RC Circuits - Linear outputs and inputs
Consider a modification of the RC filter that we discussed in class, where we additionally use an inductor L > 0 in the circuit. The governing system of equations is given by LC(d2v/ dt2(t)) + RC(dv/dt (t)) + v(t) = u(t), where u us the input voltage, t ∈ T ⊂ R, L, C, R > 0, and v is the output voltage. Provide answers to the followings:
(i) Find the output v in terms of u (take the initial conditions as you wish). Note that the solution to this second-order linear differential equation, which I suggest writing as a system of linear equations, will depend on the parameters R, C, and L; in particular, there will be three cases that you need to consider.
(ii) Is the solution linear in input? Can the output, in the cases above, be obtained as a convolution?
(iii) Now, select some values for R, L, and C as you wish. For each case of your solution above, using Matlab or whatever software you wish, plot the output of an input voltage that is constant for positive times, and zero otherwise. Do you believe that the properties you have observed are particular to your choices of R, L, and C? Give a justification
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Exercise 4) Consider the RC network shown, where v(t) is the input voltage and ve(t) is...
Exercise 4) Consider the RC network shown, where v(t) is the input voltage and ve(t) is the circuit output voltage. R is the same for all resistors (4a) Write differential equations of the circuit in terms of the currents. Convert the equa tions to the Laplace domain (5 marks)v(oO 4b) Find the transfer function Ve(s)/V(s) (5 marks) (4c) Using the final value theorem, calculate the steady-state value of ve(t) for an unit step input of u(t), i.e., u(t)-1 V (2.5...
Please help solve while providing a detailed solution. Being given the following information, use the equations...
Please help solve while providing a detailed solution.
Being given the following information, use the equations provided to find the steady-state current in the following RLC circuit. R=82 L= 0.5H C= 0.1F E(t) = 100 cos(2t) V knowing that at t = 0, i(0) = 0 Equations: UR = Ri VL = = L- di 9 Uci dt С VR + V1 + Vc = e(t) or =V (if the source voltage is constant) dq duc i= = C- q=ſidt...
Please help solve, providing a detailed solution using the equations provided below and LaPlace transform (Use...
Please help solve, providing a detailed solution using the
equations provided below and
LaPlace transform (Use the table provided in the
link) to solve the differential equations obtained when working
through the question.
Link to the Laplace Transform Table:
https://ibb.co/TkrvbNH
Being given the following information, use the equations provided to find the steady-state current in the following RLC circuit. R=82 L= 0.5H C= 0.1F E(t) = 100 cos(2t) V knowing that at t = 0, i(0) = 0 Equations: UR...
In the circuit below, the input voltage is Vin-Vinegakcos(wt), R-20 ΚΩandC15nFw l. in al Show that the output voltage is VotVcos (wt-), where V-V n peak/V1 + (RC) b) Show that this result justifies c...
In the circuit below, the input voltage is Vin-Vinegakcos(wt), R-20 ΚΩandC15nFw l. in al Show that the output voltage is VotVcos (wt-), where V-V n peak/V1 + (RC) b) Show that this result justifies calling this circuit a high-pass filter+ c) Find an expression for the phase constant δ in terms of R,C and d) At what frequency is Vi (1/V2) Vin peak? That particular frequency is known as the 3dB frequency, or f3dB, of the circuit
In the circuit...
For the given RC circuit shown below, ys the output, and ut) is the input. Values of the components are marked on schematic i) Derive the system differential equation and transfer function Y(s)/U(s)...
For the given RC circuit shown below, ys the output, and ut) is the input. Values of the components are marked on schematic i) Derive the system differential equation and transfer function Y(s)/U(s) ii) Choose voltage across capacitors as states and derive the state equations and state matrices (A, B, C,and D). iii) Validate the states by deriving the transfer function from state matrices. iv) Choose a different set of states and derive a different state equation and state Matrix...
Consider a high-pass RC circuit shown below. Let the capacitor be initially uncharged, and the input...
Consider a high-pass RC circuit shown below. Let the capacitor be initially uncharged, and the input v;(t) be applied at t= 0. v;(t) SR v.(t) (a) Obtain the differential equation governing the input and output voltages for t > 0. (b) Using trapezoidal rule convert the same into a difference equation. (c) Obtain the response v.(t) by solving the differential equation. a sampling For numerical values of R = 80092 and C = 0.754F and for vi(t) = A coswt,...
electromagnetic 1) RC Circuits: (15 pts) (a) Use Kirchhoff's voltage law (KVL) to obtain an ordinary...
electromagnetic
1) RC Circuits: (15 pts) (a) Use Kirchhoff's voltage law (KVL) to obtain an ordinary differential equation (ODE) describing the charge vs. time function (1) for a capacitor in the discharging RC circuit shown below. Assume that at time t = 0 (right before the switch is closed) the voltage across the capacitor is V = V.. R R с V(t) С t=0 t>O Fig. 1. Fully charged RC circuit Fig. 2. Discharging RC Circuit (b) Solve the ODE...
Use MATLAB to write a function that takes inputs R, L, C, TO, Tf, DeltaT and...
Use MATLAB to write a function that takes inputs R, L, C, TO, Tf, DeltaT and f and produces as output the charge on the capacitor in a driven RLC series circuit where the input voltage source is v(t) = cos(21 ft). Here f is the frequency of the voltage source in Hz. This function returns the inverse Laplace Transform of (1) L (s2 + 2) ($2 + 2as +B) where a = # and B = Lc. Note that...
PROBLEM #2: In the circuit shown, suppose that R and C are given. The transfer function...
PROBLEM #2: In the circuit shown, suppose that R and C are given. The transfer function of the circuit is G(s)== RCs +1 The impulse response of the circuit is g(t)== Let/RC ·u,(t). RC CV.CO Given that the input voltage is v;(t)=u,(t), determine the zero-state response v.(t) for t20 in two equivalent ways: (a) Use convolution. That is, compute the integral vo(t) = [ 8(t – T )v;()dt. (b) Use Laplace transforms. That is, compute vo(t) = ('{G(s)V;(s)}.
1. For the circuit shown below, we wish to find v(t) for t>0. 1 1 a....
1. For the circuit shown below, we wish to find v(t) for t>0. 1 1 a. Find the governing equation for the voltage v using KCL at the top node using the following definitions: a = 0,W, = dr. This will get you a governing equation in the same form as that derived for the case we did in class where the R, L, and C were in series. b. What is the particular solution in this case? c. If...
Exercise 4) Consider the RC network shown, where v(t) is the input voltage and ve(t) is the circuit output voltage. R is the same for all resistors (4a) Write differential equations of the circuit in terms of the currents. Convert the equa tions to the Laplace domain (5 marks)v(oO 4b) Find the transfer function Ve(s)/V(s) (5 marks) (4c) Using the final value theorem, calculate the steady-state value of ve(t) for an unit step input of u(t), i.e., u(t)-1 V (2.5...
Please help solve while providing a detailed solution.
Being given the following information, use the equations provided to find the steady-state current in the following RLC circuit. R=82 L= 0.5H C= 0.1F E(t) = 100 cos(2t) V knowing that at t = 0, i(0) = 0 Equations: UR = Ri VL = = L- di 9 Uci dt С VR + V1 + Vc = e(t) or =V (if the source voltage is constant) dq duc i= = C- q=ſidt...
Please help solve, providing a detailed solution using the
equations provided below and
LaPlace transform (Use the table provided in the
link) to solve the differential equations obtained when working
through the question.
Link to the Laplace Transform Table:
https://ibb.co/TkrvbNH
Being given the following information, use the equations provided to find the steady-state current in the following RLC circuit. R=82 L= 0.5H C= 0.1F E(t) = 100 cos(2t) V knowing that at t = 0, i(0) = 0 Equations: UR...
In the circuit below, the input voltage is Vin-Vinegakcos(wt), R-20 ΚΩandC15nFw l. in al Show that the output voltage is VotVcos (wt-), where V-V n peak/V1 + (RC) b) Show that this result justifies calling this circuit a high-pass filter+ c) Find an expression for the phase constant δ in terms of R,C and d) At what frequency is Vi (1/V2) Vin peak? That particular frequency is known as the 3dB frequency, or f3dB, of the circuit
In the circuit...
For the given RC circuit shown below, ys the output, and ut) is the input. Values of the components are marked on schematic i) Derive the system differential equation and transfer function Y(s)/U(s) ii) Choose voltage across capacitors as states and derive the state equations and state matrices (A, B, C,and D). iii) Validate the states by deriving the transfer function from state matrices. iv) Choose a different set of states and derive a different state equation and state Matrix...
Consider a high-pass RC circuit shown below. Let the capacitor be initially uncharged, and the input v;(t) be applied at t= 0. v;(t) SR v.(t) (a) Obtain the differential equation governing the input and output voltages for t > 0. (b) Using trapezoidal rule convert the same into a difference equation. (c) Obtain the response v.(t) by solving the differential equation. a sampling For numerical values of R = 80092 and C = 0.754F and for vi(t) = A coswt,...
electromagnetic
1) RC Circuits: (15 pts) (a) Use Kirchhoff's voltage law (KVL) to obtain an ordinary differential equation (ODE) describing the charge vs. time function (1) for a capacitor in the discharging RC circuit shown below. Assume that at time t = 0 (right before the switch is closed) the voltage across the capacitor is V = V.. R R с V(t) С t=0 t>O Fig. 1. Fully charged RC circuit Fig. 2. Discharging RC Circuit (b) Solve the ODE...
Use MATLAB to write a function that takes inputs R, L, C, TO, Tf, DeltaT and f and produces as output the charge on the capacitor in a driven RLC series circuit where the input voltage source is v(t) = cos(21 ft). Here f is the frequency of the voltage source in Hz. This function returns the inverse Laplace Transform of (1) L (s2 + 2) ($2 + 2as +B) where a = # and B = Lc. Note that...
PROBLEM #2: In the circuit shown, suppose that R and C are given. The transfer function of the circuit is G(s)== RCs +1 The impulse response of the circuit is g(t)== Let/RC ·u,(t). RC CV.CO Given that the input voltage is v;(t)=u,(t), determine the zero-state response v.(t) for t20 in two equivalent ways: (a) Use convolution. That is, compute the integral vo(t) = [ 8(t – T )v;()dt. (b) Use Laplace transforms. That is, compute vo(t) = ('{G(s)V;(s)}.
1. For the circuit shown below, we wish to find v(t) for t>0. 1 1 a. Find the governing equation for the voltage v using KCL at the top node using the following definitions: a = 0,W, = dr. This will get you a governing equation in the same form as that derived for the case we did in class where the R, L, and C were in series. b. What is the particular solution in this case? c. If...
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