“Determine the currents and voltages of a circuit with a transformer of transformation ratio n. Method:
1) C and L are converted into complex reactances XL and Xc for frequency analysis of the circuit, Xc=-6j and XL=8j. The source Vs is already in complex form Vs=100j .
2) Mesh law gives Vr1+V1+Vc=Vs, i.e. 2.i1 - 6j.i3 + V1=100j {1}.
Similarly, Vc+V2+Vr2+vL=0, i.e. (10+8j).i2 - 6j.i3 + V2=0 {2}.
3) Apply the marker point rule: if i1 and i2 both enter or leave marker points, i1/i2 = -n ; if i1 enters via marker point and i2 leaves marker point, i1 /i2 = +n ; and if i1 leaves marker point and i2 enters via marker point, i1 /i2 = +n .
Similarly, if V1 and V2 are both oriented towards the positive or negative poles of the coils, V2/V1 = +n , otherwise V2/V1 = -n.
4) In the present case, i1 and i2 enter through the marking points, hence: i1/i2 = -n, i1/i2 = -1/4, i2 = -4.i1 {3}.
5) V1 and V2 are both oriented towards the positive poles, hence : V2/V1 = +n, V2/V1=1/4,
i.e. V2 = 0.25.V1 {4}.
6) Replace V2 by 0.25.V1 in equation {1}, hence: (-40-32j).i1 - 6j.i3 +0.25V1= 0 {5}.
7) The law of knots gives i1+i2+i3=0, replace i2 by 4i1, i.e. -3i1 - i3 = 0 {6}.
8) Equations {1}, {5} and {6} form a 3x3 matrix whose unknowns are: i1, i3 and V1.
9) Solving the system gives i1=0.561.phase(65.45°); i3=1.684.phase(-114.55°) and V1=95.18.phase(84.74°).
10) Equation {3} gives i2 = -4.i1, from which we deduce i2.
11) Equation {4} gives V2=0.25.V1, from which we deduce V2.
12) Ohms' law V=RI is applied to impedances Xc, XL, R1 and R2 to obtain Vc, VL, Vr1 and Vr2 across each element.
26 окт 2024
