Capacitors & Inductors (Book Notes)

Capacitors in DC Circuits

Concept and DC Behavior A capacitor is a passive element designed to store energy in its electric field. In a direct current (dc) circuit, the voltage across a capacitor is constant. Since the current through a capacitor is proportional to the time rate of change of the voltage ($i=C\frac{dv}{dt}$), the current is zero when the voltage is constant. Consequently, a capacitor acts as an open circuit to dc. However, the voltage across a capacitor cannot change abruptly.

Formulas

• Current-Voltage Relationship: $i=C\frac{dv}{dt}$
• Stored Charge: $q=Cv$
• Stored Energy: $w=\frac{1}{2}C{v}^2=\frac{q^2}{2C}$
• Series Combination: The equivalent capacitance of series-connected capacitors is the reciprocal of the sum of the reciprocals of the individual capacitances (similar to resistors in parallel). $\frac{1}{C_{eq}}=\frac{1}{C_1}+\frac{1}{C_2}+\cdots +\frac{1}{C_N}$
• Parallel Combination: The equivalent capacitance of parallel-connected capacitors is the sum of the individual capacitances (similar to resistors in series). $C_{eq}=C_1+C_2+\cdots +C_N$

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Inductors in DC Circuits

Concept and DC Behavior An inductor is a passive element designed to store energy in its magnetic field. In a dc circuit, the current flowing through an inductor is constant. Since the voltage across an inductor is directly proportional to the time rate of change of the current ($v=L\frac{di}{dt}$), the voltage is zero when the current is constant. Therefore, an inductor acts as a short circuit to dc. An important property of the inductor is that the current flowing through it cannot change abruptly.

Formulas

• Voltage-Current Relationship: $v=L\frac{di}{dt}$
• Current-Voltage Integration: $i=\frac{1}{L}{\int }_{t_0}^{t}v(\tau )d\tau +i(t_0)$
• Stored Energy: $w=\frac{1}{2}L{i}^2$
• Series Combination: The equivalent inductance of series-connected inductors is the sum of the individual inductances (similar to resistors in series). $L_{eq}=L_1+L_2+\cdots +L_N$
• Parallel Combination: The equivalent inductance of parallel inductors is the reciprocal of the sum of the reciprocals of the individual inductances (similar to resistors in parallel). $\frac{1}{L_{eq}}=\frac{1}{L_1}+\frac{1}{L_2}+\cdots +\frac{1}{L_N}$

Summary Table of Characteristics

Property	Capacitor ($C$)	Inductor ($L$)
v-i Relation	$v=\frac{1}{C}{\int }_{t_0}^{t}i(\tau )d\tau +v(t_0)$	$v=L\frac{di}{dt}$
i-v Relation	$i=C\frac{dv}{dt}$	$i=\frac{1}{L}{\int }_{t_0}^{t}v(\tau )d\tau +i(t_0)$
DC Behavior	Open circuit	Short circuit
Continuity	Voltage cannot change abruptly	Current cannot change abruptly
Stored Energy	$w=\frac{1}{2}C{v}^2$	$w=\frac{1}{2}L{i}^2$

alexander sadiku