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MT-144
NETWORK ANALYSIS
Mechatronics Engineering
(04)
1
ENERGY STORAGE ELEMENTS
(Chapter 7)
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Introduction
Capacitance
Inductance
Natural Response of RC and RL Circuits
Response to DC and AC Forcing Functions
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ENERGY STORAGE ELEMENTS
Introduction
Some Comments on the “Resistive Circuits”:
Most of the circuits examined so far, are resistive circuits because the
only elements used, besides sources, are resistances.
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The equations governing these circuits are algebraic
equations; including Kirchhoff ‘s laws and Ohm's Law.
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Since resistances can only dissipate energy, we need at least
one independent source to initiate any voltage or current in the
circuit.
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In the absence of independent sources, all voltages and
currents would be zero and the circuit would have no electrical life
of its own.
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ENERGY STORAGE ELEMENTS
Introduction…
It is now time we turn our attention to the two remaining basic elements,
capacitance and inductance.
The first distinguishing feature of these elements is that they exhibit
time-dependent characteristics, namely,
i = C (dv/dt)
… for capacitance
v = L (di/dt)
… for inductance.
and
For this reason, capacitances and inductances are said to be
dynamic elements. By contrast, a resistance is a static element
because its i-v characteristic does not involve time. Time dependence
adds a new dimension to circuit behavior, allowing for a wider variety of
functions as compared to purely resistive circuit
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ENERGY STORAGE ELEMENTS
Introduction…
The second distinguishing feature is that capacitances and inductances
can absorb, store, and then release energy, making it possible for a
circuit to have an electrical life of its own even in the absence of any
sources. For obvious reasons, capacitances and inductances are also
referred to as energy-storage elements.
The formulation of circuit equations for networks containing
capacitances and inductances still relies on the combined use of
Kirchhoff’s laws and the element laws. However, since the
characteristics of these elements depend on time, the resulting
equations are no longer plain algebraic equations; they involve time
derivatives, or integrals, or both. Generally referred to as integrodifferential equations, they are not as straightforward to solve as their
algebraic counterparts. In fact, they can be solved analytically only in a
limited number of cases.
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ENERGY STORAGE ELEMENTS
Introduction…
But, this at times, includes the cases of immense interest to us.
Even when a solution cannot be found analytically, it can be evaluated
numerically using a computer, this being the reason why computer
simulation such as SPICE plays an indispensable role in the analysis
and design of circuits containing energy storage elements. Your previous
or concurrent exposure to differential equations (ODE’s) is likely to be
helpful.
In the present chapter, after introducing the capacitance and the
inductance, we study the natural response of the basic RC and RL
circuits, that is, the response provided by the circuit using the energy
stored in its capacitance or inductance.
This study introduces us to the concept of root location in the s plane, a
powerful concept that shall be explored later at an appropriate time
during your BE Program.
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ENERGY STORAGE ELEMENTS
Introduction…
We then use the integrating factor method to investigate how circuits
containing these elements react to the application of dc signals and ac
signals.
The mathematical level is designed to provide a rigorous understanding
of the various response components, namely, the natural, forced,
transient, and steady-state components.
Even though the responses to dc and ac signals may seem particular.
They provide enough insight into the most relevant aspects of circuits
containing dynamic elements to allow the designer to predict circuit
behavior in most other situations of practical interest.
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ENERGY STORAGE ELEMENTS
7.1 Capacitance:
Capacitance represents the ability of a circuit element to store charge
in response to voltage. Circuit elements that are designed to provide
this specific function are called capacitors or condensers.
As shown in Figure 7.1(a), a capacitor consists of two conductive plates
separated by a thin insulator. Applying a voltage between the plates
causes positive charge to accumulate on the plate at higher potential
and an equal amount of negative charge to accumulate on the plate at
lower potential. The rate at which the accumulated charge varies
with the applied voltage is denoted as C and is called capacitance.
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ENERGY STORAGE ELEMENTS
7.1 Capacitance…
…(7.1)
Its SI unit is the farad (F), named for the English chemist and physicist
Michael Faraday (1791-1867). Clearly, 1 F = 1 C/ V . Figure 7.l(b) shows
the circuit symbol for capacitance, along with the reference polarities for
voltage and current.
Recall from basic electricity that capacitance depends on the insulator
type and the physical dimensions:
…(7.2)
where Ɛ is the permittivity of the insulator, S is the area of the plates,
and d is the distance between them. For vacuum space Ɛ takes on the
value Ɛo = 10-9/(36π) F/m. Other media are described in terms of the
ratio Ɛr = Ɛ / Ɛo , called the relative permittivity
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ENERGY STORAGE ELEMENTS
7.1 Capacitance…
A capacitor storing charge may be likened to a cylindrical tank storing
water. The larger its cross-sectional area, the more water the tank can
hold. Moreover, the lower the height needed to store a given amount of
water, the greater the tank's storage capacity. In general q may be some
arbitrary function of v, indicating that C may itself be a function of v. In
this case the capacitance is said to be nonlinear.
Linear Capacitances
Of particular interest is the case in which q is linearly proportional to v,
for then we must have q = Cv or
… (7.3)
with C independent of v . For obvious reasons, this type of capacitance
is said to be linear. Unless stated otherwise, we shall consider only
capacitances of this type.
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ENERGY STORAGE ELEMENTS
7.1 Capacitance…
Linear Capacitances…
Equation (7.3) allows us to find the accumulated charge in terms of the
applied voltage, or vice versa. For instance, applying 10 V across the
terminals of a 1μF capacitance causes a charge q= Cv= 10-6 x10 =10 μC
(μ Coulombs) to accumulate on the plate at higher potential, and a
charge -q = -10 μC to accumulate on the other plate. Even though the
net charge within the capacitance is always zero, we identify the charge
stored in the capacitance as that on the positive plate. We thus say that
applying 10 V across a I-μF capacitance results in a stored charge of
10 μC.
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ENERGY STORAGE ELEMENTS
7.2 Inductance:
Inductance represents the ability of a circuit element to produce
magnetic flux linkage in response to current. Circuit elements specifically
designed to provide this function are called inductors. As shown in
Figure 7.8(a), an inductor consists of a coil of insulated wire wound
around a core. Sending current down the wire creates a magnetic field
in the core and, hence, a magnetic flux φ.
If the coil has N turns, the quantity λ = Nφ is called the flux linkage and
is expressed in weber-turns. The rate at which λ varies with the applied
current is denoted as L and is called the self-inductance or simply the
inductance of the coil,
where μ is the permeability
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ENERGY STORAGE ELEMENTS
Table 7.1
Comparison of the Basic Elements:
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ENERGY STORAGE ELEMENTS
7.3 NATURAL RESPONSE OF RC AND RL CIRCUITS
The analysis of circuits containing energy-storage elements is still based
on Kirchhoff’s laws and the element laws. However, since these
elements exhibit time-dependent i-v characteristics, the resulting circuit
equations are no longer plain algebraic equations; they involve time
derivatives, or integrals, or both.
The simplest circuits are those consisting of a single energy-storage
element embedded in a linear network of sources and resistances.
However complex this network may be, we can always replace it with its
Thevenin or Norton equivalent to simplify our analysis. After this
replacement, the network reduces to either equivalent of Figure 7.14.
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ENERGY STORAGE ELEMENTS
7.3 NATURAL RESPONSE OF RC AND RL CIRCUITS…
After this replacement, the network reduces to either equivalent of
Figure 7.14, where for reasons of duality we have chosen to use the
Thevenin equivalent in the capacitive case and the Norton equivalent in
the inductive case. In either case we wish to find the voltage and current
developed by the energy-storage element in terms of the source, the
resistance, and the element itself. The manner in which this voltage or
current varies with time is referred to as the time response.
Back to
slide 19
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ENERGY STORAGE ELEMENTS
7.3 NATURAL RESPONSE OF RC AND RL CIRCUITS…
First-Order Differential Equations:
In the circuit of Figure 7.14(a) we have, by the capacitance law,
i = C dv/dt. By KVL, we also have vs = Ri + v. Eliminating i we obtain :
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ENERGY STORAGE ELEMENTS
7.3 NATURAL RESPONSE OF RC AND RL CIRCUITS…
First-Order Differential Equations:
In the circuit of Figure 7.14(b) we have, by the inductance law,
v = Ldi/dt. By KCL, we also have is = v/R + i . Eliminating v yields:
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ENERGY STORAGE ELEMENTS
7.3 NATURAL RESPONSE OF RC AND RL CIRCUITS…
First-Order Differential Equations:
, in the inductive case.
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ENERGY STORAGE ELEMENTS
7.3 NATURAL RESPONSE OF RC AND RL CIRCUITS…
First-Order Differential Equations:
Both (RC) and (L/ R) have the dimensions of time. Consequently, Ƭ is
called the time constant.
Since Equation (7.25) contains both the unknown variable and its
derivative, it is said to be a differential equation.
Moreover, it is said to be of the first order because this is the order of the
highest derivative present. Consequently, the circuits of Figure 7.14,
each containing just one energy-storage element, are said to be firstorder circuits.
Figure 7.14
A circuit containing multiple capacitances (or. inductances) is still a firstorder circuit if its topology allows for the capacitances (or inductances) to
be reduced to a single equivalent capacitance (or inductance) through
repeated usage of the parallel and series formulas.
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ENERGY STORAGE ELEMENTS
7.3 NATURAL RESPONSE OF RC AND RL CIRCUITS…
First-Order Differential Equations:
A circuit containing multiple capacitances (or. inductances) is still a firstorder circuit if its topology allows for the capacitances (or inductances) to
be reduced to a single equivalent capacitance (or inductance) through
repeated usage of the parallel and series formulas.
Despite its apparent simplicity, Equation (7.25) cannot be solved by
purely algebraic manipulations. For instance, rewriting it as:
y = x - Ƭ ( dy / dt ) brings us no closer to the solution because the righthand side contains the derivative of the unknown as part of the solution
itself. Before developing a general solution, in the next section, we will
study the special but interesting case x(t) = 0.
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ENERGY STORAGE ELEMENTS
7.3 NATURAL RESPONSE OF RC AND RL CIRCUITS…
First-Order Differential Equations:
The Source-Free or Natural Response
Letting x(t) = 0 in
… (7.25) yields:
… (7.28)
Mathematically, this equation (7.28) is referred to as the homogeneous
differential equation, and its solution y(t) is referred to as the
homogeneous solution. Since letting x(t) = 0 is equivalent to letting
vs = 0 or is = 0 in the original circuits, this particular solution is
physically referred to as the source-free response. Lacking any forcing
source, the response of the circuit is driven solely by the initial energy of
its energy-storage element.
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ENERGY STORAGE ELEMENTS
7.3 NATURAL RESPONSE OF RC AND RL CIRCUITS…
First-Order Differential Equations:
The Source-Free or Natural Response…
Letting x(t) = 0 in
… (7.25) yields:
… (7.28)
Rewriting Equation (7.28) as y(t) = - Ƭ[dy(t)/ dt] , we note that aside from
the constant -Ƭ, the unknown and its derivative must be the same. You
may recall that of all functions encountered in calculus, only the
exponential function enjoys the unique property that its derivative is still
exponential. We thus assume a solution of the type:
where e= 2.718 is the base of natural logarithms, and we seek suitable
expressions for A and s that will make this solution satisfy Equation
(7.28).
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ENERGY STORAGE ELEMENTS
7.3 NATURAL RESPONSE OF RC AND RL CIRCUITS…
First-Order Differential Equations:
The Source-Free or Natural Response...
… (7.28)
To find an expression for s, we substitute Equation (7.29) into Equation
(7.28) and obtain: Ƭs Aest + Aest = 0, or
(Ƭs + 1)Aest = 0 : {one of the factors on LHS has to be zero}
Since we are seeking a solution Aest ≠ 0, the above equality can hold
only if the expression within parentheses vanishes, so
(Ƭs + 1) = 0 … (7.30), it is called the characteristic Equation)
or s = -1/Ƭ
… (7.31) , is root of 7.30, the characteristics equation.
Since it (s) has the dimensions of the reciprocal of time, or frequency.
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ENERGY STORAGE ELEMENTS
7.3 NATURAL RESPONSE OF RC AND RL CIRCUITS…
First-Order Differential Equations:
The Source-Free or Natural Response…
(Ƭs + 1) = 0 … (7.30), it is called the characteristic Equation)
or s = -1/Ƭ
… (7.31) , is root of 7.30, the characteristics equation.
Since it has the dimensions of the reciprocal of time, or frequency, the
root is variously referred to as the natural frequency, the characteristic
frequency, or the critical frequency of the circuit.
This frequency is expressed in nepers / s (Np l s). The neper (Np) is a
dimensionless unit named for the Scottish mathematician John Napier
(1550--1617) and used to designate the unit of the exponent of est, which
is a pure number.
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ENERGY STORAGE ELEMENTS
7.3 NATURAL RESPONSE OF RC AND RL CIRCUITS…
First -Order Differential Equations: The Source-Free or Natural
Response…
Next, we wish to find an expression for A. This is done on the basis of
the initial condition y(0) in the circuit, that is, on the basis of the initial
voltage v(0) across the capacitance or the initial current i(0) through the
inductance.
These conditions, in turn, are related to the initial stored energy, which is
w(0)=(1/2)Cv2(0) for the capacitance, and w(0)=(1/2)Li2(0) for the
inductance.
Thus, letting t=0 in Equation (7.29) yields y(0)=Ae0, or A=Y(0) … (7.32)
Recall that,
s = -1/Ƭ
… (7.31)
Substituting Equations (7.3 1) and (7.32) into (7.29) finally yields:
y (t) = y(0)e-t/Ƭ ... (7.33)
As shown in Figure 7.15, y(t) is an exponentially decaying function from
the initial value y(0) to the final value y(∞) = 0
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ENERGY STORAGE ELEMENTS
7.3 NATURAL RESPONSE OF RC AND RL CIRCUITS…
First -Order Differential Equations: The Source-Free or Natural
Response…
y (t) = y(0)e-t/Ƭ ... (7.33)
As shown in Figure 7.15, y(t) is an exponentially decaying function from
the initial value y(0) to the final value y(∞) = 0. Since the decay depends
only on y(0) and s, which are peculiar characteristics of the circuit
irrespective of any particular forcing function, this solution is also
called the natural response. Thus, homogeneous solution, source-free
response, and natural response are different terms for the same function
of Equation (7.33).
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ENERGY STORAGE ELEMENTS
7.3 NATURAL RESPONSE OF RC AND RL CIRCUITS…
First -Order Differential Equations: The Source-Free or Natural
Response…
y (t) = y(0)e-t/Ƭ ... (7.33)
How a circuit manages to produce a nonzero response with a zero
forcing function is an intriguing question, but, as stated, this behavior
stems from the ability of capacitors and inductors to store energy. It is
precisely this energy that allows the circuit to sustain nonzero voltages
and currents even in the absence of any forcing source. These voltages
and currents will persist until all of the initial energy has been used up by
the resistances in the circuit.
Contrast this with a purely resistive network where, in the absence of
any driving source, each voltage and current in the circuit would at all
times be zero.
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ENERGY STORAGE ELEMENTS
7.3 NATURAL RESPONSE OF RC AND RL CIRCUITS…
First -Order Differential Equations: The Source-Free or Natural
Response…
y (t) = y(0)e-t/Ƭ ... (7.33)
The Time Constant Ƭ
Mathematically, the time constant Ƭ serves the purpose of making the
argument (t/ Ƭ) of the exponential function a dimensionless number.
Physically, it provides a measure of how rapidly the exponential decay
takes place. The significance of Ƭ can be visualized in two different
ways, as follows: Evaluating Equation (7.33) at t = Ƭ yields y(t) = y(0)e-1
= 0.37y(0), indicating that after Ƭ seconds the natural response has
decayed to 37% of its initial value. Equivalently, we can say that after Ƭ
seconds the response has accomplished 100- 37= 63% of its entire
decay. Thus, one way of interpreting the time constant is:
•
Ƭ represents the amount of time it takes for the natural
response to decay to (1/e) or to 37% of its initial value.
An alternative interpretation is ….
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ENERGY STORAGE ELEMENTS
7.3 NATURAL RESPONSE OF RC AND RL CIRCUITS…
First -Order Differential Equations: The Source-Free or Natural
Response
y (t) = y(0)e-t/Ƭ ... (7.33)
The Time Constant Ƭ
An alternative interpretation is found by considering the initial slope of
the response curve. On the one hand, this slope can be found
analytically by evaluating the derivative of Equation (7.33) at t = 0, that
is, dy(0)/dt = (-1/ Ƭ) y(0)e0 = -y(0)/Ƭ.
On the other hand, it can be found geometrically as the ratio of the
y-axis to the t-axis intercepts of the tangent to the curve at the origin.
Since the y-axis intercept occurs at y(0), it follows that the t-axis
intercept must occur at t = Ƭ , in order to make the initial slope equal to
the calculated value - y(0)/ Ƭ . Thus, an alternate interpretation for the
time constant is:
•
Ƭ represents the instant at which the tangent to the natural
response at the origin intercepts the t-axis. Either of these
viewpoints can be exploited to find Ƭ experimentally by observing
the natural response with an oscilloscope.
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ENERGY STORAGE ELEMENTS
7.3 NATURAL RESPONSE OF RC AND RL CIRCUITS…
First -Order Differential Equations: The Source-Free or Natural
Response
y (t) = y(0)e-t/Ƭ ... (7.33)
The Time Constant Ƭ
We observe that the
larger the value of Ƭ , the
slower the rate of decay
because it will take
longer for the response
to decay to 37% or,
equivalently, the tangent 0.37 y(0)
at the origin will intercept
the t-axis at a later
Ƭ1 ,
Ƭ2 ,
Ƭ3 etc
instant. Conversely, a
smaller the value of Ƭ ,
greater (rapid) the rate of
decay. This is illustrated
in Figure 7.16.
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ENERGY STORAGE ELEMENTS
7.3 NATURAL RESPONSE OF RC AND RL CIRCUITS…
The Source-Free or Natural Response:
y (t) = y(0)e-t/Ƭ ... (7.33)
Decay Times (tƐ)
It is often of interest to estimate the amount of time tƐ it takes for the
natural response to decay to a given fraction Ɛ of its initial value. By
Equation (7.33), tƐ must be such that Ɛy(0)= y(0)exp(-tƐ/Ƭ),
or Ɛ= exp(-tƐ/ Ƭ). Solving for tƐ yields: tƐ = - Ƭ lnƐ …(7.34)
Home Work: Solve Exercises 7.10 & 7.11 on page 311, of Text Book
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ENERGY STORAGE ELEMENTS
7.3 NATURAL RESPONSE OF RC AND RL CIRCUITS…
The Source-Free or Natural Response
y (t) = y(0)e-t/Ƭ
... (7.33)
Though in theory the response reaches zero only in the limit t  ∞, in
practice it is customary to regard the decay as essentially complete after
about five time constants ( 5 Ƭ ), since by this time the response has
already dropped below 1% of its initial value, which is negligible in most
cases of practical interest.
The s Plane
It is good practice to visualize the root of a characteristic equation
(Ƭs + 1) = 0
… (7.30)
as a point in a plane called the s plane. Though we shall have more to
say about this plane later (in some other courses or see Chapter 9). For
the time being we ignore the vertical axis and use points of the
horizontal axis to visualize our roots.
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ENERGY STORAGE ELEMENTS
7.3 NATURAL RESPONSE OF RC AND RL CIRCUITS…
The Source-Free or Natural Response
y (t) = y(0)e-t/Ƭ
The s Plane…
Negative roots lie on the left
portion of the vertical axis,
positive roots on the right portion.
Moreover, this axis is calibrated
in Npls. Since the root s= -1/Ƭ is
negative, it lies on the left portion
of the axis. Moreover, the farther
away the root from the origin, the
more rapid the exponential
decay. Conversely, the closer the
root to the origin, the slower the
decay.
This correspondence is depicted
in Figure 7.17(a) and (b). see 
... (7.33)
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ENERGY STORAGE ELEMENTS
7.3 NATURAL RESPONSE OF RC AND RL CIRCUITS…
The Source-Free or Natural Response
y (t) = y(0)e-t/Ƭ ... (7.33)
The s Plane…
It is interesting to note that in the limit of a root right at the origin, or
1/Ƭ  0, we have Ƭ  ∞, indicating an infinitely slow decay, as depicted
in Figure 7.17(c).
In fact, Equation (7.33) predicts y (t)= y(0)e-t/∞ = y(0), that is, a constant
natural response. By Equations Ƭ= RC …(7.26) and Ƭ= L/R …(7.27),
the condition t= ∞ is achieved by letting R= ∞ in the capacitive case, and
R= 0 in the inductive case.
As we know, when open-circuited, an ideal capacitance will retain its
initial voltage indefinitely, so v(t)= v(0) for any t > 0;
Also, when short-circuited, an ideal inductance will sustain its initial
current indefinitely, so i(t)= i(0) for any t > 0. As we know, the function
associated with this type of response is the memory function.
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ENERGY STORAGE ELEMENTS
7.3 NATURAL RESPONSE OF RC AND RL CIRCUITS…The s Plane…
.
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