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Chapter 3: Q 3.5-6E (page 121)
Derive a power balance equation for the RLand RCcircuits. (See Problem 5.) Discuss the significance of the signs of the three power terms.
The power balance equation for RL and RC circuit are and respectively.
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Use the improved Euler鈥檚 method subroutine with step size h = 0.2 to approximate the solution toat the points x = 0, 0.2, 0.4, 鈥�., 2.0. Use your answers to make a rough sketch of the solution on [0, 2].
Local versus Global Error. In deriving formula (4) for Euler鈥檚 method, a rectangle was used to approximate the area under a curve (see Figure 3.14). With
\({\bf{g(t) = f(t,f(t))}}\), this approximation can be written as \(\int\limits_{{{\bf{x}}_{\bf{n}}}}^{{{\bf{x}}_{{\bf{n + 1}}}}} {{\bf{g(t)dt}} \approx {\bf{hg(}}{{\bf{x}}_{\bf{n}}}{\bf{)}}} \)where \({\bf{h = }}{{\bf{x}}_{{\bf{n + 1}}}}{\bf{ - }}{{\bf{x}}_{\bf{n}}}\) .
Building Temperature.In Section 3.3 we modeled the temperature inside a building by the initial value problem (13)\(\frac{{{\bf{dT}}}}{{{\bf{dt}}}}{\bf{ = K}}\,\,\left[ {{\bf{M}}\,{\bf{(t) - T}}\,{\bf{(t)}}} \right]{\bf{ + H}}\,{\bf{(t) + U}}\,{\bf{(t),}}\,\,{\bf{T}}\,{\bf{(}}{{\bf{t}}_{\bf{o}}}{\bf{) = }}{{\bf{T}}_{\bf{o}}}\) , where M is the temperature outside the building, T is the temperature inside the building, H is the additional heating rate, U is the furnace heating or air conditioner cooling rate, K is a positive constant, and \({{\bf{T}}_{\bf{o}}}\) is the initial temperature at time \({{\bf{t}}_{\bf{o}}}\) . In a typical model, \({{\bf{t}}_{\bf{o}}}{\bf{ = 0}}\) (midnight),\({{\bf{T}}_{\bf{o}}}{\bf{ = 6}}{{\bf{5}}^{\bf{o}}}\), \({\bf{H}}\left( {\bf{t}} \right){\bf{ = 0}}{\bf{.1}}\), \({\bf{U(t) = 1}}{\bf{.5}}\left[ {{\bf{70 - T(t)}}} \right]\)and \({\bf{M(t) = 75 - 20cos}}\frac{{{\bf{\pi t}}}}{{{\bf{12}}}}\) . The constant K is usually between\(\frac{{\bf{1}}}{{\bf{4}}}{\bf{and}}\frac{{\bf{1}}}{{\bf{2}}}\), depending on such things as insulation. To study the effect of insulating this building, consider the typical building described above and use the improved Euler鈥檚 method subroutine with\({\bf{h = }}\frac{{\bf{2}}}{{\bf{3}}}\) to approximate the solution to (13) on the interval \(0 \le {\bf{t}} \le 24\) (1 day) for \({\bf{k = 0}}{\bf{.2,}}\,{\bf{0}}{\bf{.4}}\), and 0.6.
The Taylor method of order 2 can be used to approximate the solution to the initial value problem\({\bf{y' = y,y(0) = 1}}\) , at x= 1. Show that the approximation \({{\bf{y}}_{\bf{n}}}\)obtained by using the Taylor method of order 2 with the step size \(\frac{{\bf{1}}}{{\bf{n}}}\) is given by the formula\({{\bf{y}}_{\bf{n}}}{\bf{ = }}{\left( {{\bf{1 + }}\frac{{\bf{1}}}{{\bf{n}}}{\bf{ + }}\frac{{\bf{1}}}{{{\bf{2}}{{\bf{n}}^{\bf{2}}}}}} \right)^{\bf{n}}}\). The solution to the initial value problem is\({\bf{y = }}{{\bf{e}}^{\bf{x}}}\), so \({{\bf{y}}_{\bf{n}}}\)is an approximation to the constant e.
If the resistance in the RLcircuit of Figure 3.13(a) is zero, show that the current I (t) is directly proportional to the integral of the applied voltage E(t). Similarly, show that if the resistance in the RCcircuit of Figure 3.13(b) is zero, the current is directly proportional to the derivative of the applied voltage.
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