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Understanding RC Circuits: Transient and Forced Responses

This guide explores the fundamental aspects of RC circuits, focusing on their transient and forced responses. It details how to analyze source-free and forced RC circuits, emphasizing key concepts like time constants, initial conditions, and solving first-order differential equations using Kirchhoff's voltage and current laws. Examples illustrate the process of finding voltage and current over time, considering scenarios such as capacitor initial voltages and stable conditions. This resource is ideal for students and professionals seeking to deepen their understanding of RC circuit behavior.

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Understanding RC Circuits: Transient and Forced Responses

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  1. INC 112 Basic Circuit Analysis Week 8 RC Circuits

  2. RC Circuits • The response of RC circuits can be categorized into two parts: • Transient Response • Forced Response • Transient response comes from the dynamic of R,C. • Forced response comes from the voltage source.

  3. Source-Free RC Circuits Capacitor has some energy stored so that The initial voltage at t=0 is V0 Initial condition Find i(t) from R, C

  4. Compare with the solution of RL circuits. The solution of RC circuits can be obtained with the same method. Source-free RC Source-free RL

  5. v(t) V0 t i(t) V0/R or t

  6. Time Constant The product RC is time constant for RC circuits Unit: second

  7. Forced RC Circuits C has an initial voltage of 0 from Use KVL, we got

  8. From Differentiate both sides Solve first-order differential equation Where I0 is the initial current of the circuit

  9. Natural Response Force Response C has an initial voltage = 0, But from KVL, therefore,and So,

  10. vC(t) i(t) V/R V t t vR(t) Note: Capacitor’s voltage cannot abruptly change V t

  11. How to Solve Problems? (RC) • Start by finding the voltage of the capacitor first • Assume the response that we want to find is in form of • Find the time constant τ (may use Thevenin’s) • Solve for k1, k2 using initial conditions and • status at the stable point • From the voltage, find other values that the problem ask • using KCL, KVL

  12. Example Switch open for a long time before t=0, find and sketch i(t) First, we start by finding vc(t) The initial condition of C is vc(0) = 1V The stable conditionof C is vc(∞) = 3V

  13. Assume vc(t) in form of Find the time constant after t=0 by Thevenin’s, viewing C as a load Therefore, the time constant is

  14. Find k1, k2 using vc(0) = 1, vc(∞) = 3 At t=0, vc(0) = 1 V At t = ∞, vc(∞) = 3 V Therefore, k1=3, k2 = -2 We can find i(t) by using Ohm’s law on the resistor

  15. i(t) 4A 2A t

  16. Example The switch was opened for a long time before t=0, Find i(t) Start with vc(t) The initial condition of C is vc(0) = 5V The final stable conditionof C comes from voltage divider, which is vc(∞) = 5*(1/1+0.5) = 3.33V

  17. Assume vc(t) in form of Find the time constant after t=0 by Thevenin’s, viewing C as a load Therefore, the time constant is

  18. Find k1, k2 using vc(0) = 5, vc(∞) = 3.33 At t=0, vc(0) = 5 V At t = ∞, vc(∞) = 3.33 V Therefore, k1=3.33, k2 = 1.66 We can find i(t) by using Ohm’s law on the resistor

  19. i(t) 1.66mA t vc(t) 5V 3.33V t

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