The global consensus on carbon neutrality has catalyzed the rapid integration of renewable energy generation systems into electrical power grids. As the primary interface for connecting these sources, such as wind and photovoltaic power, to the grid, the proportion of grid-connected inverters (GCIs) has increased dramatically. Voltage source inverters (VSIs) employing pulse width modulation (PWM) are widely adopted due to their relatively simple circuitry and ease of control. A critical protective measure in these inverters is the implementation of a dead-time between the complementary gate signals of the upper and lower switches in each phase leg, which is essential to prevent shoot-through faults. However, this necessary dead-time introduces non-idealities, distorting the output voltage and current waveforms. One of the most significant consequences is the generation of low-order harmonic components, often referred to as dead-time harmonics. Since the output filters of GCIs are primarily designed to attenuate high-frequency switching harmonics, they offer limited attenuation for these low-frequency dead-time components. In power systems with a high penetration of power electronics, the interaction between these inherent dead-time harmonics and the grid can exacerbate harmonic pollution and increase resonance risks. Therefore, developing an accurate harmonic source model for GCIs that accounts for dead-time effects is crucial for precise harmonic power flow analysis and system stability assessment.
Extensive research has been conducted on dead-time harmonic characteristics, primarily focusing on open-loop inverter operation. Analytical methods using single or double Fourier series have been employed to quantify the harmonic spectrum of the output voltage. The dead-time effect can be modeled as a voltage error pulse sequence opposing the current direction, which, when approximated and decomposed, reveals the introduction of fundamental and odd-order harmonic components. For a balanced three-phase system, the interaction between phases results in the prominent presence of harmonics of orders 6k±1 (where k is a positive integer), such as the 5th, 7th, 11th, and 13th. The magnitude of these harmonics is proportional to the switching frequency ($$f_c$$), dead-time duration ($$t_d$$), and DC-link voltage ($$U_{dc}$$). While open-loop analyses provide valuable insights, they fail to capture the dynamic behavior of a grid connected inverter under closed-loop control, which is the standard operating condition in practical applications.
In closed-loop modeling, a common approach in existing literature is to treat the dead-time harmonic voltage as a constant disturbance voltage source superimposed at the inverter’s output terminals. This disturbance is then incorporated into the system’s control transfer functions or equivalent circuit models to derive the resulting grid current harmonics. A significant limitation of these models is the assumption that a dead-time voltage at a specific frequency will only produce a current response at that same frequency. However, this assumption overlooks a critical phenomenon intrinsic to the standard dq-control structure used in most grid connected inverter systems: frequency coupling.
Frequency coupling arises from asymmetries in the dq-control framework. Even under symmetric three-phase input conditions, asymmetries exist in the phase-locked loop (PLL) (which only uses the q-axis voltage), the power outer loop (different controllers for active and reactive power), and the current inner loop structure. When a harmonic voltage disturbance of a particular sequence and frequency enters this asymmetric system, the control interactions can generate an additional harmonic component at a different, coupled frequency. The coupling relationship is systematic: a positive-sequence disturbance at frequency $$f_x > f_0$$ (where $$f_0$$ is the fundamental frequency) will produce both a positive-sequence response at $$f_x$$ and a negative-sequence response at $$f_x – 2f_0$$. Conversely, a negative-sequence disturbance at $$f_x$$ will produce a negative-sequence response at $$f_x$$ and a positive-sequence response at $$f_x + 2f_0$$.
This frequency coupling effect has profound implications for dead-time harmonic modeling in a grid connected inverter. Since dead-time naturally produces a spectrum containing both positive-sequence (e.g., 7th, 13th) and negative-sequence (e.g., 5th, 11th) harmonics, these components do not exist in isolation. For instance, the 7th harmonic (positive sequence, $$f_{7} = 7f_0$$) will, through the frequency coupling mechanism of the closed-loop control, generate a coupled negative-sequence component at $$f_{7} – 2f_0 = 5f_0$$ (the 5th harmonic). Similarly, the inherent dead-time 5th harmonic (negative sequence) will generate a coupled positive-sequence component at the 7th harmonic frequency. This creates a mutual interaction where adjacent (6k±1) dead-time harmonics are coupled, meaning a complete model must consider the vector superposition of the original dead-time harmonic and the coupled harmonic contributed by its adjacent counterpart. Existing models that treat dead-time as a fixed disturbance source fail to account for this coupling, leading to potential inaccuracies in predicting the true harmonic emission spectrum of a grid connected inverter.
Mechanism of Dead-Time Harmonic Coupling
The dead-time effect modifies the ideal PWM switching instants. The net error voltage caused by dead-time over a fundamental half-cycle can be approximated as a square wave with an amplitude given by:
$$ U_{D} = f_c t_d U_{dc} $$
The Fourier decomposition of this error voltage yields its harmonic composition. For three-phase systems, the primary low-frequency spectral lines are at harmonic orders $$n = 6k \pm 1$$.
The control-induced frequency coupling effect transforms the harmonic propagation through the grid connected inverter system. The standard vector control structure in the synchronous (dq) reference frame, while effective for fundamental component regulation, introduces the asymmetries mentioned earlier. When harmonic components at frequencies $$f_{x1} = (6k+1)f_0$$ (positive sequence) and $$f_{x2} = (6k-1)f_0$$ (negative sequence) are present at the inverter terminals, they are sampled and processed by the control system. The PLL, sensitive to voltage harmonics, produces a phase angle perturbation. The harmonic voltages and currents are transformed into the dq-domain using this perturbed angle, then processed through the asymmetric power and current controllers. The resulting dq-reference voltages, after inverse transformation back to the abc frame, contain not only the original frequency components but also the coupled components due to the multiplicative effects of the transformations and the asymmetric gains. This process is summarized in Table 1, which defines the frequency coupling relationship for harmonic disturbances.
| Input Voltage Disturbance | Current Response Frequencies |
|---|---|
| Positive Sequence, frequency $$f_x$$ | Positive Sequence at $$f_x$$ and Negative Sequence at $$f_x – 2f_0$$ |
| Negative Sequence, frequency $$f_x$$ | Negative Sequence at $$f_x$$ and Positive Sequence at $$f_x + 2f_0$$ |
Consequently, the interaction is bidirectional. The inherent dead-time voltage at $$f_{x1}$$ (e.g., 7th) produces a coupled current component at $$f_{x2}$$ (e.g., 5th). This coupled component at $$f_{x2}$$, when fed back through the system, can in turn produce a component at $$f_{x1}$$, interacting with the original dead-time component at that frequency. Therefore, the final harmonic current at the point of common coupling (PCC) for any 6k±1 order is the phasor sum of: 1) the “original frequency” response directly excited by the dead-time voltage at that order, and 2) the “coupled frequency” response originating from the dead-time voltage at the adjacent order.
Modeling the Dead-Time Coupled Harmonic Source
To establish an accurate model, we consider a three-phase grid connected inverter with LCL filter and standard cascaded control loops (PLL, power outer loop, current inner loop with decoupling and feedforward). The modeling process analytically traces the propagation of harmonic voltages through every relevant stage.
Step 1: Single-Frequency Dead-Time Voltage Analysis
We first analyze the system’s response when only one frequency component of dead-time voltage is present. Assume a positive-sequence dead-time voltage $$U_{Dx1}$$ at frequency $$f_{x1} = (6k+1)f_0$$ is present at the inverter output terminals. According to the frequency coupling principle, the PCC voltage and current will contain components at both $$f_{x1}$$ and its coupled frequency $$f_{x2} = f_{x1} – 2f_0 = (6k-1)f_0$$. Let the PCC phase-a voltage be:
$$ u_{an} = u_{an0} + u_{anx1} + u_{anx2} $$
with $$u_{anx1} = U_{mnx1} \cos(\omega_{x1} t + \varphi_{unx1})$$ and $$u_{anx2} = U_{mnx2} \cos(\omega_{x2} t + \varphi_{unx2})$$.
The sampled voltage undergoes a delay (typically 1.5 sampling periods, $$1.5T_s$$) before being processed by the PLL. The PLL, perturbed by the harmonic voltages, outputs an angle $$\theta_{PLL} = \theta_0 + \theta_x$$, where $$\theta_x$$ is a small harmonic component at frequency $$\omega_{x1} – \omega_0$$, derivable from the PLL’s linearized model:
$$ \boldsymbol{\theta}_x = \frac{H_{PLL}(j(\omega_{x1}-\omega_0)) e^{-j1.5T_s(\omega_{x1}-\omega_0)} ( -j \boldsymbol{U}_{a}^{nx1} + \boldsymbol{U}_{a}^{nx2} )}{j(\omega_{x1}-\omega_0) + U_{mn0} H_{PLL}(j(\omega_{x1}-\omega_0))} $$
Here, $$H_{PLL}(s)$$ is the PLL transfer function, and bold symbols denote phasors.
Using the Park transformation $$P(\theta_{PLL})$$, the PCC voltage phasors in the dq-frame for the harmonic frequencies are derived. The transformation can be separated into a fundamental part $$P(\theta_0)$$ and a perturbation part due to $$\theta_x$$. The resulting dq harmonic voltages are:
$$
\begin{aligned}
\boldsymbol{U}_{dnx} &= e^{-j1.5T_s(\omega_{x1}-\omega_0)} \boldsymbol{U}_{a}^{nx1} + e^{-j1.5T_s(\omega_{x1}-\omega_0)} \boldsymbol{U}_{a}^{nx2*} \\
\boldsymbol{U}_{qnx} &= -j e^{-j1.5T_s(\omega_{x1}-\omega_0)} \boldsymbol{U}_{a}^{nx1} + j e^{-j1.5T_s(\omega_{x1}-\omega_0)} \boldsymbol{U}_{a}^{nx2*} – j U_{mn0} \boldsymbol{\theta}_x
\end{aligned}
$$
Similar transformations apply to the PCC current and filter capacitor current phasors.
The power outer loop calculates active and reactive power (P, Q). Extracting the harmonic components $$P_x$$ and $$Q_x$$ from these, and considering the outer loop controller transfer function $$H_{out}(s)$$, the harmonic references for the dq currents are obtained:
$$
\begin{bmatrix}
\boldsymbol{I}_{d}^{refx} \\
\boldsymbol{I}_{q}^{refx}
\end{bmatrix} = – H_{out}(j(\omega_{x1}-\omega_0)) \begin{bmatrix}
\boldsymbol{P}_x \\
\boldsymbol{Q}_x
\end{bmatrix}
$$
The current inner loop, including decoupling (coefficient $$K_i$$), voltage feedforward (coefficient $$K_n$$), and capacitor current feedback (coefficient $$K_c$$), processes the error between the reference and measured currents. Using its transfer function $$H_{in}(s)$$, the dq harmonic reference voltages are:
$$
\begin{aligned}
\boldsymbol{U}_{d}^{refx} &= H_{in}(j(\omega_{x1}-\omega_0)) (\boldsymbol{I}_{d}^{refx} – \boldsymbol{I}_{d}^{nx} – K_c \boldsymbol{I}_{d}^{Cx}) – K_i \boldsymbol{I}_{q}^{nx} + K_n \boldsymbol{U}_{d}^{nx} \\
\boldsymbol{U}_{q}^{refx} &= H_{in}(j(\omega_{x1}-\omega_0)) (\boldsymbol{I}_{q}^{refx} – \boldsymbol{I}_{q}^{nx} – K_c \boldsymbol{I}_{q}^{Cx}) + K_i \boldsymbol{I}_{d}^{nx} + K_n \boldsymbol{U}_{q}^{nx}
\end{aligned}
$$
These reference voltages are transformed back to the abc frame using the inverse Park transformation $$P^{-1}(\theta_{PLL})$$, yielding the three-phase modulation voltage harmonic phasors $$\boldsymbol{U}_{a}^{sx1}$$ and $$\boldsymbol{U}_{a}^{sx2}$$. This step encapsulates the coupling effect, resulting in expressions of the form:
$$
\begin{aligned}
\boldsymbol{U}_{a}^{sx1} &= A_{11} \boldsymbol{I}_{a}^{nx1} + A_{12} \boldsymbol{U}_{a}^{nx1} + A_{13} \boldsymbol{U}_{a}^{nx2} + A_{14} \boldsymbol{\theta}_x \\
\boldsymbol{U}_{a}^{sx2} &= A_{21} \boldsymbol{I}_{a}^{nx2} + A_{22} \boldsymbol{U}_{a}^{nx2} + A_{23} \boldsymbol{U}_{a}^{nx1} + A_{24} \boldsymbol{\theta}_x
\end{aligned}
$$
where coefficients $$A_{ij}$$ are complex functions of system and control parameters.
The modulation process relates the modulation voltage to the actual inverter bridge output voltage. The effective PWM gain $$K_{pwm}$$ must account for the fact that closed-loop control compensates for the fundamental voltage drop caused by dead-time, slightly increasing the modulation index:
$$ K_{pwm} = \frac{U_{p0}}{U_{s0}} \approx \frac{U_{p0}}{U_{p0} + U_{D0}} $$
where $$U_{p0}$$, $$U_{s0}$$, and $$U_{D0}$$ are the fundamental amplitudes of the inverter output voltage, modulation voltage, and dead-time error voltage, respectively. Thus:
$$ \boldsymbol{U}_{a}^{px1} = K_{pwm} \boldsymbol{U}_{a}^{sx1}, \quad \boldsymbol{U}_{a}^{px2} = K_{pwm} \boldsymbol{U}_{a}^{sx2} $$
Finally, the circuit equations of the LCL filter and the grid impedance ($$R_g + j\omega L_g$$) are applied. The relationship between the inverter output voltage (including the dead-time source $$\boldsymbol{U}_{Dx1}$$), the PCC voltage, and the currents through the filter branches is established via Kirchhoff’s laws. For frequency $$f_{x1}$$:
$$ \boldsymbol{U}_{a}^{px1} – \boldsymbol{U}_{a}^{cx1} – \boldsymbol{U}_{Dx1} = (R_1 + j\omega_{x1} L_1) \boldsymbol{I}_{a}^{px1} $$
$$ \boldsymbol{U}_{a}^{cx1} = \frac{1}{1/R_C + j\omega_{x1} C} \boldsymbol{I}_{a}^{Cx1} $$
$$ \boldsymbol{U}_{a}^{cx1} – \boldsymbol{U}_{a}^{nx1} = (R_2 + j\omega_{x1} L_2) \boldsymbol{I}_{a}^{nx1} = (R_g + j\omega_{x1} L_g) \boldsymbol{I}_{a}^{nx1} $$
$$ \boldsymbol{I}_{a}^{px1} = \boldsymbol{I}_{a}^{nx1} + \boldsymbol{I}_{a}^{Cx1} $$
Similar equations hold for frequency $$f_{x2}$$, but without the $$\boldsymbol{U}_{Dx2}$$ term in this step (as we are only considering $$\boldsymbol{U}_{Dx1}$$ as the source).
By substituting the expressions for $$\boldsymbol{U}_{a}^{px1}$$ and $$\boldsymbol{U}_{a}^{px2}$$ from the control derivation into the circuit equations, and solving the system, we can derive the PCC harmonic voltages $$\boldsymbol{U}_{a}^{nx1}$$ and $$\boldsymbol{U}_{a}^{nx2}$$ as functions of the dead-time source $$\boldsymbol{U}_{Dx1}$$. This leads to a matrix relationship:
$$
\begin{bmatrix}
\boldsymbol{U}_{a}^{nx1} \\
\boldsymbol{U}_{a}^{nx2}
\end{bmatrix} = \frac{1}{A_2 A_4 – A_1 A_3} \begin{bmatrix}
-A_3 & A_5 \\
A_4 & -A_2
\end{bmatrix} \boldsymbol{U}_{Dx1}
$$
where $$A_1$$ to $$A_5$$ are coefficients derived from the interconnected control and circuit equations. Consequently, the PCC harmonic currents are:
$$
\begin{bmatrix}
\boldsymbol{I}_{a}^{nx1} \\
\boldsymbol{I}_{a}^{nx2}
\end{bmatrix} = \frac{1}{A_2 A_4 – A_1 A_3} \begin{bmatrix}
-A_3/(R_g + j\omega_{x1} L_g) & A_5/(R_g + j\omega_{x1} L_g) \\
A_4/(R_g + j\omega_{x2} L_g) & -A_2/(R_g + j\omega_{x2} L_g)
\end{bmatrix} \boldsymbol{U}_{Dx1}
$$
Step 2: Incorporation of the Adjacent Dead-Time Frequency
The analysis in Step 1 considered only the dead-time source $$\boldsymbol{U}_{Dx1}$$ at $$f_{x1}$$. An identical analytical procedure must be performed for the case where only the adjacent dead-time source $$\boldsymbol{U}_{Dx2}$$ at $$f_{x2} = (6k-1)f_0$$ is present. Following the same method, we obtain another set of coefficients $$B_{ij}$$ and $$B_1$$ to $$B_5$$, leading to the PCC current responses:
$$
\begin{bmatrix}
\boldsymbol{I}_{a}^{‘nx2} \\
\boldsymbol{I}_{a}^{‘nx1}
\end{bmatrix} = \frac{1}{B_2 B_4 – B_1 B_3} \begin{bmatrix}
-B_3/(R_g + j\omega_{x2} L_g) & B_5/(R_g + j\omega_{x2} L_g) \\
B_4/(R_g + j\omega_{x1} L_g) & -B_2/(R_g + j\omega_{x1} L_g)
\end{bmatrix} \boldsymbol{U}_{Dx2}
$$
Note that the indices are swapped because $$\boldsymbol{U}_{Dx2}$$ at negative sequence $$f_{x2}$$ primarily drives a response at $$f_{x2}$$ (original) and couples to $$f_{x1}$$.
Step 3: Superposition to Form the Complete Coupled Model
Since the system is linearized around the operating point for small-signal harmonic analysis, superposition holds. The total PCC harmonic current at each frequency is the phasor sum of the contributions from both dead-time sources:
$$
\begin{aligned}
\boldsymbol{I}_{a}^{pccx1} &= \boldsymbol{I}_{a}^{nx1} + \boldsymbol{I}_{a}^{‘nx1} \\
\boldsymbol{I}_{a}^{pccx2} &= \boldsymbol{I}_{a}^{nx2} + \boldsymbol{I}_{a}^{‘nx2}
\end{aligned}
$$
This final set of equations constitutes the dead-time coupled harmonic source model for the grid connected inverter. It explicitly accounts for the mutual coupling between adjacent 6k±1 dead-time harmonic voltages via the control system’s frequency coupling effect. The model is linear and algebraic for a given operating point and harmonic frequency pair, making it suitable for integration into frequency-domain harmonic analysis tools.
Model Verification and Analysis
The established model was verified using a hardware-in-the-loop (HIL) simulation platform, where the power circuit was simulated in a real-time simulator (RT-Box) and the control algorithms were executed on a physical DSP controller (TMS320F28069). The parameters of a 5 MW, 10 kV grid connected inverter with an LCL filter were used. The dead-time effect introduced noticeable 5th and 7th harmonic currents at the PCC. The model’s predictions for the magnitudes of these harmonics were compared against measurements from the HIL platform under varying dead-time and switching frequency conditions. The results, shown in Table 2 and Table 3, demonstrate a close match, validating the model’s accuracy.
| Dead-Time (μs) | 5th Harmonic Current (Model) [A] | 5th Harmonic Current (HIL) [A] | 7th Harmonic Current (Model) [A] | 7th Harmonic Current (HIL) [A] |
|---|---|---|---|---|
| 8 | 9.77 | 9.53 | 9.45 | 8.94 |
| 10 | 12.23 | 11.80 | 11.57 | 11.15 |
| 12 | 14.69 | 14.15 | 13.53 | 12.96 |
| Switching Freq. (Hz) | 5th Harmonic Current (Model) [A] | 5th Harmonic Current (HIL) [A] | 7th Harmonic Current (Model) [A] | 7th Harmonic Current (HIL) [A] |
|---|---|---|---|---|
| 3800 | 11.60 | 11.20 | 10.95 | 10.35 |
| 4100 | 12.53 | 12.10 | 11.85 | 11.35 |
| 4400 | 13.45 | 13.00 | 12.72 | 12.30 |
Further validation involved comparing the proposed model against an established reference model that treats dead-time as a constant harmonic voltage source without considering frequency coupling. The comparison, performed in both a simple RL-grid and a radial test network, confirmed that the proposed coupled model provides significantly better agreement with time-domain simulation and HIL results, as seen in Table 4. The error in the traditional model stems from its neglect of the coupled harmonic component.
| Scenario | Harmonic Order | Ref. Model (Const. Source) [A] | Proposed Coupled Model [A] | Simulation/HIL Result [A] |
|---|---|---|---|---|
| Simple Grid, t_d=10μs | 5th | 8.41 | 12.23 | 11.80 |
| Simple Grid, t_d=10μs | 7th | 4.79 | 11.57 | 11.15 |
| Radial Network, t_d=10μs | 5th | 10.93 | 12.14 | 11.74 |
| Radial Network, t_d=10μs | 7th | 6.07 | 11.18 | 10.04 |
An important practical consideration is the coexistence of dead-time harmonics and grid background harmonics. The linear nature of the derived model implies that their effects should be superimposable. This was confirmed by introducing 0.1 pu background harmonics at the 5th (negative sequence) and 7th (positive sequence) orders into the HIL test while applying dead-time. The total measured harmonic current closely matched the phasor sum of the model-predicted dead-time harmonic current and the separately calculated background harmonic response current from a frequency-coupling model for background harmonics, validating the superposition principle.
Influence of Parameters on Dead-Time Harmonic Characteristics
Applying the validated model, the influence of various operational and control parameters on the dead-time coupled harmonic emission was analyzed.
1. Power Setpoints: Variations in active ($$P_{ref}$$) and reactive ($$Q_{ref}$$) power references cause only minor changes in the dead-time harmonic current magnitude (typically within ±5%). The effect is negligible for most studies.
2. Grid Impedance: The grid inductance ($$L_g$$) has a pronounced effect, as illustrated in Table 5. While the “original frequency” component of the response remains relatively stable, the “coupled frequency” component contributed by the control interaction increases significantly with weaker grids (higher $$L_g$$). This means the overall dead-time harmonic current, being the sum of both, becomes more substantial in weak grid conditions, and the role of frequency coupling in the model becomes even more critical.
| Grid Inductance L_g (mH) | 5th Harmonic: Original Comp. [A] | 5th Harmonic: Coupled Comp. [A] | Total 5th Harmonic [A] |
|---|---|---|---|
| 1.0 | 8.15 | 1.22 | 9.37 |
| 3.0 | 8.05 | 4.18 | 12.23 |
| 5.0 | 7.95 | 6.85 | 14.80 |
3. Control Loop Parameters: The parameters of the current inner loop ($$H_{in}(s) = K_{p_i} + K_{i_i}/s$$) and power outer loop ($$H_{out}(s) = K_{p_o} + K_{i_o}/s$$) directly influence the harmonic response. Increasing the proportional gains ($$K_{p_i}, K_{p_o}$$) generally leads to an increase in the dead-time harmonic current magnitude. The integral gains have a weaker but similar influence. In contrast, the parameters of the PLL have a minimal impact on the dead-time harmonic output, as its primary influence is on the coupling pathway for voltage-sourced disturbances rather than the current-controlled response to the dead-time voltage source.
Conclusion
This paper establishes a comprehensive closed-loop harmonic source model for a grid connected inverter that accurately captures the dead-time effect while accounting for the crucial frequency coupling phenomenon inherent in dq-control structures. The model demonstrates that dead-time harmonics at adjacent 6k±1 orders are not independent but interact through the control system, resulting in a final PCC harmonic current that is the phasor sum of original and coupled components. The model is linear and suitable for integration into system-level harmonic analysis.
Validation via hardware-in-the-loop simulation confirms the model’s superior accuracy over traditional constant-disturbance-source models, especially under weak grid conditions where coupling effects are amplified. The analysis further reveals that while power setpoints and PLL parameters have minor influence, grid strength and current/power controller gains significantly affect the magnitude of dead-time harmonic emissions. This model provides a valuable tool for predicting the precise harmonic emission spectrum of grid connected inverter-based resources, facilitating more accurate assessment of harmonic propagation and resonance risks in modern power systems with high power electronic penetration.