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Transcript
Transient Analysis DC Steady-State ELEC 308 Elements of Electrical Engineering Dr. Ron Hayne Images Courtesy of Allan Hambley and Prentice-Hall Transient Analysis Scope of study: Circuits that contain sources, switches, resistances, inductances, and capacitances Transients: Time-varying currents and voltages resulting from sudden application of sources, usually due to switching Transient Analysis: Involves using circuit concepts from Chapters 1 & 2 Current-voltage relationships for inductances and capacitances involve derivatives and integrals Create circuit equations that are differential equations ELEC 308 2 First-Order RC or RL Circuits First-Order RC Circuits Contains DC sources, resistances, and a SINGLE capacitance First-Order RL Circuits Contains DC sources, resistances, and a SINGLE inductance Frequently used in timing applications Due to time constant ELEC 308 3 First-Order Circuit Algorithm 1. Apply KCL, KVL, and/or Ohm’s Law to write the circuit equation. 2. If equation contains integrals, differentiate each term to produce a PURE differential equation. 3. Assume a solution of the form K1+K2est. 4. Substitute the solution into the diff. eqn. to determine the values of K1 and s. 5. Use the initial conditions to determine the value of K2. 6. Write the final solution. ELEC 308 4 Discharge of a Capacitance ELEC 308 5 Discharge of a Capacitance Writing a KCL equation at top node after switch closed, we have : dvC t vC t C 0 dt R Multiplyin g by the resistance R : dvC t RC vC t 0 dt General form of solution is given by vC t Ke st ELEC 308 6 Discharge of a Capacitance Substitute our general solution : RCKse st Ke st 0 Solving for s : 1 RCs 1 s RC ELEC 308 7 Discharge of a Capacitance Use our initial conditions to solve for K : vC 0 Vi Ke s 0 K Vi Substituin g this in our solution gives us our complete solution : vC t Vi e t RC ELEC 308 8 Time Constant Time interval τ = RC is called the time constant of the circuit After t=5τ, vC(t)≈0 ELEC 308 9 Charging a Capacitance ELEC 308 10 Charging a Capacitance Writing KCL equation at node between resistance & capacitanc e, dvC t vC t Vs C 0 dt R Rearrangin g, dvC t RC vC t Vs dt General form of solution is given by vC t K1 K 2 e st ELEC 308 11 Charging a Capacitance Substitute our general solution and solve for s and K1 : 1 s RC and K1 Vs ELEC 308 12 Charging a Capacitance Use our initial conditions to solve for K 2 : vC 0 0 Vs K 2 e s 0 K 2 Vs Thus the complete solution : vC t Vs Vs e t RC Vs Vs e t First term is STEADY-STATE RESPONSE Or FORCED RESPONSE Second term is TRANSIENT RESPONSE ELEC 308 13 Charging a Capacitance ELEC 308 14 DC Steady State Transient terms in the expressions for current and voltages in RLC circuits decay to zero with time For DC sources, steady-state currents and voltages are CONSTANT For steady-state conditions with DC sources: CAPACITANCES behave like OPEN circuits INDUCTANCES behave like SHORT circuits ELEC 308 15 Capacitance in DC Steady-State Remember current through a capacitance: dvC t iC t C dt If voltage is constant, current is _________. CAPACITANCE behaves just like an ____ circuit ELEC 308 16 Inductance in DC Steady-State Remember voltage across an inductance: diL t v L t L dt If current is constant, voltage is _________. INDUCTANCE behaves just like a ______ circuit ELEC 308 17 Steady-State DC Analysis Find vx and ix for t >> 0 ELEC 308 18 Exercise 4.3 Find va and ia for t >> 0 ELEC 308 19 RL Transient Analysis Find i(t) and v(t) ELEC 308 20 RL Transient Analysis Time interval τ = L/R is called the time constant of the circuit After t=5τ, i (t)≈2 ELEC 308 21 Example 4.3 Find i(t) and v(t) ELEC 308 22 Example 4.3 ELEC 308 23 Exercise 4.6 Find i(t) and v(t) ELEC 308 24 Summary Transient Analysis First Order RC Circuits First Order RL Circuits DC Steady State ELEC 308 25