Consider the circuit shown inFigure P4.49. The voltage source is known as a ramp function, which is defined by
Figure P4.49
Assume that
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- Consider the circuit shown in Figure P4.55. a. Write the differential equation for v(t).b. Find the time constant and the form of the complementary solution.c. Usually, for an exponential forcing function like this, we would try a particular solution ofthe form vp(t) = K exp (−10t). Why doesn’t that work in this case?d. Find the particular solution. [Hint: Try a particular solution of the form vp(t)=K t exp (−10t). How ]e. Find the complete solution for v(t).arrow_forwardConsider the circuit shown in Figure P4.70. a. Write the differential equation for v(t). b. Find the damping coefficient, the natural frequency, and the form of the complementary solution. c. Usually, for a sinusoidal forcing function, we try a particular solution of the form v p ( t)=A cos( 10 4 t )+B sin( 10 4 t ). Why doesn’t that work in this case? d. Find the particular solution. [Hint: Try a particular solution of the form v p ( t)=At cos( 10 4 t )+B t sin( 10 4 t ). ] e. Find the complete solution for v(t).arrow_forwardDue to components not shown in the figure, the circuit of Figure P4.41 has i L ( 0 )= I i . a. Write an expression for i L (t) for t≥0. b. Find an expression for the power delivered to the resistance as a function of time. c. Integrate the power delivered to the resistance from t=0 to t=∞, and show that the result is equal to the initial energy stored in the inductancearrow_forward
- We know that the capacitor shown in Figure P4.11 is charged to a voltage of 10 V priorto t=0.a. Find expressions for the voltage across the capacitor vC(t) and the voltage across theresistor vR(t) for all time.b. Find an expression for the power delivered to the resistor.c. Integrate the power from t=0 to t=∞ to find the energy delivered.d. Show that the energy delivered to the resistor is equal to the energy stored in thecapacitor prior to t=0.arrow_forwardConsider the circuit shown in Figure P4.50. The initial current in the inductor is i s ( 0+)=0. Write the differential equation for i s(t) and solve. [Hint: Try a particular solution of the form i sp ( t )=A cos( 300t )+B sin( 300t ).]arrow_forwardDetermine expressions for and sketch i s ( t ) to scale versus time for −0.2≤t≤1.0 s for the circuit of Figure P4.37.arrow_forward
- For the circuit shown in Figure P4.29, the switch is closed for a long time prior to t=0.Find expressions for vC(t) and sketch it to scale for −80≤t≤160 ms.arrow_forwardThe initial voltage across the capacitor shown in Figure P4.3 is v C ( 0+ )=−10 V. Find anexpression for the voltage across the capacitor as a function of time. Also, determine the time t0at which the voltage crosses zero.arrow_forwardConsider the circuit shown in Figure P4.18. Prior to t=0, v 1 =100 V, and v 2 =0.a. Immediately after the switch is closed, what is the value of the current [i.e., what is thevalue of i( 0+ ) ]?b. Write the KVL equation for the circuit in terms of the current and initial voltages. Take thederivative to obtain a differential equation.c. What is the value of the time constant in this circuit?d. Find an expression for the current as a function of time.e. Find the value that v2 approaches as t becomes very large.arrow_forward
- The capacitor model we have used so far has beentreated as an ideal circuit element. A more accuratemodel for a capacitor is shown in Figure P4.67. Theideal capacitor, C, has a large “leakage” resistance, RC,in parallel with it. RC models the leakage currentthrough the capacitor. R1 and R2 represent the leadwire resistances, and L1 and L2 represent the lead wireinductances.a. If C = 1 μF, RC = 100 MΩ, R1 = R2 = 1 μΩ andL1 = L2 = 0.1 μH, find the equivalent impedanceseen at the terminals a and b as a function offrequency ω.b. Find the range of frequencies for which Zab iscapacitive, i.e., Xab > 10|Rab.Hint: Assume that RC is is much greater than 1/wC so thatyou can replace RC by an infinite resistance in part b.arrow_forwardWhat is a Linear Circuit? Simply we can say that the linear circuit is an electric circuit (Links to an external site.) and the parameters of this circuit are resistance, capacitance, inductance and etc are constant. Or we can say the parameters of the circuits are not changed with respect to the voltage and current is called the linear circuit. What is a Non-Linear Circuit? The non-linear circuit is also an electric circuit and the parameters of this circuit differ with respect to the current and the voltage. Or in the electric circuit, the parameters like waveforms, resistance, inductance and etc are not constant is called as Non- linear circuit. Question: Is it possible to apply superposition theorem to nonlinear circuit? If yes, why? and if no, why?arrow_forwardWrite the differential equation for i L(t) and find the complete solution for the circuit of Figure P4.45. [Hint: Try a particular solution of the form i Lp ( t )=A e −t .]arrow_forward
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