Other Passive Components

So far, we’ve only looked at resistors as passive components. Now, we introduce capacitors and inductors.

Capacitors

A capacitor is a device that stores charge. Supercapacitors are sometimes used as a short-term replacement for batteries (Halper and Ellenbogen [2006]).

Physical Characteristics

The relationship between the voltage across a capacitor and the charge stored by the capacitor is

\[ Q=VC \]

where \(Q\) is the charge measured in coulombs, \(V\) is the applied voltage measured in volts, and \(C\) is the capacitance of the capacitor measured in Farads.

Fig. 48 Schematic representation of a capacitor.

Just like resistance has an inverse quantity of conductance, capacitance has an inverse quantity of elastance.

Table 5 Capacitor Units

Quantity	Unit	Abbreviation	Variable
Charge	Coulomb	C	\(Q,q\)
Voltage	Volt	V	\(V,v\)
Capacitance	Farad	F	\(C\)
Elastance	Daraf	F\(^{-1}\)	\(D\)

Equivalent Capacitance: Series

Two elements connected in series share one node exclusively.

When two capacitors are in series they can be redrawn as a single capacitor.

The elastances add. Recall that

\[ D=\frac{1}{C} \]

so

\[ \frac{1}{C_S}=\frac{1}{C_1}+\frac{1}{C_2} \]

Solving for \(C_S\) and adding additional capacitors

\[ C_S=\frac{1}{\frac{1}{C_1}+\frac{1}{C_2}+\dots+\frac{1}{C_N}} \]

The value of two capacitors in series is commonly expressed as

\[ C_S=\frac{C_1C_2}{C_1+C_2} \]

which is similar to the way the equivalent resistors of resistors in parallel is calculated (1).

Equivalent Capacitance: Parallel

Two elements are in parallel when they are connected to the same two nodes.

When two capacitors are in parallel they can be redrawn as a single capacitor

Voltage/Current Relationship

All of circuit analysis can be related back to the three fundamental laws: KVL, KCL, and Ohm’s Law. As new passive components are introduced we must reconsider the relationship between voltage and current. The capacitor equation takes the place of Ohm’s law.

We can start with the charge equation introduced earlier and restated here as a time-domain function.

\[ q(t)=Cv(t) \]

taking the derivative of both sides leads to

\[ \frac{dq(t)}{dt}=C\frac{dv(t)}{dt} \]

Recall from the section on fundamentals the the time derivative of charge is current, \(i(t)=\frac{dq(t)}{dt}\), alternatively stated as the flow rate of charge. Therefore the derivative form of the capacitor equation can be stated as

\[ i(t)=C\frac{dv(t)}{dt}\]

An alternate form of the equation is used often. Taking the integral of each side of the derivative form results in the integral form of the capacitor equation:

(17)\[v(t)=\frac{1}{C}\int_{t_0}^t i(\tau)d\tau + v(t_0)\]

For capacitors, these two forms of the capacitor equation take the place of Ohm’s Law in the methods of analysis previously introduced.

We shall see in the section Capacitor how we can make the connection to Ohm’s Law even more direct.

Inductors

An inductor is a device that stores energy in magnetic flux.

Physical Characteristics

The relationship between the current through an inductor and the magnetic flux in the inductor is

\[ \phi = L I \]

where \(\phi\) is the flux measured in webers, \(I\) is the current measured in amperes, and \(L\) is the inductance of the inductor measured in Henries (Photon [2013]).

Fig. 49 Schematic representation of an inductor.

Just like resistance has an inverse quantity of conductance, inductance has an inverse quantity of reluctance.

Table 6 Inductor Units

Quantity	Unit	Abbreviation	Variable
Flux	Weber	Wb	\(\phi\)
Current	Ampere	A	\(I,i\)
Inductance	Henry	H	\(L\)
Reluctance	Inverse Henry	H\(^{-1}\)	\(\cal{R}\)

Equivalent Inductance: Series

Two elements connected in series share one node exclusively.

When two inductors are in series they can be redrawn as a single inductor

Equivalent Inductance: Parallel

Two elements are in parallel when they are connected to the same two nodes.

When two inductors are in parallel they can be redrawn as a single inductor

The reluctances add. Recall that

\[ \mathcal{R}=\frac{1}{L} \]

so

\[ \frac{1}{L_P}=\frac{1}{L_1}+\frac{1}{L_2} \]

Solving for \(L_P\) and adding additional inductors

\[ L_P=\frac{1}{\frac{1}{L_1}+\frac{1}{L_2}+\dots+\frac{1}{L_N}} \]

The value of two inductors in parallel is commonly expressed as

\[ L_P=\frac{L_1L_2}{L_1+L_2} \]

Voltage/Current Relationship

The derivative form of the inductor equation can be stated as

\[ v(t)=L\frac{di(t)}{dt}\]

An alternate form of the equation is used often. Taking the integral of each side of the derivative form results in the integral form of the inductor equation:

(18)\[i(t)=\frac{1}{L}\int_{t_0}^t v(\tau)d\tau + i(t_0)\]

For inductors, these two forms of the inductor equation take the place of Ohm’s Law in the methods of analysis previously introduced.

We shall see in the section Inductor how we can make the connection to Ohm’s Law even more direct.

Realistic Models of Passive Components

What we have looked at above and earlier are ideal resistors, capacitors, and inductors. In reality, each of these devices often exhibits aspects of the other two also. Different models for real devices have been developed. They are as follows.

A Realistic Resistor Model

Horowitz et al. (Horowitz and Hill [2020]) presents the following model for a realistic resistor.

Fig. 50 A realistic resistor model from Horowitz and Hill [2020].

Table 7 Realistic resistor model components.

Quantity	Description
\(R\)	The nominal resistance.
\(L_\ell\)	The lead inductance.
\(L_S\)	The series inductance.
\(C_P\)	The parallel or shunt capacitance.

A Realistic Capacitor Model

Inc. [2009] presents the following model for a realistic capacitor.

Fig. 51 A realistic capacitor model from Inc. [2009].

Table 8 Realistic capacitor model components.

Quantity	Description
\(R_S\)	The series resistance.
\(L_S\)	The series inductance.
\(C\)	The nominal capacitance.

A Realistic Inductor Model

Inc. [2009] presents the following model for a realistic inductor.

Fig. 52 A realistic inductor model from Inc. [2009].

Table 9 Realistic inductor model components.

Quantity	Description
\(R_S\)	The series resistance.
\(L\)	The nominal inductance.
\(C_P\)	The parallel or shunt capacitance.

Next Steps: Analyzing Circuits with Reactive Components

Both the capacitor and inductor equations result in systems of equations that require knowledge of differential equations. We can approach this type of circuit analysis in two ways: 1) head-on by solving the systems of differential equations or 2) restating the equations using something called a phasor and solving it algebraically. Both approaches are presented here. I encourage you to read both sections if you have studied differential equations. You can skip the next section and proceed the section on phasors if you have not .

References

HE06

Marin S. Halper and James C. Ellenbogen. Supercapacitors: A Brief Overview. https://www.mitre.org/sites/default/files/pdf/06_0667.pdf, 2006. [Online; accessed 14-March-2023].

HH20(1,2)

Paul Horowitz and Winfield Hill. The Art of Electronics : the X-Chapters. Cambridge University Press, 2020.

Inc09(1,2,3,4)

Agilent Technologies Inc. Agilent Impedance Measurement Handbook : A guide to measurement technology and techniques. Agilent Technologies, Inc., 2009.

Pho13

The Photon. Inductor equivalent of capacitor's charge. https://electronics.stackexchange.com/a/68361/4338, 2013. [Online; accessed 14-March-2023].