1. The converter above is used to supply three identical, 50-Hz AC loads, represented by y,...

  

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1. The converter above is used to supply three identical, 50-Hz AC loads, represented by Ży, from a DC source (with Vin = 100V) using three half-bridge circuits. Assume the load impedance Ży has the following frequency characteristic: (182 Zy (jw)=1+jl!,w 2πf. , otherwise. li (1) Va,g;0 V6.9:0 Vc,9;0 = lag:0 Ib.g:0 Teg:0 = Vin 4||||||| -T HI|HI|H ,W=0 +1 + fag(t) + Vag(t) -- انت -T 27 T A₂ e., the load looks like an open circuit at all frequencies except DC and fs = 50 Hz. Throughout his problem, the following general notation is used to refer to the Fourier series components f the signal xyz (t) and the switching functions q.(t): "s,z(l) = Xu,z:0+ -√2X, ;k COS (KW₂l + 99,-k) = Xy=0+ +ΣReal {√2 (Xyk) eh,l}, k=1 k=1 D₁ = 4:(t) = 0,5 for all switches in this converter 1 -T T 9A11-94₂ (₁ , switch is on at time t o, switch is off at time t Iso, note the following switching function relationships (as shown in the plots): JA (1) =9A,(-1.) (²-77) qc, (t)= qA₂ (₁-27) a) Determine the DC component, fundamental frequency phasor, and third harmonic phasor for the three load voltages Vag(t), Vbg(t), and veg(t). Provide all phasors in polar (magnitude and angle) form. Va,g;1 = 7B₁ (t) = GA₂ = INT B₂) - Ib,g;1 = Tc.g;1 Flor 90₁ 190₂ T 3 ieg() +(t)- Zy Vay (1) 141 iag(t) +(t)- 2T 3 T V 9:3 V c.9¹ b) Determine the DC component, fundamental frequency phasor, and third harmonic phasor for the three load currents iag(t), ibg(t), and icg(t). Provide all phasors in polar (magnitude and angle) form. Tag:1 - Eles R₁ 27 1-GC₂ Zy Vag(t) B₂ 3 3 c) Determine the current in(t). d) Determine the power factor for the leftmost (phase "A") load in the circuit (i.e., compute the power factor using Vag(t) and ia,g(t)). lenge e) Determine the DC component Vabo and the fundamental phasor Va,b;1 for the voltage Vab(t) = Vag(t) - Vbg(t), as labeled on the figure below. Provide the phasor in polar (magnitude and angle) form. ic.g(l) Zv Ur.g(t) ib,g(t) Xy, zk= Xyk@y, zk 4i, =27f₂ V 6,9;3= Vb.g;3= Tag:3 = Ib,g;3= Tega f) [BONUS] Assume a different topology is being considered, as shown below, with the same switching functions (qA1(t), qAz(t), etc.) as given in part (a). Find Va,go, Va,g:1, Va,b;0, and Va,b;1, the DC components and first harmonic phasors of vag(t) and va,b(t). Provide all phasors in polar (magnitude and angle) form. Zr vb.g(t) + ic,g(t) + Zy Uca(t) 1. The converter above is used to supply three identical, 50-Hz AC loads, represented by Ży, from a DC source (with Vin = 100V) using three half-bridge circuits. Assume the load impedance Ży has the following frequency characteristic: (182 Zy (jw)=1+jl!,w 2πf. , otherwise. li (1) Va,g;0 V6.9:0 Vc,9;0 = lag:0 Ib.g:0 Teg:0 = Vin 4||||||| -T HI|HI|H ,W=0 +1 + fag(t) + Vag(t) -- انت -T 27 T A₂ e., the load looks like an open circuit at all frequencies except DC and fs = 50 Hz. Throughout his problem, the following general notation is used to refer to the Fourier series components f the signal xyz (t) and the switching functions q.(t): "s,z(l) = Xu,z:0+ -√2X, ;k COS (KW₂l + 99,-k) = Xy=0+ +ΣReal {√2 (Xyk) eh,l}, k=1 k=1 D₁ = 4:(t) = 0,5 for all switches in this converter 1 -T T 9A11-94₂ (₁ , switch is on at time t o, switch is off at time t Iso, note the following switching function relationships (as shown in the plots): JA (1) =9A,(-1.) (²-77) qc, (t)= qA₂ (₁-27) a) Determine the DC component, fundamental frequency phasor, and third harmonic phasor for the three load voltages Vag(t), Vbg(t), and veg(t). Provide all phasors in polar (magnitude and angle) form. Va,g;1 = 7B₁ (t) = GA₂ = INT B₂) - Ib,g;1 = Tc.g;1 Flor 90₁ 190₂ T 3 ieg() +(t)- Zy Vay (1) 141 iag(t) +(t)- 2T 3 T V 9:3 V c.9¹ b) Determine the DC component, fundamental frequency phasor, and third harmonic phasor for the three load currents iag(t), ibg(t), and icg(t). Provide all phasors in polar (magnitude and angle) form. Tag:1 - Eles R₁ 27 1-GC₂ Zy Vag(t) B₂ 3 3 c) Determine the current in(t). d) Determine the power factor for the leftmost (phase "A") load in the circuit (i.e., compute the power factor using Vag(t) and ia,g(t)). lenge e) Determine the DC component Vabo and the fundamental phasor Va,b;1 for the voltage Vab(t) = Vag(t) - Vbg(t), as labeled on the figure below. Provide the phasor in polar (magnitude and angle) form. ic.g(l) Zv Ur.g(t) ib,g(t) Xy, zk= Xyk@y, zk 4i, =27f₂ V 6,9;3= Vb.g;3= Tag:3 = Ib,g;3= Tega f) [BONUS] Assume a different topology is being considered, as shown below, with the same switching functions (qA1(t), qAz(t), etc.) as given in part (a). Find Va,go, Va,g:1, Va,b;0, and Va,b;1, the DC components and first harmonic phasors of vag(t) and va,b(t). Provide all phasors in polar (magnitude and angle) form. Zr vb.g(t) + ic,g(t) + Zy Uca(t)

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