ADNAN MENDERES UNIVERSITY FACULTY OF ENGINEERING Department of

ADNAN MENDERES UNIVERSITY FACULTY OF ENGINEERING Department of Electrical and Electronics Engineering EE 201 – Circuit Theory I 2018 – 2019 Fall Assoc. Prof. Dr. Olcay ÜZENGİ AKTÜRK W-XII CH-7

Ch 7: First Order Circuits The Natural Response of An RL Circuit (Source-Free RL Circuit) Ø A source-free RL circuit occurs when its dc source is suddenly disconnected. Ø The energy already stored in the inductor/capacitor is released to the resistors. EE 201 -Circuit Theory I, Assoc. Prof. Dr. Olcay ÜZENGİ AKTÜRK, 2018 -2019 Fall

Ch 7: First Order Circuits The Natural Response of An RL Circuit (Source-Free RL Circuit) Ø Ø Assume that the independent current source generates a constant current of A, and that the switch has been in a closed position for a long time. Long time means that all currents and voltages have reached a constant value. Ø Thus only constant, or dc, currents can exist in the circuit just prior to the switch's being opened, and therefore the inductor appears as a short circuit (Ldi/dt = 0) prior to the release of the stored energy. EE 201 -Circuit Theory I, Assoc. Prof. Dr. Olcay ÜZENGİ AKTÜRK, 2018 -2019 Fall

Ch 7: First Order Circuits The Natural Response of An RL Circuit (Source-Free RL Circuit) Ø Assume that the independent current source generates a constant current of A, and that the switch has been in a closed position for a long time. EE 201 -Circuit Theory I, Assoc. Prof. Dr. Olcay ÜZENGİ AKTÜRK, 2018 -2019 Fall

Ch 7: First Order Circuits The Natural Response of An RL Circuit (Source-Free RL Circuit) Ø If at t=0, the switch is opened Ø Now the problem becomes of finding v(t) and i(t) for t 0 EE 201 -Circuit Theory I, Assoc. Prof. Dr. Olcay ÜZENGİ AKTÜRK, 2018 -2019 Fall

Ch 7: First Order Circuits The Natural Response of An RL Circuit (Source-Free RL Circuit) Ø by using KVL Ø This equation is known as a first order ordinary differential equation. The highest order derivative appearing in the equation is 1; hence the term firstorder EE 201 -Circuit Theory I, Assoc. Prof. Dr. Olcay ÜZENGİ AKTÜRK, 2018 -2019 Fall

Ch 7: First Order Circuits The Natural Response of An RL Circuit (Source-Free RL Circuit) Ø We know that an instantaneous change of current cannot occur in an inductor. Ø The current starts from an initial value I(0) and decreases exponentially toward zero as t increases Ø The coefficient of t—namely, R/L—determines the rate at which the current (or voltage) approaches zero. The reciprocal of this ratio is the time constant of the circuit denoted by . EE 201 -Circuit Theory I, Assoc. Prof. Dr. Olcay ÜZENGİ AKTÜRK, 2018 -2019 Fall

Ch 7: First Order Circuits The Natural Response of An RL Circuit (Source-Free RL Circuit) (tangent at t=0) EE 201 -Circuit Theory I, Assoc. Prof. Dr. Olcay ÜZENGİ AKTÜRK, 2018 -2019 Fall

Ch 7: First Order Circuits The Natural Response of An RL Circuit (Source-Free RL Circuit) Ø There will be a jump in voltage at t=0, so v(0) is NOT DEFINED!! DEFINED EE 201 -Circuit Theory I, Assoc. Prof. Dr. Olcay ÜZENGİ AKTÜRK, 2018 -2019 Fall

Ch 7: First Order Circuits The Natural Response of An RL Circuit (Source-Free RL Circuit) Ø Power & Energy Ø The energy delivered to the resistor during any interval of time after the switch has been opened is; EE 201 -Circuit Theory I, Assoc. Prof. Dr. Olcay ÜZENGİ AKTÜRK, 2018 -2019 Fall

Ch 7: First Order Circuits The Natural Response of An RL Circuit (Source-Free RL Circuit) Ø Example: 7. 1(from textbook) The switch in the circuit has been closed for a long time before it is opened at t = 0. Find Also find the percentage of the total energy stored in the 2 H inductor that is dissipated in the 10 Ω resistor. EE 201 -Circuit Theory I, Assoc. Prof. Dr. Olcay ÜZENGİ AKTÜRK, 2018 -2019 Fall

Ch 7: First Order Circuits The Natural Response of An RL Circuit (Source-Free RL Circuit) Example: The switch in the circuit has been closed for a long time. At t = 0, the switch is opened. Calculate i(t) for t > 0. EE 201 -Circuit Theory I, Assoc. Prof. Dr. Olcay ÜZENGİ AKTÜRK, 2018 -2019 Fall

Ch 7: First Order Circuits The Natural Response of An RL Circuit (Source-Free RL Circuit) Example: Assuming that i(0) = 10 A, calculate i(t) and ix (t) in the circuit. EE 201 -Circuit Theory I, Assoc. Prof. Dr. Olcay ÜZENGİ AKTÜRK, 2018 -2019 Fall

Ch 7: First Order Circuits The Natural Response of An RC Circuit (Source-Free RC Circuit) Ø The natural response of an RC circuit is analogous to that of an RL circuit. Ø Assume that the switch has been in position “a“ for a long time, and the capacitor C to reach a steady-state condition. Ø We know that a capacitor behaves as an open circuit in the presence of a constant voltage. (Cdv/dt = 0) Ø The important point is that when the switch is moved from position a to position b (at t = 0), the voltage on the capacitor is Vg. EE 201 -Circuit Theory I, Assoc. Prof. Dr. Olcay ÜZENGİ AKTÜRK, 2018 -2019 Fall

Ch 7: First Order Circuits The Natural Response of An RC Circuit (Source-Free RC Circuit) Ø Therefore the problem reduces to solving the circuit shown in following figure EE 201 -Circuit Theory I, Assoc. Prof. Dr. Olcay ÜZENGİ AKTÜRK, 2018 -2019 Fall

Ch 7: First Order Circuits The Natural Response of An RC Circuit (Source-Free RC Circuit) ² By using KCL EE 201 -Circuit Theory I, Assoc. Prof. Dr. Olcay ÜZENGİ AKTÜRK, 2018 -2019 Fall

Ch 7: First Order Circuits The Natural Response of An RC Circuit (Source-Free RC Circuit) Ø We know that an instantaneous change of voltagecannot occur in a capacitor. and Ø The voltage starts from an initial value v(0) and decreases exponentially toward zero as t increases Ø (tangent at t=0) The coefficient of t—namely, RC—determines the rate at which the oltage (or current) approaches zero. The reciprocal of this ratio is the time constant of the circuit denoted by . EE 201 -Circuit Theory I, Assoc. Prof. Dr. Olcay ÜZENGİ AKTÜRK, 2018 -2019 Fall

Ch 7: The Natural Response of An RC Circuit (Source-Free RC Circuit) EE 201 -Circuit Theory I, Assoc. Prof. Dr. Olcay ÜZENGİ AKTÜRK, 2018 -2019 Fall

Ch 7: First Order Circuits The Natural Response of An RC Circuit (Source-Free RL Circuit) Example: The switch in the circuit has been in position x for a long time. At t=0, the switch moves instantaneously to position b. Find, Also find the total energy dissipated in the 60 kΩ resistor. EE 201 -Circuit Theory I, Assoc. Prof. Dr. Olcay ÜZENGİ AKTÜRK, 2018 -2019 Fall

Ch 7: First Order Circuits Example: Let vc(0) = 15 V. Find vc(t), vx(t) , and ix(t) for t > 0 EE 201 -Circuit Theory I, Assoc. Prof. Dr. Olcay ÜZENGİ AKTÜRK, 2018 -2019 Fall

Ch 7: First Order Circuits Example: If the switch opens at t = 0, find v(t) for t ≥ 0 and wc(0). EE 201 -Circuit Theory I, Assoc. Prof. Dr. Olcay ÜZENGİ AKTÜRK, 2018 -2019 Fall

Ch 7: First Order Circuits Example: The switch in circuit in Fig. has been closed for a long time, and it is opened at t=0. Find v(t) for. Calculate the initial energy stored in the capacitor. EE 201 -Circuit Theory I, Assoc. Prof. Dr. Olcay ÜZENGİ AKTÜRK, 2018 -2019 Fall

Ch 7: First Order Circuits The Step Response of RL and RC Circuits Ø We are now ready to discuss the problem of finding the currents and voltages generated in first-order RL or RC circuits when either dc voltage or current sources are suddenly applied The response of a circuit to the sudden application of a constant voltage or current source is referred to as the step response of the circuit. EE 201 -Circuit Theory I, Assoc. Prof. Dr. Olcay ÜZENGİ AKTÜRK, 2018 -2019 Fall

Ch 7: First Order Circuits The Step Response of RL and RC Circuits Ø Energy stored in the inductor at the time the switch is closed is given in terms of a nonzero initial current i(0). Ø The task is to find the expressions for the current in the circuit and for the voltage across the inductor after the switch has been closed. Ø We use circuit analysis to derive the differential equation that describes the circuit in terms of the variable of interest, and then we use elementary calculus to solve the equation. EE 201 -Circuit Theory I, Assoc. Prof. Dr. Olcay ÜZENGİ AKTÜRK, 2018 -2019 Fall

Ch 7: First Order Circuits The Step Response of RL and RC Circuits Ø After the switch has been closed, KVL requires that; Using textbook EE 201 -Circuit Theory I, Assoc. Prof. Dr. Olcay ÜZENGİ AKTÜRK, 2018 -2019 Fall

Ch 7: First Order Circuits The Step Response of RL and RC Circuits Ø After these calculations, the step response of an RL circuit can be obtained as; Ø When the initial energy in the inductor is zero, EE 201 -Circuit Theory I, Assoc. Prof. Dr. Olcay ÜZENGİ AKTÜRK, 2018 -2019 Fall

Ch 7: First Order Circuits The Step Response of an RL Circuit EE 201 -Circuit Theory I, Assoc. Prof. Dr. Olcay ÜZENGİ AKTÜRK, 2018 -2019 Fall

Ch 7: First Order Circuits The Step Response of an RL Circuit Ø The voltage across an inductor is L di/dt, hence, for t>0, Ø If the initial current is zero, EE 201 -Circuit Theory I, Assoc. Prof. Dr. Olcay ÜZENGİ AKTÜRK, 2018 -2019 Fall

Ch 7: First Order Circuits Example : (7. 5 from textbook) The switch in the circuit has been in position a for a long time. At t = 0, the switch moves from position a to position b. The switch is a make-before-break type; that is, the connection at position b is established before the connection at position a is broken, so there is no interruption of current through the inductor. EE 201 -Circuit Theory I, Assoc. Prof. Dr. Olcay ÜZENGİ AKTÜRK, 2018 -2019 Fall

Solution

Ch 7: example Find i(t) in the circuit for t > 0. Assume that the switch has been closed for a long time. Answer: EE 201 -Circuit Theory I, Assoc. Prof. Dr. Olcay ÜZENGİ AKTÜRK, 2018 -2019 Fall