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 3082

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 3083

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 K 1 +K 2 e st. 4.Substitute the solution into the diff. eqn. to determine the values of K 1 and s. 5.Use the initial conditions to determine the value of K 2. 6.Write the final solution. ELEC 3084

Discharge of a Capacitance ELEC 3085

Discharge of a Capacitance ELEC 3086

Discharge of a Capacitance ELEC 3087

Discharge of a Capacitance ELEC 3088

Time Constant  Time interval τ = RC is called the time constant of the circuit  After t=5τ, v C (t)≈0 ELEC 3089

Charging a Capacitance ELEC 30810

Charging a Capacitance ELEC 30811

Charging a Capacitance ELEC 30812

Charging a Capacitance  First term is STEADY-STATE RESPONSE Or FORCED RESPONSE  Second term is TRANSIENT RESPONSE ELEC 30813

Charging a Capacitance ELEC 30814

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 30815

Capacitance in DC Steady-State  Remember current through a capacitance:  If voltage is constant, current is _________.  CAPACITANCE behaves just like an ____ circuit ELEC 30816

Inductance in DC Steady-State  Remember voltage across an inductance:  If current is constant, voltage is _________.  INDUCTANCE behaves just like a ______ circuit ELEC 30817

Steady-State DC Analysis  Find v x and i x for t >> 0 ELEC 30818

Exercise 4.3  Find v a and i a for t >> 0 ELEC 30819

RL Transient Analysis  Find i(t) and v(t) ELEC 30820

RL Transient Analysis  Time interval τ = L/R is called the time constant of the circuit  After t=5τ, i (t)≈2 ELEC 30821

Example 4.3  Find i(t) and v(t) ELEC 30822

Example 4.3 ELEC 30823

Exercise 4.6  Find i(t) and v(t) ELEC 30824

Summary  Transient Analysis First Order RC Circuits First Order RL Circuits  DC Steady State ELEC 30825