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45

Inverter Theory

Field: Power Electronics

Sequence of Expressions

Let S=(sa,sb,sc)∈{0,1}3\mathbf{S} = (s_a, s_b, s_c) \in \{0, 1\}^3 be the switching state vector, where sis_i is the state of phase ii. The instantaneous output voltage vector V⃗out(t)\vec{V}_{out}(t) is defined by the linear combination of the DC link voltage VdcV_{dc} and the switching states: \vec{V}_{out}(t) = V_{dc} \cdot \begin{pmatrix} s_a - s_b \ s_b - s_c \ s_c - s_a \bigg/ 2 \biggend{pmatrix} where the components are defined relative to the phase voltages va,vb,vcv_{a}, v_{b}, v_{c}.
Define the desired output voltage vector V⃗ref=(Vrefα,Vrefβ)\vec{V}_{ref} = (V_{ref\alpha}, V_{ref\beta}) in the α−β\alpha-\beta plane. The modulation index μ\mu and the maximum achievable voltage magnitude Vmax=Vdc/3V_{max} = V_{dc}/\sqrt{3} determine the reference vector: V⃗ref=μVmax(cos⁡(ωt+ϕ)sin⁡(ωt+ϕ)) \vec{V}_{ref} = \mu V_{max} \begin{pmatrix} \cos(\omega t + \phi) \\ \sin(\omega t + \phi) \end{pmatrix} The optimal switching times are found by mapping V⃗ref\vec{V}_{ref} to the nearest achievable space vector V⃗sv\vec{V}_{sv} such that V⃗sv≈V⃗ref\vec{V}_{sv} \approx \vec{V}_{ref}.
Let VdcV_{dc} be the DC bus voltage and Vc(t)V_{c}(t) be the high-frequency triangular carrier signal. The switching signal si(t)s_i(t) for phase ii is generated by comparing the modulating signal mi(t)m_i(t) (e.g., sinusoidal reference) with Vc(t)V_{c}(t): si(t)={1if mi(t)>Vc(t)0if mi(t)<Vc(t)undefinedif mi(t)=Vc(t) s_i(t) = \begin{cases} 1 & \text{if } m_i(t) > V_{c}(t) \\ 0 & \text{if } m_i(t) < V_{c}(t) \\ \text{undefined} & \text{if } m_i(t) = V_{c}(t) \end{cases} The output voltage is then approximated by the duty cycle δi=Vdc2TonTsw\delta_i = \frac{V_{dc}}{2} \frac{T_{on}}{T_{sw}}.
Let SH,i(t)S_{H,i}(t) and SL,i(t)S_{L,i}(t) be the high-side and low-side switching signals for phase ii, respectively. The dead time constraint mandates that the signals cannot be simultaneously active (high) or simultaneously inactive (low) in a manner that causes a short circuit: SH,i(t)⋅SL,i(t)=0∀t S_{H,i}(t) \cdot S_{L,i}(t) = 0 \quad \forall t Furthermore, the control must ensure a minimum non-zero time interval τd\tau_d where both switches are off (or in a high-impedance state) to prevent shoot-through.
Define the switching period TswT_{sw} as the inverse of the switching frequency fswf_{sw}: Tsw=1fsw T_{sw} = \frac{1}{f_{sw}} The switching action is modeled by the periodic function s(t)s(t) with period TswT_{sw}. The total switching losses PswP_{sw} are proportional to fswf_{sw} and the switching energy EswE_{sw}: Psw∝fswEsw P_{sw} \propto f_{sw} E_{sw} The design objective is to minimize PswP_{sw} subject to constraints on output ripple ΔVdc\Delta V_{dc}.
Let vout(t)v_{out}(t) be the synthesized AC voltage waveform. The Total Harmonic Distortion (THD) is defined as the ratio of the Root Mean Square (RMS) voltage of all harmonic components (excluding the fundamental) to the RMS fundamental component voltage V1V_{1}: THD=Vh,rmsV1,rms=∑h=2∞Vh2V1 \text{THD} = \frac{V_{h, rms}}{V_{1, rms}} = \frac{\sqrt{\sum_{h=2}^{\infty} V_{h}^2}}{V_{1}} where VhV_{h} is the RMS amplitude of the hh-th harmonic component, and V1V_{1} is the fundamental component amplitude.
Consider the instantaneous power PP extracted from a source (e.g., solar array) as a function of the operating current II and voltage VV: P(I,V)=V∙IP(I, V) = V \bullet I. The Maximum Power Point Tracking (MPPT) algorithm seeks to find the optimal operating point (Iopt,Vopt)(I_{opt}, V_{opt}) that maximizes PP. This is achieved by solving the condition: ∂P∂I∣V=Vopt=0 \frac{\partial P}{\partial I} \bigg|_{V=V_{opt}} = 0 or equivalently, by tracking the point where the instantaneous power derivative with respect to current is zero.
The DC link capacitor voltage ripple ΔVdc\Delta V_{dc} is the transient fluctuation in VdcV_{dc} caused by the ripple current Δidc\Delta i_{dc} flowing through the capacitor CC. Assuming a switching period TswT_{sw}, the ripple is approximated by: ΔVdc≈Δidc2πfswC \Delta V_{dc} \approx \frac{\Delta i_{dc}}{2 \pi f_{sw} C} where Δidc\Delta i_{dc} is the peak-to-peak ripple current, which is proportional to the output current IoutI_{out} and the switching period TswT_{sw}. The relationship is fundamentally governed by the capacitor charge balance: ∫tt+Tswidc(t)dt=CdVdcdt\int_{t}^{t+T_{sw}} i_{dc}(t) dt = C \frac{d V_{dc}}{dt}.
The Sinusoidal Pulse Generator (SPG) model approximates the output voltage vout(t)v_{out}(t) as a series of high-frequency rectangular pulses. For a fundamental frequency ω\omega and carrier frequency ωsw\omega_{sw}, the model can be represented by the Fourier series expansion: vout(t)≈Vdc∑h=1∞1hsin⁡(hωt)⋅(1−1h∣sin⁡(hωsw2Tsw)∣) v_{out}(t) \approx V_{dc} \sum_{h=1}^{\infty} \frac{1}{h} \sin(h \omega t) \cdot \left(1 - \frac{1}{h} \left| \sin(\frac{h \omega_{sw}}{2} T_{sw}) \right| \right) This formulation simplifies the analysis by treating the output as a superposition of harmonic components derived from the pulse shape.
Define the instantaneous output voltage vector V⃗out(t)\vec{V}_{out}(t) in the stationary α−β\alpha-\beta coordinate system. The vector is decomposed into components relative to the fundamental frequency ω\omega and the DC bus voltage VdcV_{dc}: V⃗out(t)=Vdc⋅(cos⁡(ωt)sin⁡(ωt))+V⃗ripple(t) \vec{V}_{out}(t) = V_{dc} \cdot \begin{pmatrix} \cos(\omega t) \\ \sin(\omega t) \end{pmatrix} + \vec{V}_{ripple}(t) The vector magnitude ∣V⃗out∣|\vec{V}_{out}| represents the instantaneous output voltage, and its projection onto the α−β\alpha-\beta plane is the key metric for modulation analysis.