:
Ro = R1 + 1 / (1 / R4 + 1 / (R2 + R3)) (1)
x = R
3 :
Ro = 10 + 1 / (1 / 40 + 1 / (20 + x)) (2)
Ro Io:
Ro = U / Io = 15 / Io
Io:
Io = I1 + I2
I1 = 5 / R3 = 5 / x
I2 = U4 / R4 = U4 / 40
U4 = U2 + U3 = U2 + 5
U2 = I1 * R2 = I1 * 20 = 100 / x
U4 = 100 / x + 5
I2 = (100 / x + 5) / 40
Io = 5 / x + (100 / x + 5) / 40 = 7.5 / x + 1/8
Ro = 15 / [7.5 / x + 1/8] = 2 / (1/x + 1 / 60)
(2)
2 / (1/x + 1 / 60) = 10 + 1 / [1 / 40 + 1 / (20 + x)]
(1/x + 1/60) :
2 = 10/x + 1/6 + (1/x + 1/60) / [1 / 40 + 1 / (20 + x)]
[ x / 40 + x / (20 + x) ]
x/20 + 2x / (20 + x) = 1/4 + 10/ (20 + x) + x / 240 + (x/6) / (20 + x) + 1 + x/60
(20 + x)
(20 + x) x/20 + 2x = (20 + x) / 4 + 10 + (20 + x) x / 240 + x/6 + (20 + x) + (20 + x) x / 60
:
(20 + x) (x/20 - 1/4 - x/240 - 1 - x/60) + x(2 -1/6) - 10 = 0
x - 5 - x/12 - 20 - x/3 + x^2 / 20 - x/4 - x^2 / 240 - x - x^2 / 60 + 2x - x/6 - 10 = 0
, :
(1/20 - 1/240 - 1/60) x^2 + (2 - 1/12 - 1/3 - 1/4 -1/6) x - (5 + 10 + 20) = 0
(7/240) x^2 + (7/6) x - 35 = 0
:
x = (- b +-SQRT(b^2 - 4ac))/2a
x = 240(-7/6 + SQRT(49/36 + 4*7*35/240)) / 14
x x > 0, ( )
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