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Question
Calculate the resultant capacitances for each of the following combinations of capacitors.
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Solution
Parallel combination of capacitor 1 and 2
Cp = C0 + C0 = 2C0
Series combination of capacitor Cp and 3
`1/"C"_"s" = 1/"C"_"p" + 1/"C"_3 = 1/(2 "C"_0) + 1/"C"_0 = ("or") 1/"C"_"s" = 3/2 "C"_0 ("or") "C"_"s" = 2/3 "C"_0`
`1/"C"_("s"_1) = 1/"C"_1 + 1/"C"_2 = 1/"C"_0 + 1/"C"_0 = 1/"C"_0` (or)
`1/"C"_("s"_1) = 2/"C"_0 ("or") "C"_("S"_1) = "C"_0/2`
Similarly 3 and 4 are series combination
`1/"C"_("s"_1) = 1/"C"_3 + 1/"C"_4 = 1/"C"_0 + 1/"C"_0 = 2/"C"_0 ("or") "C"_("S"_2) = "C"_0/2`
`"C"_("S"_1) and "C"_("S"_2)` are in parallel combination
`"C"_"p" = "C"_("S"_1) + "C"_("S"_2) = "C"_0/2 + "C"_0/2 ("or") "C"_"p" = (2"C"_0)/2 "C"_"p" = "C"_0`- Capacitor 1, 2 and 3 are in parallel combination
Cp = C0 + C0 + C0 = 3C0
Cp = 3C0
- Capacitar C1 and C2 are in combination
`1/"C"_("s"_1) = ("C"_1 + "C"_2)/("C"_1"C"_2)`
`"C"_("s"_1) = ("C"_1"C"_2)/("C"_1 + "C"_2)`
Similarly C3 and C4 are in series combination
`1/"C"_("s"_2) = 1/"C"_3 + 1/"C"_4 = ("C"_3 + "C"_4)/("C"_3"C"_4)`
`"C"_("s"_2) = ("C"_3"C"_4)/("C"_3 + "C"_4)`
`"C"_("s"_1) and "C"_("s"_2)` are in parallel combination across RS:
CP = `"C"_("S"_1) + "C"_("S"_2)`
`= ("C"_1"C"_2)/("C"_1 + "C"_2) + ("C"_3"C"_4)/("C"_3 + "C"_4)`
`= ("C"_1"C"_2 ("C"_3 + "C"_4) + "C"_3"C"_4("C"_1 + "C"_2))/(("C"_1 + "C"_2)("C"_3 + "C"_4))`
`"C"_"P" = ("C"_1"C"_2"C"_3 + "C"_1"C"_2"C"_4 + "C"_3"C"_4"C"_1 + "C"_3"C"_4"C"_2)/(("C"_1 + "C"_2)("C"_3 + "C"_4))` - Capacitor 1 and 2 are series combination
`1/"C"_("s"_1) = 1/"C"_1 + 1/"C"_2 = 1/"C"_0 + 1/"C"_0 = 1/"C"_0`
`1/"C"_("s"_1) = 2/"C"_0` (or) `"C"_"s"_1 = "C"_0/2`
Similarly 3 and 4 are series combination
`1/"C"_("s"_1) = 2/"C"_0 ("or") "C"_("s"_2) = "C"_0/2`
Three capacitors are in parallel combination
`"C"_"P" = "C"_0/2 + "C"_0 + "C"_0/2 + (2"C"_0)/2 + "C"_0/2 = (4"C"_0)/2`
`"C"_"P" = 2 "C"_0`
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