Impedance Modeling and Analysis of Grid-Connected Voltage-Source Converters
并网电压源转换器的阻抗建模和分析

Index Terms-Converter stability, grid-connected converters, harmonic resonance, impedance modeling.
Index Terms-转换器稳定性、并网转换器、谐波谐振、阻抗建模。

THREE-PHASE voltage-source converters (VSCs) are the basic building blocks for many applications in power systems, including grid integration of renewable energy [1] and energy storage [2], high-voltage dc transmission [3], as well as flexible ac transmission systems [4]. They are commonly referred to as grid-connected VSC in this paper. As for other power electronic circuits, external behavior of such VSC can be characterized by the impedances measured at the dc and the ac terminals. Depending on the direction of power flow, the ac terminal impedance can be considered the input impedance (in rectification mode) or the output impedance (in inversion mode), and will be simply referred to as the impedance in this study.
三相电压源转换器 （VSC） 是电力系统中许多应用的基本组成部分，包括可再生能源 [1] 和储能 [2] 的电网集成、高压直流输电 [3] 以及灵活的交流输电系统 [4]。在本文中，它们通常被称为并网 VSC。对于其他电力电子电路，这种 VSC 的外部行为可以通过在直流和交流端子处测量的阻抗来表征。根据潮流的方向，交流端子阻抗可以被认为是输入阻抗（在整流模式下）或输出阻抗（在反转模式下），在本研究中将简称为阻抗。

One important use of the impedance of a grid-connected VSC is in the analysis of stability and resonance between the converter and the grid, including that with the filter of the converter [5]. In particular, it was shown in [6] that a grid-connected VSC used for grid integration of renewable energy can be modeled as a current source in parallel with an impedance, and the invertergrid system stability can be determined by applying the Nyquist stability criterion [7] to the ratio between the grid impedance and the VSC impedance.
并网 VSC 阻抗的一个重要用途是分析转换器和电网之间的稳定性和谐振，包括与转换器滤波器的稳定性和谐振 [5]。特别是，[6] 表明，用于可再生能源并网的并网 VSC 可以建模为与阻抗并联的电流源，并且可以通过将奈奎斯特稳定性准则 [7] 应用于电网阻抗和 VSC 阻抗之间的比率来确定逆变器电网系统的稳定性。

Most grid-connected VSCs use current control in a rotating ( $dq$$dq$dqd q ) reference frame [8], which is synchronized to the fundamental component of the grid voltages by means of a phase-locked loop (PLL) [9]. Both the $dq$$dq$dqd q-domain current con-
大多数并网 VSC 在旋转 （ $dq$$dq$dqd q ） 参考系 [8] 中使用电流控制，该参考系通过锁相环 （PLL） [9] 与电网电压的基本分量同步。 $dq$$dq$dqd q -domain current con-

trol and the PLL-based grid synchronization introduce nonlinearities which cannot be removed by reduced-order modeling techniques [10]. One method to deal with the control nonlinearities is to transform the converter model into the $dq$$dq$dqd q reference frame [11]. This method, however, has several limitations and disadvantages, as discussed in [12]. The harmonic linearization method [13] overcomes these limitations by modeling threephase VSC impedance directly in the phase domain.
trol 和基于 PLL 的网格同步引入了非线性，这些非线性无法通过降阶建模技术消除 [10]。处理控制非线性的一种方法是将转换器模型转换为 $dq$$dq$dqd q 参考系 [11]。然而，这种方法有几个局限性和缺点，如 [12] 所述。谐波线性化方法 [13] 通过在相域中直接模拟三相 VSC 阻抗来克服这些限制。

This paper applies the harmonic linearization technique to develop impedance models of three-phase VSCs with PLL-based grid synchronization. A key step in the development of the impedance models is the linearization of the gridsynchronization scheme. Since there exist several synchronization schemes [14], the approach taken here is to consider a basic PLL, and show how it can be incorporated into the impedance models. Possible variations are reviewed to highlight their modeling approach. The rest of this paper is organized as follows: Section II develops impedance models assuming perfect knowledge of the grid voltage angle. Section III shows how to model the PLL, and the approach to incorporate it into the impedance models. Section IV includes verifications of the proposed impedance models from both impedance measurements and their application in analysis of harmonic resonance. Section V concludes this paper.
本文应用谐波线性化技术开发了具有基于 PLL 的电网同步的三相 VSC 的阻抗模型。阻抗模型开发的一个关键步骤是网格同步方案的线性化。由于存在多种同步方案 [14]，这里采用的方法是考虑一个基本的 PLL，并展示如何将其整合到阻抗模型中。回顾可能的变体以突出其建模方法。本文的其余部分组织如下：第二部分开发了阻抗模型，假设对电网电压角有完美的了解。第 III 节展示了如何对 PLL 进行建模，以及将其合并到阻抗模型中的方法。第 IV 节包括对阻抗测量中提出的阻抗模型的验证及其在谐波谐振分析中的应用。第五节对本文进行了总结。

The three-phase VSC considered in this paper is depicted in Fig. 1. Phase voltages are denoted as $v_a,v_b$$v_a,v_b$v_(a),v_(b)v_{a}, v_{b}, and $v_c$$v_c$v_(c)v_{c}, while phase currents as $i_a,i_b$$i_a,i_b$i_(a),i_(b)i_{a}, i_{b}, and $i_c$$i_c$i_(c)i_{c}. Considering the large dc bus capacitors, and the lower than fundamental frequency control bandwidth of the dc bus voltage, $V_{\mathrm{dc}}$$V_{\mathrm{dc}}$V_(dc)V_{\mathrm{dc}} is assumed constant in this study. For the same reason, the active and reactive parts of the current references ( $I_{\mathrm{dr}}$$I_{\mathrm{dr}}$I_(dr)I_{\mathrm{dr}} and $I_{\mathrm{qr}}$$I_{\mathrm{qr}}$I_(qr)I_{\mathrm{qr}} ) are assumed constant. In the time domain, the phase voltage with a small-signal perturbation can be written as
本文中考虑的三相 VSC 如图 1 所示。相电压表示为 $v_a,v_b$$v_a,v_b$v_(a),v_(b)v_{a}, v_{b} ， 和 $v_c$$v_c$v_(c)v_{c} ，而相电流表示 $i_a,i_b$$i_a,i_b$i_(a),i_(b)i_{a}, i_{b} 为 、 和 $i_c$$i_c$i_(c)i_{c} 。考虑到大型直流母线电容，以及直流母线电压低于基频的控制带宽， $V_{\mathrm{dc}}$$V_{\mathrm{dc}}$V_(dc)V_{\mathrm{dc}} 在本研究中假设是恒定的。出于同样的原因，current references （ $I_{\mathrm{dr}}$$I_{\mathrm{dr}}$I_(dr)I_{\mathrm{dr}} 和 $I_{\mathrm{qr}}$$I_{\mathrm{qr}}$I_(qr)I_{\mathrm{qr}} ） 的 active 和 reactive 部分被假定为 constant。在时域中，具有小信号扰动的相电压可以写为

where $V_1$$V_1$V_(1)V_{1} corresponds to the magnitude of the fundamental voltage at frequency $f_1,V_p$$f_1,V_p$f_(1),V_(p)f_{1}, V_{p} with ${\varphi }_{\mathrm{vp}}$${\varphi }_{\mathrm{vp}}$phi_(vp)\phi_{\mathrm{vp}} correspond to the magnitude and phase of the positive-sequence perturbation at frequency $f_p$$f_p$f_(p)f_{p}, and $V_n$$V_n$V_(n)V_{n} with ${\varphi }_{\mathrm{vn}}$${\varphi }_{\mathrm{vn}}$phi_(vn)\phi_{\mathrm{vn}} correspond to the magnitude and phase of the negative-sequence perturbation at frequency $f_n$$f_n$f_(n)f_{n}. Other phase voltages can be inferred from (1). In the frequency domain, (1) can be written as follows:
其中 $V_1$$V_1$V_(1)V_{1} 对应于频率 $f_1,V_p$$f_1,V_p$f_(1),V_(p)f_{1}, V_{p} 处基波电压的大小 对应于 ${\varphi }_{\mathrm{vp}}$${\varphi }_{\mathrm{vp}}$phi_(vp)\phi_{\mathrm{vp}} 频率处正序扰动的大小和相位 $f_p$$f_p$f_(p)f_{p} ， $V_n$$V_n$V_(n)V_{n} 与 ${\varphi }_{\mathrm{vn}}$${\varphi }_{\mathrm{vn}}$phi_(vn)\phi_{\mathrm{vn}} 对应于频率处负序扰动的大小和相位 $f_n$$f_n$f_(n)f_{n} 。其他相电压可以从 （1） 中推断出来。在频域中，（1） 可以写成如下：

Fig. 1. Block diagram of three-phase VSC for grid-connected applications.
图 1.用于并网应用的三相 VSC 框图。
where ${\mathbf{V}}_p=\left(V_p/2\right)e^{\pm j{\varphi }_{\mathrm{vp}}}$${\mathbf{V}}_p=\left(V_p/2\right)e^{\pm j{\varphi }_{\mathrm{vp}}}$V_(p)=(V_(p)//2)e^(+-jphi_(vp))\mathbf{V}_{p}=\left(V_{p} / 2\right) e^{ \pm j \phi_{\mathrm{vp}}} and others follow the same notation. The current response to the voltage perturbation can be found from the converter averaged model
where ${\mathbf{V}}_p=\left(V_p/2\right)e^{\pm j{\varphi }_{\mathrm{vp}}}$${\mathbf{V}}_p=\left(V_p/2\right)e^{\pm j{\varphi }_{\mathrm{vp}}}$V_(p)=(V_(p)//2)e^(+-jphi_(vp))\mathbf{V}_{p}=\left(V_{p} / 2\right) e^{ \pm j \phi_{\mathrm{vp}}} 和其他 API 遵循相同的表示法。电流对电压扰动的响应可以从转换器平均模型中找到

where $m_a,m_b$$m_a,m_b$m_(a),m_(b)m_{a}, m_{b}, and $m_c$$m_c$m_(c)m_{c} are the modulating (reference) signals for the pulse width modulation (PWM), and $K_m$$K_m$K_(m)K_{m} is the modulator gain. The relationship between duty ratios and the modulating signal is taken as follows:
其中 $m_a,m_b$$m_a,m_b$m_(a),m_(b)m_{a}, m_{b} ， 和 $m_c$$m_c$m_(c)m_{c} 是脉宽调制 （PWM） 的调制 （参考） 信号， 是 $K_m$$K_m$K_(m)K_{m} 调制器增益。占空比和调制信号之间的关系如下：

where $d_{\mathrm{a}1}$$d_{\mathrm{a}1}$d_(a1)d_{\mathrm{a} 1} and $d_{\mathrm{a}2}$$d_{\mathrm{a}2}$d_(a2)d_{\mathrm{a} 2} are the duty ratios of ${\mathrm{S}}_{\mathrm{a}1}$${\mathrm{S}}_{\mathrm{a}1}$S_(a1)\mathrm{S}_{\mathrm{a} 1} and ${\mathrm{S}}_{\mathrm{a}2}$${\mathrm{S}}_{\mathrm{a}2}$S_(a2)\mathrm{S}_{\mathrm{a} 2}, respectively. Other phases follow the same convention.
其中 $d_{\mathrm{a}1}$$d_{\mathrm{a}1}$d_(a1)d_{\mathrm{a} 1} 和 $d_{\mathrm{a}2}$$d_{\mathrm{a}2}$d_(a2)d_{\mathrm{a} 2} 分别是 ${\mathrm{S}}_{\mathrm{a}1}$${\mathrm{S}}_{\mathrm{a}1}$S_(a1)\mathrm{S}_{\mathrm{a} 1} 和 ${\mathrm{S}}_{\mathrm{a}2}$${\mathrm{S}}_{\mathrm{a}2}$S_(a2)\mathrm{S}_{\mathrm{a} 2} 的占空比。其他阶段遵循相同的约定。

In order to solve (3) for impedance in the frequency domain, the sequence components in the modulating signals should be found as functions of the voltage and current perturbations. Then, positive-sequence impedance is defined as the ratio of ${\mathbf{V}}_p$${\mathbf{V}}_p$V_(p)\mathbf{V}_{p} to $-{\mathbf{I}}_p$$-{\mathbf{I}}_p$-I_(p)-\mathbf{I}_{p}, and negative-sequence impedance is defined as the ratio of ${\mathbf{V}}_n$${\mathbf{V}}_n$V_(n)\mathbf{V}_{n} to $-{\mathbf{I}}_n$$-{\mathbf{I}}_n$-I_(n)-\mathbf{I}_{n}. Coupling should also be examined. Both phase- and $dq$$dq$dqd q-domain current control strategies will be considered here, which use Park’s transformation defined as follows:
为了求解 （3） 频域中的阻抗，调制信号中的序列分量应作为电压和电流扰动的函数。然后，正序阻抗定义为 ${\mathbf{V}}_p$${\mathbf{V}}_p$V_(p)\mathbf{V}_{p} 的比值 ， $-{\mathbf{I}}_p$$-{\mathbf{I}}_p$-I_(p)-\mathbf{I}_{p} 负序阻抗定义为 ${\mathbf{V}}_n$${\mathbf{V}}_n$V_(n)\mathbf{V}_{n} 的 $-{\mathbf{I}}_n$$-{\mathbf{I}}_n$-I_(n)-\mathbf{I}_{n} 比值。还应检查耦合。这里将考虑相域 $dq$$dq$dqd q 和域电流控制策略，它们使用 Park 变换，定义如下：

An inductive output filter is assumed in Fig. 1. Additional filter elements (such as in the case when an LCL is used) can be handled by simply modifying (3).
图 1 中假设有一个电感输出滤波器。额外的过滤元件（例如在使用 LCL 的情况下）可以通过简单地修改 （3） 来处理。

Fig. 2. Block diagram of a phase-domain current controller.
图 2.相域电流控制器的框图。

Fig. 2 depicts a phase-domain current controller. To find the frequency-domain response of the controller to the harmonic perturbation, first neglect the PLL dynamics, such that ${\theta }_{\text{PLL}}(t)={\theta }_1(t)\equiv 2\pi f_{1}t$${\theta }_{\text{PLL }}(t)={\theta }_1(t)\equiv 2\pi f_{1}t$theta_("PLL ")(t)=theta_(1)(t)-=2pif_(1)t\theta_{\text {PLL }}(t)=\theta_{1}(t) \equiv 2 \pi f_{1} t. Hence the reference currents $i_{\text{ar}},i_{\text{br}}$$i_{\text{ar }},i_{\text{br }}$i_("ar "),i_("br ")i_{\text {ar }}, i_{\text {br }}, and $i_{\mathrm{cr}}$$i_{\mathrm{cr}}$i_(cr)i_{\mathrm{cr}} are not affected by the perturbation. As a result, the sequence components in the modulating signals may be found as follows:
图 2 描述了一个相域电流控制器。要找到控制器对谐波扰动的频域响应，首先忽略 PLL 动力学，使得 ${\theta }_{\text{PLL}}(t)={\theta }_1(t)\equiv 2\pi f_{1}t$${\theta }_{\text{PLL }}(t)={\theta }_1(t)\equiv 2\pi f_{1}t$theta_("PLL ")(t)=theta_(1)(t)-=2pif_(1)t\theta_{\text {PLL }}(t)=\theta_{1}(t) \equiv 2 \pi f_{1} t 。因此参考电流 $i_{\text{ar}},i_{\text{br}}$$i_{\text{ar }},i_{\text{br }}$i_("ar "),i_("br ")i_{\text {ar }}, i_{\text {br }} ， $i_{\mathrm{cr}}$$i_{\mathrm{cr}}$i_(cr)i_{\mathrm{cr}} 不受扰动的影响。因此，调制信号中的序列分量可以找到如下：
${\mathbf{M}}_a[f]=\{\begin{array}{ll}-H_i(s)G_i(s){\mathbf{I}}_p+K_f(s)G_v(s){\mathbf{V}}_p,& f=\pm f_p\\ -H_i(s)G_i(s){\mathbf{I}}_n+K_f(s)G_v(s){\mathbf{V}}_n,& f=\pm f_n\end{array}$${\mathbf{M}}_a[f]=\left\{\begin{array}{l}-H_i(s)G_i(s){\mathbf{I}}_p+K_f(s)G_v(s){\mathbf{V}}_p,\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}f=\pm f_p\\ -H_i(s)G_i(s){\mathbf{I}}_n+K_f(s)G_v(s){\mathbf{V}}_n,\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}f=\pm f_n\end{array}\right.$M_(a)[f]={[-H_(i)(s)G_(i)(s)I_(p)+K_(f)(s)G_(v)(s)V_(p)",",f=+-f_(p)],[-H_(i)(s)G_(i)(s)I_(n)+K_(f)(s)G_(v)(s)V_(n)",",f=+-f_(n)]:}\mathbf{M}_{a}[f]= \begin{cases}-H_{i}(s) G_{i}(s) \mathbf{I}_{p}+K_{f}(s) G_{v}(s) \mathbf{V}_{p}, & f= \pm f_{p} \\ -H_{i}(s) G_{i}(s) \mathbf{I}_{n}+K_{f}(s) G_{v}(s) \mathbf{V}_{n}, & f= \pm f_{n}\end{cases}
where $H_i(s)$$H_i(s)$H_(i)(s)H_{i}(s) is a current control compensator, $K_f(s)$$K_f(s)$K_(f)(s)K_{f}(s) is a feedforward gain, and
其中 $H_i(s)$$H_i(s)$H_(i)(s)H_{i}(s) 是电流控制补偿器， $K_f(s)$$K_f(s)$K_(f)(s)K_{f}(s) 是前馈增益，并且

models the current sampling delay, with $T_i$$T_i$T_(i)T_{i} representing the sampling interval and ${\omega }_i$${\omega }_i$omega_(i)\omega_{i} its ADC prefilter cutoff frequency. Similarly
对当前采样延迟进行建模， $T_i$$T_i$T_(i)T_{i} 表示采样间隔及其 ${\omega }_i$${\omega }_i$omega_(i)\omega_{i} ADC 预滤波器截止频率。同样地

models the voltage sampling delay, with $T_v$$T_v$T_(v)T_{v} representing the sampling interval, ${\omega }_v$${\omega }_v$omega_(v)\omega_{v} its ADC prefilter cutoff frequency, and ${\omega }_{\mathrm{tv}}$${\omega }_{\mathrm{tv}}$omega_(tv)\omega_{\mathrm{tv}} its transducer delay. Since ${\mathbf{V}}_p$${\mathbf{V}}_p$V_(p)\mathbf{V}_{p} does not result in any negativesequence response in ${\mathbf{M}}_a$${\mathbf{M}}_a$M_(a)\mathbf{M}_{a}, and ${\mathbf{V}}_n$${\mathbf{V}}_n$V_(n)\mathbf{V}_{n} does not result in any positivesequence response either, sequence components are decoupled from each other. Introducing (7) in the frequency-domain version of (3), impedance models can be found as follows:
对电压采样延迟进行建模， $T_v$$T_v$T_(v)T_{v} 表示采样间隔、 ${\omega }_v$${\omega }_v$omega_(v)\omega_{v} ADC 预滤波器截止频率和 ${\omega }_{\mathrm{tv}}$${\omega }_{\mathrm{tv}}$omega_(tv)\omega_{\mathrm{tv}} 传感器延迟。由于在 ${\mathbf{M}}_a$${\mathbf{M}}_a$M_(a)\mathbf{M}_{a} 中不会导致任何 negativesequence 响应，也不会 ${\mathbf{V}}_n$${\mathbf{V}}_n$V_(n)\mathbf{V}_{n} 产生任何 positivesequence 响应，因此 ${\mathbf{V}}_p$${\mathbf{V}}_p$V_(p)\mathbf{V}_{p} sequence 组件彼此解耦。在 （3） 的频域版本中引入 （7），阻抗模型如下：

where $Z_p(s)$$Z_p(s)$Z_(p)(s)Z_{p}(s) and $Z_n(s)$$Z_n(s)$Z_(n)(s)Z_{n}(s) denote positive-sequence and negativesequence impedances, respectively.
其中 $Z_p(s)$$Z_p(s)$Z_(p)(s)Z_{p}(s) 和 $Z_n(s)$$Z_n(s)$Z_(n)(s)Z_{n}(s) 分别表示正序和负序阻抗。

Fig. 3 depicts a $dq$$dq$dqd q-domain current controller. Recall that currents $i_d$$i_d$i_(d)i_{d} and $i_q$$i_q$i_(q)i_{q} are outputs of a $dq$$dq$dqd q-domain transformation, which in the frequency domain involves a convolution of the frequency components in the phase currents, with the frequency components in Park’s transformation. Taking ${\theta }_{\text{PLL}}(t)={\theta }_1(t)$${\theta }_{\text{PLL }}(t)={\theta }_1(t)$theta_("PLL ")(t)=theta_(1)(t)\theta_{\text {PLL }}(t)=\theta_{1}(t), the frequency components in Park’s transformation are easy to
图 3 描述了一个 $dq$$dq$dqd q -domain 电流控制器。回想一下，currents $i_d$$i_d$i_(d)i_{d} 和 $i_q$$i_q$i_(q)i_{q} 是域变换的输出 $dq$$dq$dqd q ，在频域中涉及相电流中频率分量与 Park 变换中的频率分量的卷积。取 ${\theta }_{\text{PLL}}(t)={\theta }_1(t)$${\theta }_{\text{PLL }}(t)={\theta }_1(t)$theta_("PLL ")(t)=theta_(1)(t)\theta_{\text {PLL }}(t)=\theta_{1}(t) ，Park 变换中的频率分量很容易

Fig. 3. Block diagram of a $dq$$dq$dqd q-domain current controller.
图 3. $dq$$dq$dqd q -domain 电流控制器的框图。

TABLE I  表 I
Frequency Components in $DQ$$DQ$DQD Q-Domain Current Controller Output NEGLECTING PLL DYNAMICS
域电流控制器输出中的 $DQ$$DQ$DQD Q 频率分量忽略 PLL 动态

derive, and the result of the convolution is as follows:
derive，卷积的结果如下：

where $I_1$$I_1$I_(1)I_{1} and ${\varphi }_{\mathrm{i}1}$${\varphi }_{\mathrm{i}1}$phi_(i1)\phi_{\mathrm{i} 1} correspond to the amplitude and phase of the fundamental current. Sampling at the fundamental frequency is neglected since $G_i(\pm j2\pi f_1)\approx 1$$G_i\left(\pm j2\pi f_1\right)\approx 1$G_(i)(+-j2pif_(1))~~1G_{i}\left( \pm j 2 \pi f_{1}\right) \approx 1. From the control block diagram, ${\mathbf{C}}_d$${\mathbf{C}}_d$C_(d)\mathbf{C}_{d} and ${\mathbf{C}}_q$${\mathbf{C}}_q$C_(q)\mathbf{C}_{q} can be obtained as linear combinations of (11) and (12) using $H_i(s)$$H_i(s)$H_(i)(s)H_{i}(s) and the decoupling gain $K_d$$K_d$K_(d)K_{d}. A convolution of the frequency components in ${\mathbf{C}}_d$${\mathbf{C}}_d$C_(d)\mathbf{C}_{d} and ${\mathbf{C}}_q$${\mathbf{C}}_q$C_(q)\mathbf{C}_{q} with the frequency components in the inverse Park’s transformation gives ${\mathbf{C}}_a,{\mathbf{C}}_b$${\mathbf{C}}_a,{\mathbf{C}}_b$C_(a),C_(b)\mathbf{C}_{a}, \mathbf{C}_{b}, and ${\mathbf{C}}_c$${\mathbf{C}}_c$C_(c)\mathbf{C}_{c}. Table I shows the possible combinations to consider in the convolution. Note that ${\mathbf{V}}_p$${\mathbf{V}}_p$V_(p)\mathbf{V}_{p} does not result in any negative-sequence response at $f_p$$f_p$f_(p)f_{p}, and ${\mathbf{V}}_n$${\mathbf{V}}_n$V_(n)\mathbf{V}_{n} does not result in any positive-sequence response at $f_n$$f_n$f_(n)f_{n}, which means there is no impedance coupling. The nonlinear coupling at $\pm (f_p-$$\pm \left(f_p-\right.$+-(f_(p)-:}\pm\left(f_{p}-\right. $2f_1)$$\left.2f_1\right)${:2f_(1))\left.2 f_{1}\right) and $\pm (f_n+2f_1)$$\pm \left(f_n+2f_1\right)$+-(f_(n)+2f_(1))\pm\left(f_{n}+2 f_{1}\right) is neglected for impedance modeling in the phase domain. Combining the controller output with the voltage feedforward yields the modulating signals to introduce in the frequency-domain version of (3), which can be solved for sequence impedances as follows:
其中 $I_1$$I_1$I_(1)I_{1} 和 ${\varphi }_{\mathrm{i}1}$${\varphi }_{\mathrm{i}1}$phi_(i1)\phi_{\mathrm{i} 1} 对应于基波电流的幅度和相位。基频采样被忽略了，因为 $G_i(\pm j2\pi f_1)\approx 1$$G_i\left(\pm j2\pi f_1\right)\approx 1$G_(i)(+-j2pif_(1))~~1G_{i}\left( \pm j 2 \pi f_{1}\right) \approx 1 。从控制框图中， ${\mathbf{C}}_d$${\mathbf{C}}_d$C_(d)\mathbf{C}_{d} 可以 ${\mathbf{C}}_q$${\mathbf{C}}_q$C_(q)\mathbf{C}_{q} 得到 （11） 和 （12） 的线性组合，使用 $H_i(s)$$H_i(s)$H_(i)(s)H_{i}(s) 和去耦增益 $K_d$$K_d$K_(d)K_{d} 。在逆 Park 变换中 ${\mathbf{C}}_d$${\mathbf{C}}_d$C_(d)\mathbf{C}_{d} 和 ${\mathbf{C}}_q$${\mathbf{C}}_q$C_(q)\mathbf{C}_{q} 与 频率分量 的卷积得到 ${\mathbf{C}}_a,{\mathbf{C}}_b$${\mathbf{C}}_a,{\mathbf{C}}_b$C_(a),C_(b)\mathbf{C}_{a}, \mathbf{C}_{b} 、 和 ${\mathbf{C}}_c$${\mathbf{C}}_c$C_(c)\mathbf{C}_{c} 。表 I 显示了卷积中要考虑的可能组合。请注意，这 ${\mathbf{V}}_p$${\mathbf{V}}_p$V_(p)\mathbf{V}_{p} 不会在 处 $f_p$$f_p$f_(p)f_{p} 产生任何负序响应，也不会 ${\mathbf{V}}_n$${\mathbf{V}}_n$V_(n)\mathbf{V}_{n} 在 处 $f_n$$f_n$f_(n)f_{n} 产生任何正序响应，这意味着没有阻抗耦合。在 和 $\pm (f_n+2f_1)$$\pm \left(f_n+2f_1\right)$+-(f_(n)+2f_(1))\pm\left(f_{n}+2 f_{1}\right) 处 $\pm (f_p-$$\pm \left(f_p-\right.$+-(f_(p)-:}\pm\left(f_{p}-\right. $2f_1)$$\left.2f_1\right)${:2f_(1))\left.2 f_{1}\right) 的非线性耦合在相域中的阻抗建模中被忽略。将控制器输出与电压前馈相结合，产生在 （3） 的频域版本中引入的调制信号，该信号可以按如下方式求解序列阻抗：
$Z_p(s)=\frac{K_m{V}_{\mathrm{dc}}[H_i(s-j2\pi f_1)-j{K}_d]G_i(s)+sL}{1-K_m{V}_{\mathrm{dc}}{K}_f(s)G_v(s)}$$Z_p(s)=\frac{K_m{V}_{\mathrm{dc}}\left[H_i\left(s-j2\pi f_1\right)-j{K}_d\right]G_i(s)+sL}{1-K_m{V}_{\mathrm{dc}}{K}_f(s)G_v(s)}$Z_(p)(s)=(K_(m)V_(dc)[H_(i)(s-j2pif_(1))-jK_(d)]G_(i)(s)+sL)/(1-K_(m)V_(dc)K_(f)(s)G_(v)(s))Z_{p}(s)=\frac{K_{m} V_{\mathrm{dc}}\left[H_{i}\left(s-j 2 \pi f_{1}\right)-j K_{d}\right] G_{i}(s)+s L}{1-K_{m} V_{\mathrm{dc}} K_{f}(s) G_{v}(s)}
$Z_n(s)=\frac{K_m{V}_{\mathrm{dc}}[H_i(s+j2\pi f_1)+j{K}_d]G_i(s)+sL}{1-K_m{V}_{\mathrm{dc}}{K}_f(s)G_v(s)}$$Z_n(s)=\frac{K_m{V}_{\mathrm{dc}}\left[H_i\left(s+j2\pi f_1\right)+j{K}_d\right]G_i(s)+sL}{1-K_m{V}_{\mathrm{dc}}{K}_f(s)G_v(s)}$Z_(n)(s)=(K_(m)V_(dc)[H_(i)(s+j2pif_(1))+jK_(d)]G_(i)(s)+sL)/(1-K_(m)V_(dc)K_(f)(s)G_(v)(s))Z_{n}(s)=\frac{K_{m} V_{\mathrm{dc}}\left[H_{i}\left(s+j 2 \pi f_{1}\right)+j K_{d}\right] G_{i}(s)+s L}{1-K_{m} V_{\mathrm{dc}} K_{f}(s) G_{v}(s)}.