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CAPACITORS AND INDUCTORS

Module by: Dinh Sy Hien

Summary: We shall introduce two important passive linear circuit elements: the capacitor and the inductor. With the introduction of capacitors and inductors, we will be able analyze more important and practical circuits. We begin by introducing capacitors and describing how to combine them in series or in parallel. Later, we do the same for inductors. As typical applications, we explore how capacitors are combined with op amp to form integrators, differentiators.

INTRODUCTION

So far we have limited our study to resistive circuits. In this chapter, we shall introduce two new and important passive linear circuit elements: the capacitor and the inductor. Unlike resistors, which dissipate energy, capacitors and inductors do not dissipate but store energy, which can be retrieved at a later time. For this reason, capacitors and inductors are called storage elements.
The application of resistive circuits is quite limited. With the introduction of capacitors and inductors in this chapter, we will be able analyze more important and practical circuits. Be assured that the circuit analysis techniques covered in chapters 3 and 4 are equally applicable to circuits with capacitors and inductors.
We begin by introducing capacitors and describing how to combine them in series or in parallel. Later, we do the same for inductors. As typical applications, we explore how capacitors are combined with op amp to form integrators, differentiators.

CAPACITORS

A capacitor is a passive element designed to store energy in its electric field. Besides resistors, capacitors are the most common components. Capacitors are used extensively in electronics, communications, computers, and power systems. For example, they are used in the tuning circuits of radio receivers and as dynamic memory elements in computer systems.
A capacitor is typically constructed as depicted in Figure 1.
Figure 1: A typical capacitor.
A capacitor consists of two conducting plates separated by an insulator (or dielectric).
In many practical applications, the plates may be aluminum foil while the dielectric may be air, ceramic, paper, or mica.
When a voltage source v is connected to the capacitor, as in Figure 2, the source deposits a positive charge q on one plate and a negative charge – q on the other. The capacitor is said to store the electric charge. The amount of charge stored, represented by q, is directly proportional to the applied voltage v so that
q = Cv q = Cv size 12{q= ital "Cv"} {} (1)
Where C, the constant of proportionality, is known as the capacitance of the capacitor. The unit of capacitance is the farad (F) in honor of the English physicist Michael Faraday (1791-1867). From Equation 1, we may derive the following definition.
Capacitance is the ratio of the charge on one plate of a capacitor to the voltage difference between the two plates, measured in farads (F).
Note that from Equation 1 that 1 farad = 1 coulomb/volt.
Figure 2: A capacitor with applied voltage V.
Although the capacitance C of capacitor is the ratio of the charge q per plate to the applied voltage v, it does not depend on q or v. it depends on the physical dimensions of the capacitor. For example, for the parallel plate capacitor shown in Figure 1, the capacitance is given by
C = εA d C = εA d size 12{C= { {εA} over {d} } } {} (2)
Where A is the surface area of each plate, d is the distance between the plates, and εε size 12{ε} {} is the permittivity of the dielectric material between the plates. Although Equation 2 applies to only parallel-plate capacitors, we may infer from it that, in general, three factors determine the value of the capacitance:
  1. The surface area of the plate – the larger the area, the greater the capacitance.
  2. The spacing between the plates – the smaller the spacing, the greater the capacitance.
  3. The permittivity of the material – the higher the permittivity, the greater the capacitance.
Capacitors are commercially available in different values and types. Typically, capacitors have values in the picofarad (pF) to microfarad ( μμ size 12{μ} {}F) range. They are described by the dielectric material they are made of and by whether they are of fixed or variable type. Figure 3 shows the circuit symbols for fixed and variable capacitors. Note that according to the passive sign convention, current is considered to flow into the positive terminal of the capacitor when the capacitor is being charged, and out of the positive terminal when the capacitor is discharging.
Figure 3: Circuit symbols for capacitors: a) fixed capacitor, b) variable capacitor.
Figure 4 shows common types of fixed-value capacitors. Polyester capacitors are light in weight, stable, and their change with temperature is predictable. Instead of polyester, other dielectric materials such as mica and polystyrene may be used. Film capacitors are rolled and housed in metal or plastic films. Electrolytic capacitors produce very high capacitance.
Figure 4: Fixed capacitors: a) polyester capacitor, b) ceramic capacitor, c) electrolytic capacitor.
Figure 5 shows the most common types of variable capacitors. The capacitance of a trimmer (or padder) capacitor or a glass piston capacitor is varied by turning the screw. The trimmer capacitor is often placed in parallel with another capacitor so that the equivalent capacitance can be varied by turning slightly. The capacitance of the variable air capacitor (meshed plate) is varied by turning the shaft. Variable capacitors are used in radio receivers allowing one to tune to various stations. In addition, capacitors are used to block dc, pass ac, shift phase, store energy, start motors, and suppress noise.
Figure 5: Variable capacitors: a) trimmer capacitor, b) film trim capacitor.
To obtain the current-voltage relationship of the capacitor, we take the derivative of both sides of Equation 1. Since
i = dq dt i = dq dt size 12{i= { { ital "dq"} over { ital "dt"} } } {} (3)
Differentiating both sides of Equation 1 gives
i = C dv dt i = C dv dt size 12{i=C { { ital "dv"} over { ital "dt"} } } {} (4)
This is current-voltage relationship for a capacitor, assuming the passive sign convention. The relationship is illustrated in Figure 6 for a capacitor whose capacitance is independent of voltage. Capacitors that satisfy Equation 4 are said to be linear. For a nonlinear capacitor, the plot of the current-voltage relationship is not a straight line. We will assume linear capacitors in this book.
Figure 6: Current voltage relationship of a capacitor.
The voltage-current relation of the capacitor can be obtained by integrating both sides of Equation 4. We get
v = 1 C ∫ − ∞ t idt v = 1 C ∫ − ∞ t idt size 12{v= { {1} over {C} } Int cSub { size 8{ - infinity } } cSup { size 8{t} } { ital "idt"} } {} (5)
or
v = 1 C ∫ t 0 t idt + v ( t 0 ) v = 1 C ∫ t 0 t idt + v ( t 0 ) size 12{v= { {1} over {C} } Int cSub { size 8{t rSub { size 6{0} } } } cSup {t} { ital "idt"+v \( t rSub { size 8{0} } \) } } {} (6)
where v(t0)=q(t0)/Cv(t0)=q(t0)/C size 12{v \( t rSub { size 8{0} } \) = {q \( t rSub { size 8{0} } \) } slash {C} } {} is the voltage across the capacitor at time t0t0 size 12{t rSub { size 8{0} } } {}. Equation 6 shows that capacitor voltage depends on the past history of the capacitor current. Hence, the capacitor has memory – a properly that is often exploited.
The instantaneous power delivered to the capacitor is
p = vi = Cv dv dt p = vi = Cv dv dt size 12{p= ital "vi"= ital "Cv" { { ital "dv"} over { ital "dt"} } } {} (7)
The energy stored in the capacitor is therefore
w = ∫ − ∞ t pdt = C ∫ − ∞ t v dv dt dt = C ∫ − ∞ t vdv = 1 2 Cv 2 ∣ − ∞ t w = ∫ − ∞ t pdt = C ∫ − ∞ t v dv dt dt = C ∫ − ∞ t vdv = 1 2 Cv 2 ∣ − ∞ t size 12{w= Int cSub { size 8{ - infinity } } cSup { size 8{t} } { ital "pdt"} =C Int cSub { size 8{ - infinity } } cSup { size 8{t} } {v { { ital "dv"} over { ital "dt"} } ital "dt"=C Int cSub { size 8{ - infinity } } cSup { size 8{t} } { ital "vdv"= { {1} over {2} } } } ital "Cv" rSup { size 8{2} } \rline rSub { size 8{ - infinity } } rSup { size 8{t} } } {} (8)
We note that v(-∞) = 0, because the capacitor was uncharged at t=−∞t=−∞ size 12{t= - infinity } {}. Thus,
w = 1 2 Cv 2 w = 1 2 Cv 2 size 12{w= { {1} over {2} } ital "Cv" rSup { size 8{2} } } {} (9)
Using Equation 1, we may rewrite Equation 9 as
w = q 2 2C w = q 2 2C size 12{w= { {q rSup { size 8{2} } } over {2C} } } {} (10)
Equation 6 or Equation 10 represents the energy stored in the electric field that exists between the plates of the capacitor. This energy can be retrieved, since an ideal capacitor cannot dissipate energy. In fact, the word capacitor is derived from this element’s capacity to store energy in the electric field.
We should note the following important properties of a capacitor:
1. Note from Equation 4 that when the voltage across a capacitor is not changing with time (i.e., dc voltage), the current through the capacitor is zero. Thus,
A capacitor is an open circuit to dc.
However, if a battery (dc voltage) is connected across a capacitor, the capacitor changes.
2. The voltage on the capacitor must be continuous.
The voltage on a capacitor cannot charge abruptly.
The capacitor resists an abrupt change in the voltage across it. According to Equation 4, a discontinuous change in voltage requires an infinite current, which is physically impossible. For example, the voltage across a capacitor may take the form shown in Figure 7(a), whereas it is not physically possible for the capacitor voltage to take the form shown in Figure 7(b) because of the abrupt changes. Conversely, the current through a capacitor can change instantaneously.
  1. The ideal capacitor does not dissipate energy. It takes power from the circuit when storing energy in its field and returns previously stored energy when delivering power to the circuit.
  2. A real, nonideal capacitor has a parallel-model leakage resistance, as shown in Figure 8. The leakage resistance may be as high as 100 M ΩΩ size 12{ %OMEGA } {} and can be neglected for most practical applications. For this reason, we will assume ideal capacitors in this book.
Figure 7: Voltage across a capacitor: a) allowed, b) not allowable, an abrupt change is not possible.
Figure 8: Circuit model of a nonideal capacitor.

SERIES AND PARALELL CAPACITORS

We know from resistive circuit that series-parallel combination is a powerful tool for reducing circuits. This technique can be extended to series-parallel connections of capacitors, which are sometimes encountered. We desire to replace these capacitors by a single equivalent capacitor CeqCeq size 12{C rSub { size 8{ ital "eq"} } } {}.
In order to obtain the equivalent capacitor CeqCeq size 12{C rSub { size 8{ ital "eq"} } } {} of N capacitors in parallel, consider the circuit in Figure 9(a). The equivalent circuit is in Figure 9(b). Note that the capacitors have the same voltage v across them. Applying KCL to Figure 9(a),
i = i 1 + i 2 + i 3 + . . . + i N i = i 1 + i 2 + i 3 + . . . + i N size 12{i=i rSub { size 8{1} } +i rSub { size 8{2} } +i rSub { size 8{3} } + "." "." "." +i rSub { size 8{N} } } {} (11)
But ik=Ckdv/dtik=Ckdv/dt size 12{i rSub { size 8{k} } =C rSub { size 8{k} } { ital "dv"} slash { ital "dt"} } {}. Hence,
i = C 1 dv dt + C 2 dv dt + C 3 dv dt + ... + C N dv dt ∑ k = 1 N C k dv dt = C eq dv dt i = C 1 dv dt + C 2 dv dt + C 3 dv dt + ... + C N dv dt ∑ k = 1 N C k dv dt = C eq dv dt alignl { stack { size 12{i=C rSub { size 8{1} } { { ital "dv"} over { ital "dt"} } +C rSub { size 8{2} } { { ital "dv"} over { ital "dt"} } +C rSub { size 8{3} } { { ital "dv"} over { ital "dt"} } + dotsaxis +C rSub { size 8{N} } { { ital "dv"} over { ital "dt"} } } {} # = left ( Sum cSub { size 8{k=1} } cSup { size 8{N} } {C rSub { size 8{k} } } right ) { { ital "dv"} over { ital "dt"} } =C rSub { size 8{ ital "eq"} } { { ital "dv"} over { ital "dt"} } {} } } {} (12)
Where
C eq = C 1 + C 2 + C 3 + . . . + C N C eq = C 1 + C 2 + C 3 + . . . + C N size 12{C rSub { size 8{ ital "eq"} } =C rSub { size 8{1} } +C rSub { size 8{2} } +C rSub { size 8{3} } + "." "." "." +C rSub { size 8{N} } } {} (13)
The equivalent capacitance of N parallel-connected capacitors is the sum of the individual capacitors.
We observe that capacitors in parallel combine in the same manner as resistors in series.
Figure 9: a) Parallel connected N capacitor, b) equivalent circuit for the parallel capacitors.
We now obtain CeqCeq size 12{C rSub { size 8{ ital "eq"} } } {} of N capacitors connected in series by comparing the circuit in Figure 10(a) with the equivalent circuit in Figure 10(b). Note that the same current i flows (and consequently the same charge) through the capacitors. Applying KVL to the loop in Figure 10(a),
v = v 1 + v 2 + v 3 + . . . + v N v = v 1 + v 2 + v 3 + . . . + v N size 12{v=v rSub { size 8{1} } +v rSub { size 8{2} } +v rSub { size 8{3} } + "." "." "." +v rSub { size 8{N} } } {} (14)
But vk=1Ck∫t0titdt+vkt0vk=1Ck∫t0titdt+vkt0 size 12{v rSub { size 8{k} } = { {1} over {C rSub { size 8{k} } } } Int cSub { size 8{t rSub { size 6{0} } } } cSup {t} {i left (t right ) ital "dt"} size 12{+v rSub {k} left ( size 12{t rSub {0} } right )}} {}, therefore,
v = 1 C 1 ∫ t 0 t i t dt + v 1 t 0 + 1 C 2 ∫ t 0 t i t dt + v 2 t 0 + ... + 1 C N ∫ t 0 t i t dt + v N t 0 = 1 C 1 + 1 C 2 + ... + 1 C N ∫ t 0 t i t dt + v 1 t 0 + v 2 t 0 + ... + v N t 0 = 1 C eq ∫ t 0 t i t dt + v t 0 v = 1 C 1 ∫ t 0 t i t dt + v 1 t 0 + 1 C 2 ∫ t 0 t i t dt + v 2 t 0 + ... + 1 C N ∫ t 0 t i t dt + v N t 0 = 1 C 1 + 1 C 2 + ... + 1 C N ∫ t 0 t i t dt + v 1 t 0 + v 2 t 0 + ... + v N t 0 = 1 C eq ∫ t 0 t i t dt + v t 0 alignl { stack { size 12{v= { {1} over {C rSub { size 8{1} } } } Int cSub { size 8{t rSub { size 6{0} } } } cSup {t} {i left (t right ) ital "dt"} size 12{+v"" lSub {1} left ( size 12{t rSub {0} } right )} size 12{+ { {1} over {C"" lSub {2} } } Int cSub {t rSub { size 6{0} } } cSup {t} { size 12{i left (t right ) ital "dt"} } } size 12{+v rSub {2} left ( size 12{t rSub {0} } right )} size 12{+ dotsaxis + { {1} over {C rSub {N} } } Int cSub {t"" lSub { size 6{0} } } cSup {t} { size 12{i left (t right ) ital "dt"+v rSub {N} left ( size 12{t rSub {0} } right )} } }} {} # size 12{ {}= left ( { {1} over {C rSub { size 8{1} } } } + { {1} over {C rSub { size 8{2} } } } + dotsaxis + { {1} over {C"" lSub { size 8{N} } } } right ) Int cSub { size 8{t"" lSub { size 6{0} } } } cSup {t} {i left (t right ) ital "dt"} size 12{+v"" lSub {1} left ( size 12{t rSub {0} } right )} size 12{+v rSub {2} left ( size 12{t rSub {0} } right )} size 12{+ dotsaxis +v rSub {N} left ( size 12{t rSub {0} } right )}} {} # size 12{ {}= { {1} over {C rSub { size 8{ ital "eq"} } } } Int cSub { size 8{t rSub { size 6{0} } } } cSup {t} {i left (t right ) ital "dt"} size 12{+v left (t"" lSub {0} right )}} {} } } {} (15)
Where
1 C eq = 1 C 1 + 1 C 2 + 1 C 3 + ... + 1 C N 1 C eq = 1 C 1 + 1 C 2 + 1 C 3 + ... + 1 C N size 12{ { {1} over {C rSub { size 8{ ital "eq"} } } } = { {1} over {C rSub { size 8{1} } } } + { {1} over {C rSub { size 8{2} } } } + { {1} over {C rSub { size 8{3} } } } + dotsaxis + { {1} over {C rSub { size 8{N} } } } } {} (16)
The initial voltage v(t0)v(t0) size 12{v \( t rSub { size 8{0} } \) } {} across CeqCeq size 12{C rSub { size 8{ ital "eq"} } } {} is required by KVL to be the sum of the capacitor voltages at t0t0 size 12{t rSub { size 8{0} } } {}. Or according to Equation 15,
v ( t 0 ) = v 1 ( t 0 ) + v 2 ( t 0 ) + . . . + v N ( t 0 ) v ( t 0 ) = v 1 ( t 0 ) + v 2 ( t 0 ) + . . . + v N ( t 0 ) size 12{v \( t rSub { size 8{0} } \) =v rSub { size 8{1} } \( t rSub { size 8{0} } \) +v rSub { size 8{2} } \( t rSub { size 8{0} } \) + "." "." "." +v rSub { size 8{N} } \( t rSub { size 8{0} } \) } {}
Thus, according to Equation 16,
The equivalent capacitor of series connected capacitors is the reciprocal of the sum of the reciprocals of the individual capacitances.
Figure 10: a) Series connected N capacitors, b) equivalent circuit for the series capacitor.
Note that capacitors in series combine in the same manner as resistors in parallel. For N = 2 (i.e., two capacitors in series), Equation 16 becomes
C eq = C 1 C 2