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 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 non- linear. Reverse biased PN junction of a diode acts as a non-linear capacitor. C depends upon insulator type and physical dimensions Where 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, 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. OR

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. ForC:i = C dv/dtFor 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, v s = Ri + v. Eliminating ‘i’ ; By the Inductive law; v = L di / dt. Applying KCL in the RL circuit, i s = 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 v s (t) or i s (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 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

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.

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 v s or i s 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

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

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