Circuits and Analog Electronics Ch 3 Basic RL

Circuits and Analog Electronics Ch 3 Basic RL and RC Circuits 3. 1 First-Order RC Circuits 3. 2 First-Order RL Circuits 3. 3 Examples References: References Hayt-Ch 5, 6; Gao-Ch 5;

Ch 3 Basic RL and RC Circuits 3. 1 First-Order RC Circuits Key Words: Words Transient Response of RC Circuits, Time constant

Ch 3 Basic RL and RC Circuits 3. 1 First-Order RC Circuits • Used for filtering signal by blocking certain frequencies and passing others. e. g. low-pass filter • 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 1 st order differential equation. Ideal Linear Capacitor Energy stored A capacitor is an energy storage device memory device.

Ch 3 Basic RL and RC Circuits 3. 1 First-Order RC Circuits + vs(t) + - vr(t) - R + vc(t) C - • One capacitor and one resistor • The source and resistor may be equivalent to a circuit with many resistors and sources.

Ch 3 Basic RL and RC Circuits 3. 1 First-Order RC Circuits Transient Response of RC Circuits Switch is thrown to 1 KVL around the loop: Initial condition Called time constant

Ch 3 Basic RL and RC Circuits 3. 1 First-Order RC Circuits Time Constant R=2 k C=0. 1 F RC

Ch 3 Basic RL and RC Circuits 3. 1 First-Order RC Circuits Transient Response of RC Circuits Switch to 2 Initial condition

Ch 3 Basic RL and RC Circuits 3. 1 First-Order RC Circuits Time Constant R=2 k C=0. 1 F

Ch 3 Basic RL and RC Circuits 3. 1 First-Order RC Circuits

Ch 3 Basic RL and RC Circuits 3. 2 First-Order RL Circuits Key Words: Words Transient Response of RL Circuits, Time constant

Ch 3 Basic RL and RC Circuits 3. 2 First-Order RL Circuits Ideal Linear Inductor The rest of the circuit i(t) + L v(t) Energy stored: • One inductor and one resistor • The source and resistor may be equivalent to a circuit with many resistors and sources.

Ch 3 Basic RL and RC Circuits 3. 2 First-Order RL Circuits Transient Response of RL Circuits Switch to 1 KVL around the loop: Initial condition Called time constant

Ch 3 Basic RL and RC Circuits 3. 2 First-Order RL Circuits Time constant . i (t) 0 t • Indicate new fast i (t) will drop to zero precisely. • It is the amount of time for i (t) to drop zero if it is dropping at the initial rate.

Ch 3 Basic RL and RC Circuits 3. 2 First-Order RL Circuits Transient Response of RL Circuits Switch to 2 Initial condition

Ch 3 Basic RL and RC Circuits 3. 2 First-Order RL Circuits Transient Response of RL Circuits Input energy to L L export its energy and it is dissipated by R

Ch 3 Basic RL and RC Circuits Summary Initial Value （t=0） RL Circuits Source (0 state) Sourcefree (0 input) RC Circuits Source (0 state) Sourcefree (0 input) Steady Value （ t ） time constant

Ch 3 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

Ch 3 Basic RL and RC Circuits Summary • How to determine initial conditions for a transient circuit. When a sudden change occurs, only two types of quantities will remain the same as before the change. – IL(t), inductor current – Vc(t), capacitor voltage • Find these two types of the values before the change and use them as the initial conditions of the circuit after change.

Ch 3 Basic RL and RC Circuits 3. 3 Examples About Calculation for The Initial Value i i. C i. L t=0 i(0+) i. C(0+) + _ i. L(0+) + v. C(0+)=4 V v. L(0+) - 1 A

Ch 3 Basic RL and RC Circuits 3. 3 Examples (Analyzing an RC circuit or RL circuit) Method 1 1) Thévenin Equivalent. (Draw out C or L) Simplify the circuit Veq , Req 2) Find Leq(Ceq), and = Leq/Req ( = Ceq. Req) 3) Substituting Leq(Ceq) and to the previous solution of differential equation for RC (RL) circuit.

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

Ch 3 Basic RL and RC Circuits 3. 3 Examples (Analyzing an RC circuit or RL circuit) Method 3 (step-by-step) Given f（0＋），thus In general, Initial Steady 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:

Ch 3 Basic RL and RC Circuits 3. 3 Examples P 3. 1 v. C(0) = 0, Find v. C(t) for t 0. Method 3: Apply Thevenin theoren :

Ch 3 Basic RL and RC Circuits 3. 3 Examples P 3. 2 v. C(0)=0, Find vo, v. C(t) for t 0. Apply Thevenin theoren :