Mechatronics Engineering MT-144 NETWORK ANALYSIS Mechatronics Engineering (04)

ENERGY STORAGE ELEMENTS (Chapter 7) Introduction Capacitance Inductance Natural Response of RC and RL Circuits Response to DC and AC Forcing Functions

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. The equations governing these circuits are algebraic equations; including Kirchhoff ‘s laws and Ohm's Law. Since resistances can only dissipate energy, we need at least one independent source to initiate any voltage or current in the circuit. In the absence of independent sources, all voltages and currents would be zero and the circuit would have no electrical life of its own.

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 and v = L (di/dt) … for inductance. 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

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 integro-differential 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.

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.

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.

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.

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

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.

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.

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

ENERGY STORAGE ELEMENTS Table 7.1 Comparison of the Basic Elements:

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.

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

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 :

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:

ENERGY STORAGE ELEMENTS 7.3 NATURAL RESPONSE OF RC AND RL CIRCUITS… First-Order Differential Equations: , in the inductive case.

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 first-order circuits. A circuit containing multiple capacitances (or. inductances) is still a first-order 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. Figure 7.14

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 first-order 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 right-hand 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.

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.

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).

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.

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.

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

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).

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.

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 ….

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.

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 at the origin will intercept the t-axis at a later instant. Conversely, a smaller the value of Ƭ , greater (rapid) the rate of decay. This is illustrated in Figure 7.16. 0.37 y(0) Ƭ1 , Ƭ2 , Ƭ3 etc

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

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.

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… 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 

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.

ENERGY STORAGE ELEMENTS 7.3 NATURAL RESPONSE OF RC AND RL CIRCUITS…The s Plane… .