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LINEAR CIRCUIT
ANALYSIS
EE-111
ENGR. IMRAN AZIZ
Chapter 7: Energy Storage Elements
• Capacitance
• Inductance
• Natural Response of RC and RL Circuits
• Response to DC and AC Forcing Functions
Introduction:
• Besides sources, we’ve so far studied circuits containing only
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resistors.
Since resistor can only dissipate energy, we need an
independent source in the circuit to initiate any v or i in circuit.
Now we’ll turn our attention towards two remaining elements;
capacitance and inductance.
These elements exhibit time dependent characteristics; so
called dynamic elements, in contrast resistor is a static element.
L and C can absorb, store and release energy, so circuit may
have electrical life even in the absence of any source. These
devices are also called energy storage elements.
Circuits containing L and C shall also be analyzed with the help
of Kirchhoff’s Laws and Element Laws.
7.1 Capacitance:
• Ability of an element to store charge in response to voltage.
• Elements designed to provide this function are called capacitors.
• Two plates separated by thin insulator
• When voltage is applied, positive charge
accumulates at one plate while negative
on other one.
• The rate at which accumulated charge
varies w.r.t. applied voltage, is called C.
• 1F = 1C/V
• As q is a function of v, so, C may also be a function of v. In such
a case when C depends upon v, capacitance is said to be nonlinear.
• Reverse biased PN junction of a diode acts as a non-linear
capacitor.
• C depends upon insulator type and physical dimensions
• Where ᵋ is the permittivity of insulator, S is area of plates and d
is distance between them.
• Linear Capacitance:
• When q is linearly proportional to v, we’ll have,
OR
• With C independent of v.
• This type of capacitance is the LINEAR one. We shall consider only
capacitances of this type, unless stated otherwise.
• Above equation allows us to find accumulated charge on capacitor when
specific voltage is applied. E.g. 10V is applied across 1uF capacitor, then
charge stored on capacitor would be 10uC.
• The i-v Characteristics:
• When voltage across a capacitor is increased by dv, this causes an
increase of charge on capacitor by dq = Cdv.
• As we know that i = dq / dt. So,
• Capacitance is performing operation of voltage differentiation.
• More rapid is the change in voltage, more is the current.
• If current through a capacitor is zero, voltage must be constant or zero.
• Capacitors are used in DRAM of computer system to store a specific
voltage level.
• Uniform Charge/Discharge:
• If dv/dt is constant; i.e rate of change in voltage is constant; i.e capacitor is
charged / discharged at constant rate, then current i is constant through
capacitor.
• When rate of change in voltage is constant w.r.t. time, derivative can be
replaced with finite differences
• Engineers often remember this as
• This relation holds only when I is kept constant.
• The v-i Characteristics:
• Equation already discussed can also be rearranged to find a relation for
voltage across capacitor.
• Where v(0) is the constant of integration and is the voltage across
capacitor at t = 0. v(0) is related to initially stored charge in capacitor q(0)
• Capacitance is said to perform the operation of current integration.
• When current is constant; i(t) = I, and taking v(0) = 0;
• Which shows that constant current through capacitance yields linear
voltage or voltage ramp. The rate of ramp is
• Generally, when a capacitance is applied with symmetric
square wave current, triangular voltage waveform is developed,
which has the same frequency as the applied waveform.
• DC Component for a given freq, C and current depends upon
initial conditions v(0).
• The triangular wave of voltage for given example has a dc
component of v = 2.5V
• DC Component would have been zero if v(0) = -2.5 V.
• Capacitive Energy:
• The process of charging a capacitor involves expenditure of energy.
• As energy is found by integrating power. In case of Capacitance, p = v . i =
v . Cdv/dt. So,
• Integration yields
• For instance, energy stored in 1uF capacitor that has been charged to 10V,
is 50uJ.
• Unlike resistances, which can only absorb energy to dissipate it as heat,
capacitances can absorb energy from a circuit, store it, and then return it at
a latter time.
• Capacitances are also said to be nondissipative elements.
• Capacitances in Parallel:
• When two capacitances are in parallel, they share the same voltage, v,
however, current gets distributed between them.
• By applying KCL,
• Where
• Capacitances in Series:
• When two capacitances are in series, they carry the same current i.
• Their equivalent capacitance can be calculated as
• Practical Capacitors:
• Practical capacitors satisfy C = q / v as long as q and v are confined within
certain limits.
• When its connection are broken, practical capacitor wouldn’t retain the
charge indefinitely, but will gradually discharge. This is called capacitor
leakage.
• Capacitor leakage can be modeled as leakage-free capacitance in parallel
with parasitic resistance.
• To cope with leakage in DRAMs, the capacitors are periodically recharged
to their intended voltage to prevent loss of info. This operation is called
memory refreshing.
• One common fabrication technique is to use two thin sheets of metal foils
separated with an insulator.
• Common available capacitances range from microfarads to millifarads
• Capacitances arise naturally whenever current carrying conductors come in
close proximity. This unintended capacitance is called stray or parasitic
capacitance.
7.2 Inductance:
• Passing the current in wire, creates magnetic flux Ф in the core
• If there are N no of turns in coil
• Is called magnetic flux linkage.
• Ability of an element to produce magnetic flux linkage in
response to current is called inductance.
• Elements designed to provide this function are called Inductors.
• Coil of insulated wire wound around a core
• Rate at which λ varies w.r.t current, is
called Inductance L.
• L depends upon following physical dimensions
• Where ‘u’ is the permeability of core material, S is cross-
sectional area of core and ‘l’ is the length of flux path through
the core.
• The v-i Characteristics:
• When current i through coil is increased by di, flux linkage through core
also increases by N.dФ
• As
L=λ/i
• Now, Faraday’s Law describes that this change in flux linkage in turn
induces voltage
• Comparing above equations,
• Inductance is performing operation of current differentiation.
• More rapid is the change in current, more is the voltage across inductor.
• When constant current is provided to L, no voltage exists across inductor.
• The i-v Characteristics:
• Equation already discussed can also be rearranged to find a relation for
current through L.
• Where i(0) is the current though inductor at t = 0.
• Inductance is said to perform the operation of voltage integration.
• When voltage is constant; v(t) = V, and taking i(0) = 0;
• Which shows that constant voltage across inductance yields linear current
or current ramp. The rate of ramp is di / dt = V/L.
• The Principle of Duality:
• Behavior of C and L is parallel provided we interchange C with L & i with v.
• The behavior is called Duality.
• For C:
i = C dv/dt
For L: v = L di/dt.
• In other words, L does to i what C does to v and vice versa.
• L differentiates i, while C differentiates v. Likewise, L integrates v while C
integrates i.
• C responds to triangular voltage waveform to square current waveform,
while L responds triangular current waveform to square voltage wave.
• This duality principle shall be used with studying transients and AC
response of L and C circuits.
• Inductive Energy:
• The process of establishing magnetic flux inside an inductor involves
expenditure of energy.
• As energy is found by integrating power. In case of Inductance, p = v.i = i .
Ldi/dt. So,
• Integration yields
• This result could also have been found by using duality principle;
interchange C with L & i with v
• Inductive energy is stored in the form of potential energy in the magnetic
filed inside the core.
• As current is increased from zero to some non-zero value, energy is
absorbed from external source and stored in the inductance.
• As current is decreased back to zero, magnetic filed collapses and
accumulated energy is returned back to circuit.
• Inductances in Series and Parallel:
• Practical Inductors:
• Practical inductors satisfy the characteristics approximately. One of the
main limitation is resistance of windings.
• For a practical resistor, only when L di/dt >> Ri,
be regarded as ideal.
• Of the three basic elements, inductor is least ideal
• Due to its heaviness, bulkiness and the fact that it can’t be fabricated on
integrated circuits, makes this element least popular in modern electronics.
• Inductors are still used in power supply designs and high frequency signal
processing.
• All circuit elements exhibit a small amount of inductance due to the fact that
current through element produces magnetic flux. This unintended
inductance is called Stray or Parasitic Inductance.
• Even a plain wire exhibits stray inductance, besides stray resistance and
stray capacitance.
• Comparison of Basic Elements:
7.3 Natural Response of RC and RL Circuits
• Analysis of circuits containing energy storage elements is still
based on Kirchhoff’s Laws and Element laws.
• Since these elements exhibit time dependent characteristics, so
the equations are no more simple algebraic but involves time
derivatives or integral or both.
• We wish to find voltage / current though energy storage
element, in terms of source, resistance and element itself.
• The manner in which the voltage or current varies with time is
referred as time response.
• First Order Differential Equations:
• By the capacitive law;
• i = C dv / dt.
• Applying KVL in the RC circuit,
• vs = Ri + v.
• Eliminating ‘i’ ;
• By the Inductive law;
• v = L di / dt.
• Applying KCL in the RL circuit,
• is = v/R + i.
• Eliminating ‘v’ ;
• Both the equations are of type
• Where y(t) representing either v(t) or i(t) is unknown variable.
• While x(t) representing either vs(t) or is(t) is forcing function.
• Moreover,
• RC and L/R has dimension of time. So, these are called time
constant.
• As the circuit equations contain first order derivatives, so these
circuits are called first order circuits.
• These equations can’t be solved by algebraic manipulations.
Lets discuss an interesting and special case when x(t) = 0.
• The Source Free OR Natural Response:
• Let x(t) = 0. i.e source free circuit.
• This is homogeneous equation whose solution is called homogeneous soln
• This solution is physically referred as Source Free or Natural Response.
• Which shows that aside from constant –Ƭ, the unknown and its derivate
must be same. Which is true only for exponential function.
• We must assume the solution of the type
• We seek to find an expression for A and s. Lets find expression for s,
• Above equation has the root
• Since this root has dimensions of frequency, it is variously referred as
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natural freq, characteristic freq or critical freq of the circuit.
It has the unit of nepers/s where nepers (Np) is a dimensionless unit.
Next we wish to find expression for A.
This is done on the basis of initial conditions y(0) in the circuit i.e initial
voltage v(0) across Capacitance or initial current i(0) through Inductance.
Letting t = 0 in the below equation
• Which is the natural / source free response of RC or RL circuit.
• In case of RC circuit, above equation shall be
• 𝑣(𝑡) = 𝑣 0 𝑒 −𝑡/𝑇
• While for RL circuit:
• 𝑖(𝑡) = 𝑖 0 𝑒 −𝑡/𝑇
• y(t) is exponentially decaying function from initial value y(0) to final value
y(∞) = 0.
• Since this decay depends only on initial value and 𝑇, irrespective of forcing
function value, so this response is called Natural Response.
• Homogeneous solution, natural response and source free response are
different names for same function.
• The Time Constant 𝑇:
• Mathematically, 𝑇 serves the purpose to make argument t / 𝑇 a
dimensionless number.
• Physically, it tells that how rapidly decay takes place.
• Evaluating the equation at t = 𝑇:
• After 𝑇 seconds, output remains 37 percent of maximum value. OR
• After 𝑇 seconds, output decayed 63 percent of entire decay.
• 𝑇 represents instant at which tangent to curve at origin intercepts the t axis.
• Larger the value of 𝑇 , slower the rate of decay because
• Longer it will take to decay to 37 % OR
• The tangent at origin will intercept the t axis
at later instant.
• Conversely, smaller 𝑇 makes rapid decay.
• The s Plane:
• It is a good practice to analyze the root of characteristic equation as a point
in s plane.
• We’ll discuss this plane in more detail in coming chapter, here we’ll only
discuss the horizontal axis.
• Negative roots lie on left portion of axis, while +ve roots on right portion.
• Farther away the root from origin, more rapid is the decay.
• Closer the root from origin, slower is the decay.
• This is an interesting case that when 1 / 𝑇 approaches to zero. Which
means that 𝑇 approaches infinity, indicating an infinitely slow decay.
• The condition 𝑇 = ∞ is achieved by letting R = ∞ in capacitive case and R =
0 in case of inductor.
• When ideal capacitor circuit is open circuited, the capacitor will retain its
voltage indefinitely. So v(t) = v(0)
• Likewise, when ideal inductive circuit is short circuited, the ideal inductor
shall sustain its initial current indefinitely. So, i(t) = i(0)
7.4 Response to DC and AC Forcing Functions
• Having investigated the particular but important case x(t) = 0,
now we’ll turn our attention towards finding the solution in case
of arbitrary input x(t).
• We’ll one by one discuss the solutions for DC and AC forcing
functions.
• General Solution of the Differential Equation:
• For the below given circuit, we’ve found the differential equation as below:
• Multiplying both sides by
• Complete response is the sum of natural and forced response.
• Where
• Forced response depends upon x(t) i.e vs or is in RC or RL circuits.
• Depending on the form of x(t), forced response can be calculated by using
numerical techniques.
• However, there are two cases of most practically interest.
• DC Forcing Function
• AC Forcing Function
• Response to DC Forcing Function:
• A DC forcing function is of the type
• Putting this value of x(t) in forced response
• For an RC circuit
𝑣 𝑓𝑜𝑟𝑐𝑒𝑑 = 𝑉𝑠(1 − 𝑒
𝑖 𝑓𝑜𝑟𝑐𝑒𝑑 = 𝐼𝑠(1 − 𝑒
• Complete response will be
• For an RL circuit
𝑡
−𝑇
𝑡
−𝑇
)
)
• After 𝑇 seconds, output rise 63 percent of entire rise
• Below figure illustrates the complete response for the case
• Top equation holds in general for all possible values of y(0) and Y(∞) i.e
these values may be positive, negative or zero.
• When Xs = 0, the complete solution reduces to natural response.
• The Transient and DC Steady State Components:
• The complete response to DC forcing function is
• When t approaches infinite
• Then complete response will be
• Where
• Response to AC Forcing Function:
• Lets take an AC signal as forcing function
• As we’ve already found general relation for forced response; which is
• Putting the value of x(t) in forced response
• We can use the following formula from integration table
• OR
• Using trigonometric identity
• We’ll get the forced response as
• The complete response shall be
• Where
• So, equation for complete
response shall be
• The Transient and AC Steady State Components:
• The complete solution when AC is applied:
• As
• Where
THE END