Notes on Combining Parallel and Series Connected Capacitors and Inductors

Combining Capacitors and Inductors

Series Connection of Capacitors

• Concept: Voltage across capacitors in series is summed up using Kirchhoff's Voltage Law (KVL).

• Capacitance Relation:

• For three capacitors: C1, C2, C3

• Voltage relation:
  [ V{subs} = V{C1} + V{C2} + V{C3} ]

• Voltage across capacitors expressed as integrals of current:
  [ V{C1} = \frac{1}{C1} \int I(t) dt ] [ V{C2} = \frac{1}{C2} \int I(t) dt ]
  [ V_{C3} = \frac{1}{C3} \int I(t) dt ]

• Equivalent Capacitance (C_EQ):

• Relating the total voltage to an equivalent capacitance:
  [ V{subs} = \frac{1}{C{EQ}} \int I(t) dt ]

• Setting equal voltage expressions:
  [ \frac{1}{C_{EQ}} \int I(t) dt = \left( \frac{1}{C1} + \frac{1}{C2} + \frac{1}{C3} \right) \int I(t) dt ]

• Canceling integral terms yields:
  [ \frac{1}{C_{EQ}} = \frac{1}{C1} + \frac{1}{C2} + \frac{1}{C3} ]

Parallel Connection of Capacitors

• Concept: In a parallel connection, the voltage across all capacitors is the same.

• Capacitance Relation:

• KCL relation for capacitors:
  [ I{subs} = I{C1} + I{C2} + I{C3} ]

• Current through capacitors can be expressed as:
  [ I{C1} = C1 \frac{dV}{dt} ] [ I{C2} = C2 \frac{dV}{dt} ]
  [ I_{C3} = C3 \frac{dV}{dt} ]

• Equivalent Capacitance (C_EQ):

• For the equivalent, we write:
  [ I{subs} = C{EQ} \frac{dV}{dt} ]

• Equating expressions yields:
  [ C_{EQ} \frac{dV}{dt} = (C1 + C2 + C3) \frac{dV}{dt} ]

• Simplifying gives:
  [ C_{EQ} = C1 + C2 + C3 ]

• Interpretation: Think of capacitors in parallel as areas of plates adding together.

Series Connection of Inductors

• Concept: In series, the voltages across inductors also sum up using KVL.

• Inductance Relation:

• Voltage across inductors:
  [ V{subs} = V{L1} + V{L2} + V{L3} ]

• Voltage across each inductor expressed with current:
  [ V{L1} = L1 \frac{di}{dt} ] [ V{L2} = L2 \frac{di}{dt} ]
  [ V_{L3} = L3 \frac{di}{dt} ]

• Equivalent Inductance (L_EQ):

• For equivalent inductance:
  [ V{subs} = L{EQ} \frac{di}{dt} ]

• Setting expressions equal gives:
  [ L_{EQ} \frac{di}{dt} = (L1 + L2 + L3) \frac{di}{dt} ]

• Cancel yields:
  [ L_{EQ} = L1 + L2 + L3 ]

• Interpretation: Inductors in series can be understood by counting the number of turns of each coil.

Parallel Connection of Inductors

• Concept: Inductors in parallel possess the same voltage properties as capacitors in parallel.

• Inductance Relation: The current through each inductor is given by:
  [ I_{subs} = \frac{1}{L1} \int V(t) dt + \frac{1}{L2} \int V(t) dt + \frac{1}{L3} \int V(t) dt ]

• Equivalent Inductance (L_EQ):

• For the equivalent, we have:
  [ I{subs} = \frac{1}{L{EQ}} \int V(t) dt ]

• Setting equal expressions gives:
  [ \frac{1}{L_{EQ}} \int V(t) dt = \left( \frac{1}{L1} + \frac{1}{L2} + \frac{1}{L3} \right) \int V(t) dt ]

• Canceling yields:
  [ \frac{1}{L_{EQ}} = \frac{1}{L1} + \frac{1}{L2} + \frac{1}{L3} ]

• Comparison: The equations for parallel inductances align with those for parallel resistors.