Ch3 Basic RL and RC Circuits 3.1 First-Order RC Circuits 3.2 First-Order RL Circuits 3.3 Exemples Readings Readings: Gao-Ch5; Hayt-Ch5, 6 Circuits and Analog Electronics

Ch3 Basic RL and RC Circuits 3.1 First-Order RC Circuits Key Words Key Words: Transient Response of RC Circuits Time constant

Ch3 Basic RL and RC Circuits 3.1 First-Order RC Circuits Any circuit with a single energy storage element, an arbitrary number of sources, and an arbitrary number of resistors is a circuit of order 1. Any voltage or current in such a circuit is the solution to a 1st order differential equation. Ideal Linear Capacitor Energy stored A capacitor is an energy storage device  memory device.

Ch3 Basic RL and RC Circuits 3.1 First-Order RC Circuits One capacitor and one resistor The source and resistor may be equivalent to a circuit with many resistors and sources. R + - Cv s (t) + - v c (t) +- v r (t)

Ch3 Basic RL and RC Circuits 3.1 First-Order RC Circuits KVL around the loop: Initial condition Switch is thrown to 1 Called time constant Transient Response of RC Circuits

Ch3 Basic RL and RC Circuits 3.1 First-Order RC Circuits Time Constant RC R=2k C=0.1  F

Ch3 Basic RL and RC Circuits 3.1 First-Order RC Circuits Switch is thrown to 2 Initial condition Transient Response of RC Circuits

Ch3 Basic RL and RC Circuits 3.1 First-Order RC Circuits Time Constant R=2k C=0.1  F

Ch3 Basic RL and RC Circuits 3.1 First-Order RC Circuits

Ch3 Basic RL and RC Circuits 3.2 First-Order RL Circuits Key Words Key Words: Transient Response of RL Circuits Time constant

Ch3 Basic RL and RC Circuits 3.2 First-Order RL Circuits Ideal Linear Inductor i(t) + - v(t) The rest of the circuit L Energy stored: One inductor and one resistor The source and resistor may be equivalent to a circuit with many resistors and sources.

Ch3 Basic RL and RC Circuits 3.2 First-Order RL Circuits Switch is thrown to 1 KVL around the loop: Initial condition Called time constant Transient Response of RL Circuits

Ch3 Basic RL and RC Circuits 3.2 First-Order RL Circuits Switch is thrown to 2 Initial condition Transient Response of RL Circuits

Ch3 Basic RL and RC Circuits 3.2 First-Order RL Circuits Transient Response of RL Circuits

Ch3 Basic RL and RC Circuits Initial Value （ t=0 ） Steady Value （ t  ） time constant  RL Circuits Source (0 state) Source- free (0 input) RC Circuits Source (0 state) Source- free (0 input) Summary

Ch3 Basic RL and RC Circuits Summary The Time Constant For an RC circuit,  = RC For an RL circuit,  = L/R -1/  is the initial slope of an exponential with an initial value of 1 Also,  is the amount of time necessary for an exponential to decay to 36.7% of its initial value

Ch3 Basic RL and RC Circuits Summary How to determine initial conditions for a transient circuit. When a sudden change occurs, only two types quantities will definitely remain the same before and after the change. –I L (t), inductor current –Vc(t), capacitor voltage Find these two types of the values before the change and use them as initial conditions of the circuit after change.

Ch3 Basic RL and RC Circuits About Calculation for The Initial Value iCiC iLiL ii t=0 + _ 1A  + - v L(0+) v C(0+) =4V i (0+) i C(0+) i L(0+) 3.3 Exemples

Ch3 Basic RL and RC Circuits 3.3 Exemples Method 1 (Analyzing an RC circuit or RL circuit) Simplify the circuit 2) Find L eq (C eq ), and  =L eq /R eq (  =C eq R eq ) 1) Thévenin Equivalent.(Draw out C or L) V eq, R eq 3) Substituting Leq(Ceq), and  to previous solution of differential equation for RC (RL)circuit.

Ch3 Basic RL and RC Circuits 3.3 Exemples Method 2 (Analyzing an RC circuit or RL circuit) 3) Find the particular solution. 1) KVL around the loop,  the differential equation 4) The total solution is the sum of the particular and homogeneous solutions. 2) Find the homogeneous solution.

3.3 Exemples Method 3 (step-by-step) (Analyzing an RC circuit or RL circuit) 1) Draw the circuit for t=0 - and find v(0 - ), or i(0 - ) 2) Use the continuity of the capacitor voltage, or inductor current, draw the circuit for t=0 + to find v(0 + ), or i(0 + ) 3) Find v(  ), or i(  ) at steady state 4) Find the time constant  –For an RC circuit,  = RC –For an RL circuit,  = L/R 5) The solution is: Given f （ 0 ＋）， thus Initial Steady In general, Ch3 Basic RL and RC Circuits

3.3 Exemples P3.1 v C(0) =0, Find v C(t) for t  0. P3.2 v C(0) =0, Find v o, v C(t) for t  0.