Mechatronics Engineering MT-144 NETWORK ANALYSIS Mechatronics Engineering (05)

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

ENERGY STORAGE ELEMENTS 7.4 RESPONSE TO DC AND AC FORCING FUNCTIONS Having investigated the particular but important case in which x (t )= 0 in Equation (7.25), we now wish to find the solution for the case of an arbitrary forcing function x(t) . We then turn our attention to the two types of forcing functions of greatest interest, namely, the dc and ac forcing functions. The level of mathematical detail is designed to provide a clear understanding of the various response components as well as the terminology. General Solution to the Differential Equation An elegant method for solving Equation (7.25) is provided by the integrating factor approach, as follows. Multiplying both sides of Equation (7.25) by (1/Ƭ ) exp (tlƬ ) yields:

ENERGY STORAGE ELEMENTS 7.4 RESPONSE TO DC AND AC FORCING FUNCTIONS General Solution to the Differential Equation … Recognizing that the left-hand side is the derivative of a product, we can write: Replacing t with the dummy variable (ξ) and integrating both sides from 0 to t yields That is :

ENERGY STORAGE ELEMENTS 7.4 RESPONSE TO DC AND AC FORCING FUNCTIONS General Solution to the Differential Equation … where y(0) again represents the initial value of y. Finally, solving for y(t) yields the general expression for the solution, Equation 7.35 This expression indicates that the response y(t) consists of two components. The first component, which is independent of the forcing function x(t) , is the already familiar natural response. The second component, which depends on the particular forcing function x(t), is called the forced response. For obvious reasons, the sum of the two components is called the complete response. To emphasize this, Equation (7.35) is often expressed in the form of Equation 7.36:

ENERGY STORAGE ELEMENTS 7.4 RESPONSE TO DC AND AC FORCING FUNCTIONS General Solution to the Differential Equation … To emphasize this, Equation (7.35) is often expressed in the form of Equation 7.36: where, (7.37) (7.38) It is apparent that the forced response depends on the functional form of X(ξ) that is, on the manner in which vs or is in our RC and RL circuits vary with time. Depending on this form, evaluating the integral of Equation (7.38) analytically may be a formidable task, and numerical methods may have to be used instead. For now, the cases of greatest practical interest are only two, namely, the case of a constant or dc forcing function, and the case of a sinusoidal or ac forcing function.

ENERGY STORAGE ELEMENTS 7.4 RESPONSE TO DC AND AC FORCING FUNCTIONS General Solution to the Differential Equation … To emphasize this, Equation (7.35) is often expressed in the form of Equation 7.36: where, (7.37) (7.38) Response to a DC Forcing Function A dc forcing function is of the type: x(t) = XS … (7.39) , where XS is a suitable constant. This means that in the case of the RC circuit of Figure 7.14 we have vS = VS and in the RL circuit we have is = IS. Substituting Equation (7.39) into Equation (7.38) yields:

ENERGY STORAGE ELEMENTS 7.4 RESPONSE TO DC AND AC FORCING FUNCTIONS General Solution to the Differential Equation … Response to a DC Forcing Function … A dc forcing function is of the type: x(t) = XS … (7.39) , where XS is a suitable constant. This means that in the case of the RC circuit of Figure 7.14 we have vS = VS and in the RL circuit we have is = IS. Substituting Equation (7.39) into Equation (7.38) yields: or … (7.40) Recall Equation n (7.36): Substituting from (7.40) into Equation (7.36) yields the complete response to a dc forcing function,

ENERGY STORAGE ELEMENTS 7.4 RESPONSE TO DC AND AC FORCING FUNCTIONS General Solution to the Differential Equation … Response to a DC Forcing Function … Noting that in the limit t∞ this equation yields: y(∞) = y(0)e-∞ + Xs (1 – e-∞) = Xs, we can also write … (7.42) where, … (7.43) As depicted in Figure 7.18, the complete response is an exponential transition from the initial value y(0) to the final value y(∞). This transition is again governed by the time constant Ƭ, which is now visualized as the time it takes for y(t) to accomplish (100 - 37)%, or 63% of the entire transition. Alternatively, Ƭ can be visualized as the time at which the tangent to the curve at the origin intercepts the y = y(∞) asymptote. Either viewpoint can be exploited to find Ƭ experimentally by observing the response to a dc forcing function with an oscilloscope.

ENERGY STORAGE ELEMENTS 7.4 RESPONSE TO DC AND AC FORCING FUNCTIONS General Solution to the Differential Equation … Response to a DC Forcing Function … As depicted in Figure 7.18, the complete response is an exponential transition from the initial value y(0) to the final value y(∞). This transition is again governed by the time constant Ƭ, which is now visualized as the time it takes for y(t) to accomplish (100 - 37)%, or 63% of the entire transition. Alternatively, Ƭ can be visualized as the time at which the tangent to the curve at the origin intercepts the y=y(∞) asymptote. Either viewpoint can be exploited to find Ƭ experimentally by observing the response to a dc forcing function with an oscilloscope.

ENERGY STORAGE ELEMENTS 7.4 RESPONSE TO DC AND AC FORCING FUNCTIONS General Solution to the Differential Equation …Response to a DC Forcing Function… Figure 7.18 illustrates the complete response for the case y(∞) > y(0) > 0, but analogous diagrams can be constructed for all other possible cases. Just keep in mind that Equation (7.42) holds in general, regardless of whether y(0) and y(∞) are positive, negative, or zero. Note that if XS= 0, so that y(∞) = 0, the complete response reduces to the natural response. We shall use Equation (7.42) so often that it is worth committing it to memory. Moreover, when sketching transient responses, it is a good habit to show the tangent at the origin and label its intercept with the y = y(∞) asymptote explicitly. This makes it easier for the reader to identify the value of the time constant governing the transient.

ENERGY STORAGE ELEMENTS 7.4 RESPONSE TO DC AND AC FORCING FUNCTIONS General Solution to the Differential Equation … Response to a DC Forcing Function… Home work: Attempt Exercise 7.12 (page 316) of text

ENERGY STORAGE ELEMENTS 7.4 RESPONSE TO DC AND AC FORCING FUNCTIONS General Solution to the Differential Equation … Response to a DC Forcing Function… The Transient and DC Steady-State Components Equation (7.42) reveals that the response to a dc forcing function can be regarded as the sum of two components, namely, an exponentially decaying component with initial magnitude [y(0) - y(∞)], called the transient component, and a time independent component of value y(∞), called the dc steady-state component because this is the value to which the complete response will settle once the transient component has died out. To emphasize this, the complete response to a dc forcing function is often expressed as: … (7.44) where, … (7.45) … (7.46) The two portions of y(t) are shown explicitly in Figure 7.18 (next slide).

ENERGY STORAGE ELEMENTS 7.4 RESPONSE TO DC AND AC FORCING FUNCTIONS General Solution to the Differential Equation … Response to a DC Forcing Function… The Transient and DC Steady-State Components … (7.44) The two portions of y(t) are shown explicitly in Figure 7.18.

ENERGY STORAGE ELEMENTS 7.4 RESPONSE TO DC AND AC FORCING FUNCTIONS General Solution to the Differential Equation … Response to a DC Forcing Function… The Transient and DC Steady-State Components… It is interesting to note that yxsient has the same functional form as ynatural and that it consists of two terms: the natural response itself, y(0)e-t/Ƭ, and the term y(∞)e-t/Ƭ , which is brought about by the forcing function. We also note that yss has the same functional form as the forcing function, yss = y(∞) = Xs. It is common practice to refer to the complete response to a dc forcing function as simply the transient response. This response, to be studied later (see Chapters 8 and 9 of the text book), is of fundamental importance because it allows us to deduce important properties that are characteristic of the particular circuit producing it. In fact, the design specifications for circuit systems are often given in terms of transient-response parameters.

ENERGY STORAGE ELEMENTS 7.4 RESPONSE TO DC AND AC FORCING FUNCTIONS General Solution to the Differential Equation … Response to an AC Forcing Function As we shall see in greater detail in Chapter 10, an ac forcing function is a sinusoidal function of the type: X(t) = Xm cos ωt ...(7.47) where Xm is called the amplitude and ω the angular frequency of the ac signal, whose units are radians/ s (rad/s). This means that in the capacitive circuit of Figure 7.14 we have vs = Vm cosωt, and in the inductive circuit we have is = Im cosωt. Substituting Equation (7.47) into Equation (7.38) yields: