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RC Circuits

RC Circuits. circuits in which the currents vary in time rate of charging a cap depends on C and R of circuit differential equations. I. Given: R, C, q o (initial charge) Find: q(t) and I(t) when switch is closed. q. C. R. -q. 1) . (Kirchhoff’s Loop Rule). 2).

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RC Circuits

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  1. RC Circuits • circuits in which the currents vary in time • rate of charging a cap depends on C and R of circuit • differential equations

  2. I Given: R, C, qo (initial charge) Find: q(t) and I(t) when switch is closed q C R -q 1) (Kirchhoff’s Loop Rule) 2) Discharging a Capacitor (- sign because q decreases for I > 0 That is, current in circuit equals the decrease of charge on the capacitor)

  3. Combine 1) and 2) to get: I where: q = q(t) q(0) = qo q C R -q This is a differential equation for the function q(t), subject to the initial condition q(0) = q0 . We are looking for a function which is proportional to its own first derivative (since dq/dt ~ -q).

  4. Solution: RC is called the “time constant” or “characteristic time” of the circuit. Units: 1 Ω x 1 F = 1 second (show this!) Write t (“tau”) = RC, then: (discharging)

  5. q Discharging qo t 2 3 t = , q ≈ 0.37 qo = (qo/e) t = 2 , q ≈ 0.14 qo = (qo/e2) t = 3 , q ≈ 0.05 qo = (qo/e3) t  ∞ , q  0 = (qo/e∞)

  6. Draw a graph for I(t).

  7. Example 1 • A capacitor is charged up to 18 volts, and then connected across a resistor. After 10 seconds, the capacitor voltage has fallen to 12 volts. • What is the time constant RC ? • What will the voltage be after another 10 seconds (20 seconds total)? • 8V • 6V • 4V • 0

  8. Then, where t =RC. Charging a capacitor C is initially uncharged, and the switch is closed at t=0. After a long time, the capacitor has charge Qf . C R Question: What is Qf equal to? What is V(t) ? Recall, C=Q/V, so V(t)=Q(t)/C

  9. Charging a capacitor q Qf t 2 3 t = 0, q=0 t = 3 RC, q 0.95 Qf t = RC, q 0.63 Qf etc. t = 2 RC, q 0.86 Qf

  10. Draw a graph of I(t). Why is I=+dq/dt this time?

  11. Example 2 100 kΩ The capacitor is initially uncharged. After the switch is closed, find: 12 V 2 µF • Initial current • Initial voltage across the resistor • Initial voltage across the capacitor • Time for voltage across C to reach 0.63*12V • Final voltage across the resistor • Final voltage across the capacitor

  12. Solution

  13. Example 3 The circuit below contains two resistors, R1 = 2.00 kΩ and R2 = 3.00 kΩ, and two capacitors, C1 = 2.00 μF and C2 = 3.00 μF, connected to a battery with emf ε = 120 V. No charge is on either capacitor before switch S is closed. Determine the charges q1 and q2 on capacitors C1 and C2, respectively, after the switch is closed.

  14. Solution:1) reconstruct the circuit so that it becomes a simple RC circuit containing a single resistor and single capacitor 2) determine the total charge q stored in the equivalent circuit.

  15. “RC” Circuits • a capacitor takes time to charge or discharge through a resistor • “time constant” or “characteristic time” • = RC • (1 ohm) x (1 farad) = 1 second

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