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Consider the circuit from P2 in PS2, now connected to an external source as shown in figure 2. The external source is an AC source with Vac(t) = V₁ cos(Nt + p). R, L, Vo, 2, are arbitrary positive constants. ³. R HI 0000 Figure 2: 'R+LC' circuit with an external source. (a) Find the differential equation that describes the current in the inductor. You can use all what you learned about this circuit in PS2 but show again your work, including Kirchhoff equations. Remember to have sum of potentials to set the differential equation equal to V(t). Hint: just the differential equation. Do not solve it yet. (b) Find IL(t) ignoring the transient behavior. That is, get the steady response to the external source 4. Hint: one way is to set I₁(t) = A cos Nt + B sin Nt, with A, B some constants. Then, find A, B by solving the differential equation of IL(t). Some suggestions from class: i/ do not use a coefficient of 1 in the Ï term. Use instead (RLC)Ï. The algebra will have less fractions. ii/ Beware that V(t) has an arbitrary phase: p. Use cos(nt + p) = cos Nt cos - sin Nt sin o. iii/ at the end, check that A, B have units of current! (c) Given IL(t), find the voltage across the coil inductor V₁(t)? Leave the answer in terms of A, B, or whichever labels you used in the previous questions. That is, you can, but it is not required to substitute the explicit expressions of A, B found in the previous question.

Consider the circuit from P2 in PS2, now connected to an external source as shown in figure 2. The external source is an AC source with Vac(t) = V₁ cos(Nt + p). R, L, Vo, 2, are arbitrary positive constants. ³. R HI 0000 Figure 2: 'R+LC' circuit with an external source. (a) Find the differential equation that describes the current in the inductor. You can use all what you learned about this circuit in PS2 but show again your work, including Kirchhoff equations. Remember to have sum of potentials to set the differential equation equal to V(t). Hint: just the differential equation. Do not solve it yet. (b) Find IL(t) ignoring the transient behavior. That is, get the steady response to the external source 4. Hint: one way is to set I₁(t) = A cos Nt + B sin Nt, with A, B some constants. Then, find A, B by solving the differential equation of IL(t). Some suggestions from class: i/ do not use a coefficient of 1 in the Ï term. Use instead (RLC)Ï. The algebra will have less fractions. ii/ Beware that V(t) has an arbitrary phase: p. Use cos(nt + p) = cos Nt cos - sin Nt sin o. iii/ at the end, check that A, B have units of current! (c) Given IL(t), find the voltage across the coil inductor V₁(t)? Leave the answer in terms of A, B, or whichever labels you used in the previous questions. That is, you can, but it is not required to substitute the explicit expressions of A, B found in the previous question.

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
Section: Chapter Questions
Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...
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**Educational Website Transcription:** ### R+LC Circuit Analysis with an External AC Source **Figure 2: ‘R+LC’ circuit with an external source** The diagram in Figure 2 represents an R+LC circuit connected to an external AC source. The components include a resistor (R), an inductor (L), and a capacitor (C). The external source is represented by a symbol for an AC generator. #### Problem Scenarios and Explanations (a) **Differential Equation for the Inductor Current:** - Task: Derive the differential equation describing the current in the inductor. - Tips: Utilize Kirchhoff's voltage law and set the sum of potentials equal to V(t). The equation should be left in its differential form. (b) **Inductor Current IL(t) in Steady State:** - Task: Determine IL(t) ignoring transient behavior, focusing on the steady response to the AC source. - Hint: Use the expression IL(t) = A cos(Ωt) + B sin(Ωt) and solve for constants A and B. Ensure the solution has units of current. (c) **Voltage Across the Inductor VL(t):** - Task: Find the expression for VL(t) using your results from the previous part. Leave it in terms of A and B as necessary. (d) **Maximum Inductor Current IL,max:** - Task: Derive an expression for the peak current over one AC cycle. - Notes: This step involves understanding the dependence on phase φ without using derivatives. Ensure IL,max is in the correct units. (e) **External Frequency for Maximum IL,max:** - Task: Determine the external frequency Ω that maximizes IL,max (referred to as Ωmax). - Considerations: Ωmax should relate R, L, C in such a way that you verify the units are 1/s. (f) **LC Circuit and Resonant Frequency:** - Scenario: An LC circuit, where R = 0, displays simple harmonic oscillation (SHO) at its resonant frequency Ωres = 1/√LC. Discuss how R affects Ωmax in relation to Ωres. Validate the units. This problem set challenges understanding of circuit dynamics, focusing on how external AC sources interact with RLC components.
Transcribed Image Text:**Educational Website Transcription:** ### R+LC Circuit Analysis with an External AC Source **Figure 2: ‘R+LC’ circuit with an external source** The diagram in Figure 2 represents an R+LC circuit connected to an external AC source. The components include a resistor (R), an inductor (L), and a capacitor (C). The external source is represented by a symbol for an AC generator. #### Problem Scenarios and Explanations (a) **Differential Equation for the Inductor Current:** - Task: Derive the differential equation describing the current in the inductor. - Tips: Utilize Kirchhoff's voltage law and set the sum of potentials equal to V(t). The equation should be left in its differential form. (b) **Inductor Current IL(t) in Steady State:** - Task: Determine IL(t) ignoring transient behavior, focusing on the steady response to the AC source. - Hint: Use the expression IL(t) = A cos(Ωt) + B sin(Ωt) and solve for constants A and B. Ensure the solution has units of current. (c) **Voltage Across the Inductor VL(t):** - Task: Find the expression for VL(t) using your results from the previous part. Leave it in terms of A and B as necessary. (d) **Maximum Inductor Current IL,max:** - Task: Derive an expression for the peak current over one AC cycle. - Notes: This step involves understanding the dependence on phase φ without using derivatives. Ensure IL,max is in the correct units. (e) **External Frequency for Maximum IL,max:** - Task: Determine the external frequency Ω that maximizes IL,max (referred to as Ωmax). - Considerations: Ωmax should relate R, L, C in such a way that you verify the units are 1/s. (f) **LC Circuit and Resonant Frequency:** - Scenario: An LC circuit, where R = 0, displays simple harmonic oscillation (SHO) at its resonant frequency Ωres = 1/√LC. Discuss how R affects Ωmax in relation to Ωres. Validate the units. This problem set challenges understanding of circuit dynamics, focusing on how external AC sources interact with RLC components.
The image contains the following text: "L, C constant and vary R, do we need to increase or decrease R in order to get Ω<sub>max</sub> closer to Ω<sub>Res</sub>?" This text is asking a question related to electrical circuits, specifically about the relationship between resistance (R), inductance (L), and capacitance (C) in the context of resonant frequency. In this scenario, L and C are held constant while R is varied. The question is whether increasing or decreasing R will bring the maximum angular frequency (Ω<sub>max</sub>) closer to the resonant angular frequency (Ω<sub>Res</sub>).
Transcribed Image Text:The image contains the following text: "L, C constant and vary R, do we need to increase or decrease R in order to get Ω<sub>max</sub> closer to Ω<sub>Res</sub>?" This text is asking a question related to electrical circuits, specifically about the relationship between resistance (R), inductance (L), and capacitance (C) in the context of resonant frequency. In this scenario, L and C are held constant while R is varied. The question is whether increasing or decreasing R will bring the maximum angular frequency (Ω<sub>max</sub>) closer to the resonant angular frequency (Ω<sub>Res</sub>).
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