Lecture 24: Circuit Theory Revisited

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Lecture 24

Circuit Theory Revisited

24.1 Circuit Theory Revisited

Circuit theory is one of the most successful and often used theories in electrical engineering.

Its success is mainly due to its simplicity: it can capture the physics of highly complex

circuits and structures, which is very important in the computer and micro-chip industry.

Now, having understood electromagnetic theory in its full glory, it is prudent to revisit circuit

theory and study its relationship to electromagnetic theory [29, 31, 48, 59].

The two most important laws in circuit theory are Kirchoff current law (KCL) and Kirch-

hoff voltage law (KVL) [14, 45]. These two laws are derivable from the current continuity

equation and from Faraday’s law.

24.1.1 Kirchhoff Current Law

Figure 24.1: Schematics showing the derivation of Kirchhoff current law. All currents flowing

into a node must add up to zero.

237

238 Electromagnetic Field Theory

Kirchhoff current law (KCL) is a consequence of the current continuity equation, or that

∇ · J = −jω% (24.1.1)

It is a consequence of charge conservation. But it is also derivable from generalized Ampere’s

law and Gauss’ law for charge.1

First, we assume that all currents are flowing into a node as shown in Figure 24.1, and

that the node is non-charge accumulating with ω → 0. Then the charge continuity equation

becomes

∇ · J = 0 (24.1.2)

By integrating the above current continuity equation over a volume containing the node, it

is easy to show that

N ∑

Ii = 0 (24.1.3)

which is the statement of KCL. This is shown for the schematics of Figure 24.1.

24.1.2 Kirchhoff Voltage Law

Kirchhoff voltage law is the consequence of Faraday’s law. For the truly static case when

ω = 0, it is

∇ × E = 0 (24.1.4)

The above implies that E = −∇Φ, from which we can deduce that

−

E · dl = 0 (24.1.5)

For statics, the statement that E = −∇Φ also implies that we can define a voltage drop

between two points, a and b to be

Vba = −

ˆ b

E · dl =

ˆ b

∇Φ · dl = Φ(rb) − Φ(ra) = Vb − Va (24.1.6)

As has been shown before, to be exact, E = −∇Φ−∂/∂tA, but we have ignored the induction

effect. Therefore, this concept is only valid in the low frequency or long wavelength limit, or

that the dimension over which the above is applied is very small so that retardation effect

can be ignored.

A good way to remember the above formula is that if Vb > Va, then the electric field

points from point a to point b. Electric field always points from the point of higher potential

1Some authors will say that charge conservation is more fundamental, and that Gauss’ law and Ampere’s

law are consistent with charge conservation and the current continuity equation.

Circuit Theory Revisited 239

to point of lower potential. Faraday’s law when applied to the static case for a closed loop of

resistors shown in Figure 24.2 gives Kirchhoff voltage law (KVL), or that

N ∑

Vj = 0 (24.1.7)

Notice that the voltage drop across a resistor is always positive, since the voltages to the left

of the resistors in Figure 24.2 are always higher than the voltages to the right of the resistors.

This implies that internal to the resistor, there is always an electric field that points from the

left to the right.

If one of the voltage drops is due to a voltage source, it can be modeled by a negative

resistor as shown in Figure 24.3. The voltage drop across a negative resistor is opposite to

that of a positive resistor. As we have learn from the Poynting’s theorem, negative resistor

gives out energy instead of dissipates energy.

Figure 24.2: Kichhoff voltage law where the sum of all voltages around a loop is zero, which

is the consequence of static Faraday’s law.

Figure 24.3: A voltage source can also be modeled by a negative resistor.

240 Electromagnetic Field Theory

Faraday’s law for the time-varying case is

∇ × E = − ∂B

∂t (24.1.8)

Writing the above in integral form, one gets

−

E · dl = d

B · dS (24.1.9)

We can apply the above to a loop shown in Figure 24.4, or a loop C that goes from a to b to

c to d to a. We can further assume that this loop is very small compared to wavelength so

that potential theory that E = −∇Φ can be applied. Furthermore, we assume that this loop

C does not have any magnetic flux through it so that the right-hand side of the above can be

set to zero, or

−

E · dl = 0 (24.1.10)

Figure 24.4: The Kirchhoff voltage law for a circuit loop consisting of resistor, inductor, and

capacitor can also be derived from Faraday’s law at low frequency.

Circuit Theory Revisited 241

Figure 24.5: The voltage-current relation of an inductor can be obtained by unwrapping an

inductor coil, and then calculate its flux linkage.

Notice that this loop does not go through the inductor, but goes directly from c to d.

Then there is no flux linkage in this loop and thus

−

ˆ b

E · dl −

ˆ c

E · dl −

ˆ d

E · dl −

ˆ a

E · dl = 0 (24.1.11)

Inside the source or the battery, it is assumed that the electric field points opposite to the

direction of integration dl, and hence the first term on the left-hand side of the above is

positive while the other terms are negative. Writing out the above more explicitly, we have

V0(t) + Vcb + Vdc + Vad = 0 (24.1.12)

Notice that in the above, in accordance to (24.1.6), Vb > Vc, Vc > Vd, and Va > Va. Therefore,

Vcb, Vdc, and Vad are all negative quantities but V0(t) > 0. We will study the contributions

to each of the terms, the inductor, the capacitor, and the resistor more carefully next.

24.1.3 Inductor

To find the voltage current relation of an inductor, we apply Faraday’s law to a closed loop

C′ formed by dc and the inductor coil shown in the Figure 24.5 where we have unwrapped the

solenoid into a larger loop. Assume that the inductor is made of a very good conductor, so

that the electric field in the wire is small or zero. Then the only contribution to the left-hand

side of Faraday’s law is the integration from point d to point c. We assume that outside the

loop in the region between c and d, potential theory applies, and hence, E = −∇Φ. Now,

we can connect Vdc in the previous equation to the flux linkage to the inductor. When the

voltage source attempts to drive an electric current into the loop, Lenz’s law (1834)2 comes

into effect, essentially, generating an opposing voltage. The opposing voltage gives rise to

charge accumulation at d and c, and hence, a low frequency electric field at the gap.

To this end, we form a new C′ that goes from d to c, and then continue onto the wire that

leads to the inductor. But this new loop will contain the flux B generated by the inductor

2Lenz’s law can also be explained from Faraday’s law (1831).

242 Electromagnetic Field Theory

current. Thus ˛

C′

E · dl =

ˆ c

E · dl = −Vdc = − d

S′

B · dS (24.1.13)

The inductance L is defined as the flux linkage per unit current, or

L =

[ˆ

S′

B · dS

]

/I (24.1.14)

So the voltage in (24.1.13) is then

Vdc = d

dt (LI) = L dI

dt (24.1.15)

Had there been a finite resistance in the wire of the inductor, then the electric field is

non-zero inside the wire. Taking this into account, we have

E · dl = RLI − Vdc = − d

B · dS (24.1.16)

Consequently,

Vdc = RLI + L dI

dt (24.1.17)

Thus, to account for the loss of the coil, we add a resistor in the equation. The above becomes

simpler in the frequency domain, namely

Vdc = RLI + jωLI (24.1.18)

24.1.4 Capacitance

The capacitance is the proportionality constant between the charge Q stored in the capacitor,

and the voltage V applied across the capacitor, or Q = CV . Then

C = Q

V (24.1.19)

From the current continuity equation, one can easily show that in Figure 24.6,

I = dQ

dt = d

dt (CVda) = C dVda

dt (24.1.20)

Integrating the above equation, one gets

Vda(t) = 1

ˆ t

−∞

Idt′ (24.1.21)

The above looks quite cumbersome in the time domain, but in the frequency domain, it

becomes

I = jωCVda (24.1.22)

Circuit Theory Revisited 243

Figure 24.6: Schematics showing the calculation of the capacitance of a capacitor.

24.1.5 Resistor

The electric field is not zero inside the resistor as electric field is needed to push electrons

through it. As is well known,

J = σE (24.1.23)

From this, we deduce that Vcb = Vc − Vb is a negative number given by

Vcb = −

ˆ c

E · dl = −

ˆ c

σ · dl (24.1.24)

where we assume a uniform current J = ˆlI/A in the resistor where ˆl is a unit vector pointing

in the direction of current flow in the resistor. We can assumed that I is a constant along the

length of the resistor, and thus,

Vcb = −

ˆ c

Idl

σA = −I

ˆ c

σA = −IR (24.1.25)

and

R =

ˆ c

σA (24.1.26)

Again, for simplicity, we assume long wavelength or low frequency in the above derivation.

24.2 Some Remarks

In this course, we have learnt that given the sources % and J of an electromagnetic system,

one can find Φ and A, from which we can find E and H. This is even true at DC or statics.

We have also looked at the definition of inductor L and capacitor C. But clever engineering

244 Electromagnetic Field Theory

is driven by heuristics: it is better, at times, to look at inductors and capacitors as energy

storage devices, rather than flux linkage and charge storage devices.

Another important remark is that even though circuit theory is simpler that Maxwell’s

equations in its full glory, not all the physics is lost in it. The physics of the induction

term in Faraday’s law and the displacement current term in generalized Ampere’s law are

still retained. In fact, wave physics is still retained in circuit theory: one can make slow

wave structure out a series of inductors and capacitors. The lumped-element model of a

transmission line is an example of a slow-wave structure design. Since the wave is slow, it has

a smaller wavelength, and resonators can be made smaller: We see this in the LC tank circuit

which is a much smaller resonator in wavelength compared to a microwave cavity resonator

for instance. The only short coming is that inductors and capacitors generally have higher

losses than air or vacuum.

24.2.1 Energy Storage Method for Inductor and Capacitor

Often time, it is more expedient to think of inductors and capacitors as energy storage devices.

This enables us to identify stray (also called parasitic) inductances and capacitances more

easily. This manner of thinking allows for an alternative way of calculating inductances and

capacitances as well [29].

The energy stored in an inductor is due to its energy storage in the magnetic field, and it

is alternatively written, according to circuit theory, as

Wm = 1

2 LI2 (24.2.1)

Therefore, it is simpler to think that an inductance exists whenever there is stray magnetic

field to store magnetic energy. A piece of wire carries a current that produces a magnetic

field enabling energy storage in the magnetic field. Hence, a piece of wire in fact behaves

like a small inductor, and it is non-negligible at high frequencies: Stray inductances occur

whenever there are stray magnetic fields.

By the same token, a capacitor can be thought of as an electric energy storage device

rather than a charge storage device. The energy stored in a capacitor, from circuit theory, is

We = 1

2 CV 2 (24.2.2)

Therefore, whenever stray electric field exists, one can think of stray capacitances as we have

seen in the case of fringing field capacitances in a microstrip line.

24.2.2 Finding Closed-Form Formulas for Inductance and Capaci-

tance

Finding closed form solutions for inductors and capacitors is a difficult endeavor. Only certain

geometries are amenable to closed form solutions. Even a simple circular loop does not have

a closed form solution for its inductance L. If we assume a uniform current on a circular

loop, in theory, the magnetic field can be calculated using Bio-Savart law that we have learnt

Circuit Theory Revisited 245

before, namely that

H(r) =

ˆ I(r′)dl′ × ˆ R

4πR2 (24.2.3)

But the above cannot be evaluated in closed form save in terms of complicate elliptic integrals.

However, if we have a solenoid as shown in Figure 24.7, an approximate formula for the

inductance L can be found if the fringing field at the end of the solenoid can be ignored.

The inductance can be found using the flux linkage method [28, 29]. Figure 24.8 shows the

schematics used to find the approximate inductance of this inductor.

Figure 24.7: The flux-linkage method is used to estimate the inductor of a solenoid (courtesy

of SolenoidSupplier.Com).

Figure 24.8: Finding the inductor flux linkage by assuming the magnetic field is uniform

inside a long solenoid.

The capacitance of a parallel plate capacitor can be found by solving a boundary value

problem (BVP) for electrostatics. The electrostatic BVP for capacitor involves Poisson’s

equation and Laplace equation which are scalar equations [42][Thomson’s theorem].

246 Electromagnetic Field Theory

Figure 24.9: The capacitance between two charged conductors can be found by solving a

boundary value problem (BVP).

Assume a geometry of two conductors charged to +V and −V volts as shown in Figure

24.9. Surface charges will accumulate on the surfaces of the conductors. Using Poisson’s

equations, and Green’s function for Poisson’s equation, one can express the potential in

between the two conductors as due to the surface charges density σ(r). It can be expressed

Φ(r) = 1

dS′ σ(r′)

4π|r − r′| (24.2.4)

where S is the union of two surfaces S1 and S2. Since Φ has values of +V and −V on the

two conductors, we require that

Φ(r) = 1

dS′ σ(r′)

4π|r − r′| =

{

+V, r ∈ S1

−V, r ∈ S2

(24.2.5)

In the above, σ(r′), the surface charge density, is the unknown yet to be sought and it is

embedded in an integral. But the right-hand side of the equation is known. Hence, this

equation is also known as an integral equation. The integral equation can be solved by

numerical methods.

Having found σ(r), then it can be integrated to find Q, the total charge on one of the

conductors. Since the voltage difference between the two conductors is known, the capacitance

can be found as C = Q/(2V ).

24.3 Importance of Circuit Theory in IC Design

The clock rate of computer circuits has peaked at about 3 GHz due to the resistive loss, or

the I2R loss. At this frequency, the wavelength is about 10 cm. Since transistors and circuit

components are shrinking due to the compounding effect of Moore’s law, most components,

Circuit Theory Revisited 247

which are of nanometer dimensions, are much smaller than the wavelength. Thus, most of

the physics of electromagnetic signal in a circuit can be captured using circuit theory.

Figure 24.10 shows the schematics and the cross section of a computer chip at different

levels: the transistor level at the bottom-most. The signals are taken out of a transistor by

XY lines at the middle level that are linked to the ball-grid array at the top-most level of

the chip. And then, the signal leaves the chip via a package. Since these nanometer-size

structures are much smaller than the wavelength, they are usually modeled by lumped R,

L, and C elements if retardation effect can be ignored. If retardation effect is needed, it is

usually modeled by a transmission line. This is important at the package level where the

dimensions of the components are larger.

A process of parameter extraction where computer software or field solvers (software that

solve Maxwell’s equations numerically) are used to extract these lumped-element parameters.

Finally, a computer chip is modeled as a network involving a large number of transistors,

diodes, and R, L, and C elements. Subsequently, a very useful commercial software called

SPICE (Simulation Program with Integrated-Circuit Emphasis) [123], which is a computer-

aided software, solves for the voltages and currents in this network.

248 Electromagnetic Field Theory

Figure 24.10: Courtesy of Wikipedia and Intel.

The SPICE software has many capabilities, including modeling of transmission lines for

microwave engineering. Figure 24.11 shows an interface of an RF-SPICE that allows the

modeling of transmission line with a Smith chart interface.

Circuit Theory Revisited 249

Figure 24.11: SPICE is also used to solve RF problems (courtesy of EMAG Technologies

Inc.).

24.3.1 Decoupling Capacitors and Spiral Inductors

Decoupling capacitors are an important part of modern computer chip design. They can

regulate voltage supply on the power delivery network of the chip as they can remove high-

frequency noise and voltage fluctuation from a circuit as shown in Figure 24.12. Figure 24.13

shows a 3D IC computer chip where decoupling capacitors are integrated into its design.

Figure 24.12: A decoupling capacitor is essentially a low-pass filter allowing low-frequency

signal to pass through, while high-frequency signal is short-circuited (courtesy learningabout-

electronics.com).

250 Electromagnetic Field Theory

Figure 24.13: Modern computer chip design is 3D and is like a jungle. There are different

levels in the chip and they are connected by through silicon vias (TSV). IMD stands for inter-

metal dielectrics. One can see different XY lines serving as power and ground lines (courtesy

of Semantic Scholars).

Inductors are also indispensable in IC design, as they can be used as a high frequency

choke. However, designing compact inductor is still a challenge. Spiral inductors are used

because of their planar structure and ease of fabrication.

Figure 24.14: Spiral inductors are difficult to build on a chip, but by using laminal structure,

it can be integrated into the IC fabrication process (courtesy of Quan Yuan, Research Gate).

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