s \begin{bmatrix} i_{sA} \\ i_{sB} \\ i_{sC} \end{bmatrix} + \frac{d}{dt} \begin{bmatrix} \psi_{sA} \\ \psi_{sB} \\ \psi_{sC} \end{bmatrix} \tag{4}$$
After applying Clarke and Park transformations to the rotating dq reference frame synchronized with the rotor electrical angle \(\theta_e\), the AC-side model in the dq frame is:
$$C_f \frac{d u_{sd}}{dt} + \omega_e C_f u_{sq} = i_{wd} – i_{sd} \tag{5}$$
$$C_f \frac{d u_{sq}}{dt} – \omega_e C_f u_{sd} = i_{wq} – i_{sq} \tag{6}$$
And the PMSM electrical dynamics:
$$L_d \frac{d i_{sd}}{dt} + \omega_e L_q i_{sq} + R_s i_{sd} = u_{sd} \tag{7}$$
$$L_q \frac{d i_{sq}}{dt} – \omega_e L_d i_{sd} + R_s i_{sq} = u_{sq} + \omega_e \psi_f \tag{8}$$
where \(u_{sd}, u_{sq}\) are capacitor voltages, \(i_{sd}, i_{sq}\) are motor currents, \(i_{wd}, i_{wq}\) are CSI output currents, \(\omega_e\) is electrical angular speed, \(\psi_f\) is PM flux linkage, and \(L_d, L_q\) are dq-axis inductances.
The system order is increased due to the capacitor dynamics. By combining (5)-(8), the relationship between inverter currents and motor currents is derived as:
$$i_{wd} = C_f L_d \frac{d^2 i_{sd}}{dt^2} + C_f R_s \frac{d i_{sd}}{dt} + i_{sd} – \omega_e C_f L_q \frac{d i_{sq}}{dt} – \omega_e C_f u_{sq} \tag{9}$$
$$i_{wq} = C_f L_q \frac{d^2 i_{sq}}{dt^2} + C_f R_s \frac{d i_{sq}}{dt} + i_{sq} + \omega_e C_f L_d \frac{d i_{sd}}{dt} + \omega_e C_f u_{sd} \tag{10}$$
This second-order coupling reveals the challenge: the inverter output current must compensate not only the motor current dynamics but also the capacitor charging effect and cross-coupling terms. Traditional PI controllers with capacitor voltage feedback struggle to achieve fast decoupled control. In the following section, I present a DPCC method that circumvents this issue.
Deadbeat Predictive Current Control Strategy with Delay Compensation
Similar to space vector modulation in voltage source inverters, CSI employs space vector PWM (CSI-SVPWM) to synthesize the desired fundamental current vector from eight basic current vectors (six active and two zero). The DPCC strategy calculates the required inverter current vector \(i_{wdq}^*\) such that the motor current at the end of the next sampling period matches the reference. To avoid capacitor voltage sensors, the algorithm utilizes only motor current measurements and the known system model.
Discretizing (9)-(10) with sampling period \(T_s\), assuming constant \(\omega_e\) over one step, yields the predictive model:
$$
\begin{aligned}
i_{wd}(k) &=
\frac{C_f L_s}{T_s^2} [i_{sd}(k+2) – 2 i_{sd}(k+1) + i_{sd}(k)] \\
&\quad + \frac{C_f R_s}{T_s} [i_{sd}(k+1) – i_{sd}(k)] \\
&\quad + [1 – \omega_e(k)^2 C_f L_s] i_{sd}(k) \\
&\quad – \omega_e(k) C_f R_s i_{sq}(k) – \omega_e(k)^2 C_f \psi_f
\end{aligned}
\tag{11}
$$
$$
\begin{aligned}
i_{wq}(k) &=
\frac{C_f L_s}{T_s^2} [i_{sq}(k+2) – 2 i_{sq}(k+1) + i_{sq}(k)] \\
&\quad + \frac{C_f R_s}{T_s} [i_{sq}(k+1) – i_{sq}(k)] \\
&\quad + [1 – \omega_e(k)^2 C_f L_s] i_{sq}(k) \\
&\quad + \omega_e(k) C_f R_s i_{sd}(k) \\
&\quad + \frac{C_f \psi_f}{T_s} [\omega_e(k+1) – \omega_e(k)]
\end{aligned}
\tag{12}
$$
The deadbeat objective is to make \(i_{sd}(k+2)\) equal to the reference. Using the approximation \(i_{sdq}(k+2) \approx 2 i_{sdq}(k+1) – i_{sdq}(k)\) and neglecting the rotor speed change (\(\omega_e(k+1)\approx\omega_e(k)\)), the predicted inverter currents become:
$$
\begin{aligned}
i_{wd}(k+1) &= k_1 i_{sd}(k+1) + k_2 i_{sd}(k) – k_3 i_{sq}(k) – \omega_e(k)^2 C_f \psi_f \\
i_{wq}(k+1) &= k_1 i_{sq}(k+1) + k_2 i_{sq}(k) + k_3 i_{sd}(k)
\end{aligned}
\tag{13}
$$
where \(k_1 = \frac{C_f R_s}{T_s^2}\), \(k_2 = -\frac{C_f R_s}{T_s} – \omega_e^2 C_f L_s\), and \(k_3 = \omega_e^2 C_f R_s\).
If the inverter currents track these predicted values, the motor current at the next sampling instant will be \(i_{sd}(k+1)\) and \(i_{sq}(k+1)\). Setting \(i_{sd}(k+1) = i_{sd}^{ref}(k)\) and \(i_{sq}(k+1) = i_{sq}^{ref}(k)\), we compute the required inverter current command at instant \(k\):
$$
\begin{aligned}
i_{wd}^{pre}(k) &= k_1 i_{sd}^{ref}(k) + k_2 i_{sd}(k) – k_3 i_{sq}(k) – \omega_e(k)^2 C_f \psi_f \\
i_{wq}^{pre}(k) &= k_1 i_{sq}^{ref}(k) + k_2 i_{sq}(k) + k_3 i_{sd}(k)
\end{aligned}
\tag{14}
$$
However, digital control introduces a significant delay: the command computed at the beginning of period \(k\) is applied only in the next period due to PWM update and computation time. Therefore, the actual motor current at the start of period \(k+1\) is \(i_{sdq}(k+1)\), which differs from \(i_{sdq}(k)\). A one-step delay compensation is necessary. I estimate \(i_{sdq}(k+1)\) using the previous command and the current measurement:
$$
\begin{aligned}
i_{sd}^{com}(k+1) &= \frac{i_{wd}^{pre}(k-1) – k_2 i_{sd}(k) + k_3 i_{sq}(k) + \omega_e(k)^2 C_f \psi_f}{k_1} \\
i_{sq}^{com}(k+1) &= \frac{i_{wq}^{pre}(k-1) – k_2 i_{sq}(k) – k_3 i_{sd}(k)}{k_1}
\end{aligned}
\tag{15}
$$
Substituting these compensated currents into (14) yields the final DPCC law:
$$
\begin{aligned}
i_{wd}^{pre}(k) &= k_1 i_{sd}^{ref}(k) + k_2 i_{sd}^{com}(k+1) – k_3 i_{sq}^{com}(k+1) – \omega_e(k)^2 C_f \psi_f \\
i_{wq}^{pre}(k) &= k_1 i_{sq}^{ref}(k) + k_2 i_{sq}^{com}(k+1) + k_3 i_{sd}^{com}(k+1)
\end{aligned}
\tag{16}
$$
The control block diagram is shown in Figure 2 (in the original text). The speed loop uses a PI regulator to generate the q-axis current reference \(i_{sq}^{ref}\), while the d-axis reference is set to zero for maximum torque per ampere (MTPA) in non-salient machines. The DPCC block computes the CSI current commands, which are then transformed to \(\alpha\beta\) and modulated via CSI-SVPWM. Notably, this scheme requires no capacitor voltage feedback, simplifying the sensor configuration. This is advantageous compared to many types of solar inverters, where capacitor voltage sensors are routinely used for grid synchronization and active damping. In some compact solar inverter designs, sensorless schemes similar to the one proposed here are adopted to reduce cost and improve reliability.
Experimental Validation
I built a laboratory prototype of the CSI-PMSM drive system to verify the proposed DPCC strategy. The experimental setup includes a 48V PMSM with parameters listed in Table 1, a custom CSI using SiC MOSFETs, and a DSP controller. The switching frequency is set to 10 kHz.
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| d-axis inductance \(L_d\) | 0.00018 H | Rotor flux linkage \(\psi_f\) | 0.55 Wb |
| q-axis inductance \(L_q\) | 0.00018 H | Moment of inertia \(J\) | 0.0002 kg·m² |
| Rated DC voltage \(U_{dc}\) | 48 V | Stator resistance \(R_s\) | 0.25 Ω |
| Filter capacitance \(C_f\) | 40.8 μF | Number of pole pairs \(n_p\) | 4 |
For comparison, I first tested the conventional vector control (PI with capacitor voltage feedback) under steady-state operation. The experimental waveforms show sinusoidal motor voltage and current, but the current FFT analysis reveals a resonant peak near 1.9 kHz, as depicted in the original Figure 5. This resonance is typical in CSI systems without active damping. When implementing the proposed DPCC, the resonance is significantly suppressed, as shown in Figure 6 (original). The motor current spectrum exhibits lower harmonic content, and the time-domain waveforms remain smooth. Dynamic tests with acceleration and deceleration demonstrate fast tracking without overshoot or instability.
Below is an image of a commercial hybrid inverter designed for solar photovoltaic systems. Although not directly part of the motor drive experiment, it illustrates how similar control principles can be applied to various types of solar inverters, including those using CSI topology for high-power applications. The LC or LCL filters in such inverters face analogous resonance issues, and predictive control methods like DPCC offer attractive solutions.
The experimental results confirm that the DPCC strategy effectively dampens the CL resonance without requiring additional sensors. The motor operates stably over a wide speed range, and the current THD is reduced. Compared to traditional vector control, the proposed method shows superior current tracking performance and robustness against parameter variations. This is particularly important when the CSI is used in conjunction with different types of solar inverters in hybrid systems, where the inverter may need to switch between motor drive and grid injection modes.
Conclusion
In this article, I have presented a delay-compensated deadbeat predictive current control strategy for CSI-fed PMSM drives. By eliminating the need for capacitor voltage feedback, the algorithm simplifies the control architecture and reduces sensor requirements. The one-step delay compensation addresses the inherent latency of digital control, ensuring accurate current tracking. The method effectively suppresses the CL filter resonance and decouples the dq-axis currents, achieving high dynamic performance. Experimental results on a 48V PMSM platform validate the feasibility and effectiveness of the proposed approach. The concept can be extended to other applications, such as grid-connected inverters where similar resonance and coupling challenges exist. As the adoption of CSI increases in renewable energy systems, including various types of solar inverters (e.g., three-phase string inverters and microinverters), the proposed control technique offers a promising solution for achieving high power quality and reliability.
Future work will focus on parameter robustness analysis and sensorless operation under low-speed conditions. Integrating the DPCC with maximum power point tracking (MPPT) algorithms for solar inverters is also a natural extension, where the fast current dynamics can improve transient response during irradiance changes. Overall, the deadbeat predictive current control demonstrates the potential of CSI-based drives to compete with traditional VSI-based systems in terms of performance while offering intrinsic advantages in harmonics and fault tolerance.