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Transient Analysis DC Steady-State ELEC 308 Elements of Electrical Engineering Dr. Ron Hayne Images Courtesy of Allan Hambley and Prentice-Hall

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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

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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

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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

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Discharge of a Capacitance ELEC 3085

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Discharge of a Capacitance ELEC 3086

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Discharge of a Capacitance ELEC 3087

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Discharge of a Capacitance ELEC 3088

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

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Charging a Capacitance ELEC 30810

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Charging a Capacitance ELEC 30811

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Charging a Capacitance ELEC 30812

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

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Charging a Capacitance ELEC 30814

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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

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

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

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

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

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

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

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

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Example 4.3 ELEC 30823

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

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

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