Response of FirstOrder Circuits RL Circuits RC Circuits
![Response of First-Order Circuits RL Circuits RC Circuits ECE 201 Circuit Theory I 1 Response of First-Order Circuits RL Circuits RC Circuits ECE 201 Circuit Theory I 1](https://slidetodoc.com/presentation_image_h/adfca17fa27cc2b99b95d23904b82f5e/image-1.jpg)
![The Natural Response of a Circuit • The currents and voltages that arise when The Natural Response of a Circuit • The currents and voltages that arise when](https://slidetodoc.com/presentation_image_h/adfca17fa27cc2b99b95d23904b82f5e/image-2.jpg)
![Step Response • The sudden application of a DC voltage or current source is Step Response • The sudden application of a DC voltage or current source is](https://slidetodoc.com/presentation_image_h/adfca17fa27cc2b99b95d23904b82f5e/image-3.jpg)
![Circuits for Natural Response • Energy is “stored” in an inductor (a) as an Circuits for Natural Response • Energy is “stored” in an inductor (a) as an](https://slidetodoc.com/presentation_image_h/adfca17fa27cc2b99b95d23904b82f5e/image-4.jpg)
![General Configurations for RL • If the independent sources are equal to zero, the General Configurations for RL • If the independent sources are equal to zero, the](https://slidetodoc.com/presentation_image_h/adfca17fa27cc2b99b95d23904b82f5e/image-5.jpg)
![Natural Response of an RL Circuit • Consider the circuit shown. • Assume that Natural Response of an RL Circuit • Consider the circuit shown. • Assume that](https://slidetodoc.com/presentation_image_h/adfca17fa27cc2b99b95d23904b82f5e/image-6.jpg)
![What does “for a long time” Mean? • All of the currents and voltages What does “for a long time” Mean? • All of the currents and voltages](https://slidetodoc.com/presentation_image_h/adfca17fa27cc2b99b95d23904b82f5e/image-7.jpg)
![Just before t = 0 • The voltage across the inductor is equal to Just before t = 0 • The voltage across the inductor is equal to](https://slidetodoc.com/presentation_image_h/adfca17fa27cc2b99b95d23904b82f5e/image-8.jpg)
![Just after t = 0 • The current source and its parallel resistor R Just after t = 0 • The current source and its parallel resistor R](https://slidetodoc.com/presentation_image_h/adfca17fa27cc2b99b95d23904b82f5e/image-9.jpg)
![The expression for the current ECE 201 Circuit Theory I 10 The expression for the current ECE 201 Circuit Theory I 10](https://slidetodoc.com/presentation_image_h/adfca17fa27cc2b99b95d23904b82f5e/image-10.jpg)
![A first-order ordinary differential equation with constant coefficients. How do we solve it? ECE A first-order ordinary differential equation with constant coefficients. How do we solve it? ECE](https://slidetodoc.com/presentation_image_h/adfca17fa27cc2b99b95d23904b82f5e/image-11.jpg)
![ECE 201 Circuit Theory I 12 ECE 201 Circuit Theory I 12](https://slidetodoc.com/presentation_image_h/adfca17fa27cc2b99b95d23904b82f5e/image-12.jpg)
![The current in an inductor cannot change instantaneously • Let the time just before The current in an inductor cannot change instantaneously • Let the time just before](https://slidetodoc.com/presentation_image_h/adfca17fa27cc2b99b95d23904b82f5e/image-13.jpg)
![The Complete Solution ECE 201 Circuit Theory I 14 The Complete Solution ECE 201 Circuit Theory I 14](https://slidetodoc.com/presentation_image_h/adfca17fa27cc2b99b95d23904b82f5e/image-14.jpg)
![The voltage drop across the resistor ECE 201 Circuit Theory I 15 The voltage drop across the resistor ECE 201 Circuit Theory I 15](https://slidetodoc.com/presentation_image_h/adfca17fa27cc2b99b95d23904b82f5e/image-15.jpg)
![The Power Dissipated in the Resistor ECE 201 Circuit Theory I 16 The Power Dissipated in the Resistor ECE 201 Circuit Theory I 16](https://slidetodoc.com/presentation_image_h/adfca17fa27cc2b99b95d23904b82f5e/image-16.jpg)
![The Energy Delivered to the Resistor ECE 201 Circuit Theory I 17 The Energy Delivered to the Resistor ECE 201 Circuit Theory I 17](https://slidetodoc.com/presentation_image_h/adfca17fa27cc2b99b95d23904b82f5e/image-17.jpg)
![Time Constant • The rate at which the current or voltage approaches zero. ECE Time Constant • The rate at which the current or voltage approaches zero. ECE](https://slidetodoc.com/presentation_image_h/adfca17fa27cc2b99b95d23904b82f5e/image-18.jpg)
![Rewriting in terms of Time Constant ECE 201 Circuit Theory I 19 Rewriting in terms of Time Constant ECE 201 Circuit Theory I 19](https://slidetodoc.com/presentation_image_h/adfca17fa27cc2b99b95d23904b82f5e/image-19.jpg)
![Table 7. 1 page 217 of the text ECE 201 Circuit Theory I 20 Table 7. 1 page 217 of the text ECE 201 Circuit Theory I 20](https://slidetodoc.com/presentation_image_h/adfca17fa27cc2b99b95d23904b82f5e/image-20.jpg)
![Graphical Interpretation of Time Constant • Determine the time constant from the plot of Graphical Interpretation of Time Constant • Determine the time constant from the plot of](https://slidetodoc.com/presentation_image_h/adfca17fa27cc2b99b95d23904b82f5e/image-21.jpg)
![Graphical Interpretation Tangent at t = 0 intersects the time axis at the time Graphical Interpretation Tangent at t = 0 intersects the time axis at the time](https://slidetodoc.com/presentation_image_h/adfca17fa27cc2b99b95d23904b82f5e/image-22.jpg)
![Procedure to Determine the Natural Response of an RL Circuit • Find the initial Procedure to Determine the Natural Response of an RL Circuit • Find the initial](https://slidetodoc.com/presentation_image_h/adfca17fa27cc2b99b95d23904b82f5e/image-23.jpg)
- Slides: 23
![Response of FirstOrder Circuits RL Circuits RC Circuits ECE 201 Circuit Theory I 1 Response of First-Order Circuits RL Circuits RC Circuits ECE 201 Circuit Theory I 1](https://slidetodoc.com/presentation_image_h/adfca17fa27cc2b99b95d23904b82f5e/image-1.jpg)
Response of First-Order Circuits RL Circuits RC Circuits ECE 201 Circuit Theory I 1
![The Natural Response of a Circuit The currents and voltages that arise when The Natural Response of a Circuit • The currents and voltages that arise when](https://slidetodoc.com/presentation_image_h/adfca17fa27cc2b99b95d23904b82f5e/image-2.jpg)
The Natural Response of a Circuit • The currents and voltages that arise when energy stored in an inductor or capacitor is suddenly released into a resistive circuit. • These “signals” are determined by the circuit itself, not by external sources! ECE 201 Circuit Theory I 2
![Step Response The sudden application of a DC voltage or current source is Step Response • The sudden application of a DC voltage or current source is](https://slidetodoc.com/presentation_image_h/adfca17fa27cc2b99b95d23904b82f5e/image-3.jpg)
Step Response • The sudden application of a DC voltage or current source is referred to as a “step”. • The step response consists of the voltages and currents that arise when energy is being absorbed by an inductor or capacitor. ECE 201 Circuit Theory I 3
![Circuits for Natural Response Energy is stored in an inductor a as an Circuits for Natural Response • Energy is “stored” in an inductor (a) as an](https://slidetodoc.com/presentation_image_h/adfca17fa27cc2b99b95d23904b82f5e/image-4.jpg)
Circuits for Natural Response • Energy is “stored” in an inductor (a) as an initial current. • Energy is “stored” in a capacitor (b) as an initial voltage. ECE 201 Circuit Theory I 4
![General Configurations for RL If the independent sources are equal to zero the General Configurations for RL • If the independent sources are equal to zero, the](https://slidetodoc.com/presentation_image_h/adfca17fa27cc2b99b95d23904b82f5e/image-5.jpg)
General Configurations for RL • If the independent sources are equal to zero, the circuits simplify to ECE 201 Circuit Theory I 5
![Natural Response of an RL Circuit Consider the circuit shown Assume that Natural Response of an RL Circuit • Consider the circuit shown. • Assume that](https://slidetodoc.com/presentation_image_h/adfca17fa27cc2b99b95d23904b82f5e/image-6.jpg)
Natural Response of an RL Circuit • Consider the circuit shown. • Assume that the switch has been closed “for a long time”, and is “opened” at t=0. ECE 201 Circuit Theory I 6
![What does for a long time Mean All of the currents and voltages What does “for a long time” Mean? • All of the currents and voltages](https://slidetodoc.com/presentation_image_h/adfca17fa27cc2b99b95d23904b82f5e/image-7.jpg)
What does “for a long time” Mean? • All of the currents and voltages have reached a constant (dc) value. • What is the voltage across the inductor just before the switch is opened? ECE 201 Circuit Theory I 7
![Just before t 0 The voltage across the inductor is equal to Just before t = 0 • The voltage across the inductor is equal to](https://slidetodoc.com/presentation_image_h/adfca17fa27cc2b99b95d23904b82f5e/image-8.jpg)
Just before t = 0 • The voltage across the inductor is equal to zero. • There is no current in either resistor. • The current in the inductor is equal to IS. ECE 201 Circuit Theory I 8
![Just after t 0 The current source and its parallel resistor R Just after t = 0 • The current source and its parallel resistor R](https://slidetodoc.com/presentation_image_h/adfca17fa27cc2b99b95d23904b82f5e/image-9.jpg)
Just after t = 0 • The current source and its parallel resistor R 0 are disconnected from the rest of the circuit, and the inductor begins to release energy. ECE 201 Circuit Theory I 9
![The expression for the current ECE 201 Circuit Theory I 10 The expression for the current ECE 201 Circuit Theory I 10](https://slidetodoc.com/presentation_image_h/adfca17fa27cc2b99b95d23904b82f5e/image-10.jpg)
The expression for the current ECE 201 Circuit Theory I 10
![A firstorder ordinary differential equation with constant coefficients How do we solve it ECE A first-order ordinary differential equation with constant coefficients. How do we solve it? ECE](https://slidetodoc.com/presentation_image_h/adfca17fa27cc2b99b95d23904b82f5e/image-11.jpg)
A first-order ordinary differential equation with constant coefficients. How do we solve it? ECE 201 Circuit Theory I 11
![ECE 201 Circuit Theory I 12 ECE 201 Circuit Theory I 12](https://slidetodoc.com/presentation_image_h/adfca17fa27cc2b99b95d23904b82f5e/image-12.jpg)
ECE 201 Circuit Theory I 12
![The current in an inductor cannot change instantaneously Let the time just before The current in an inductor cannot change instantaneously • Let the time just before](https://slidetodoc.com/presentation_image_h/adfca17fa27cc2b99b95d23904b82f5e/image-13.jpg)
The current in an inductor cannot change instantaneously • Let the time just before switching be called t(0 -). • The time just after switching will be called t(0+). • For the inductor, ECE 201 Circuit Theory I 13
![The Complete Solution ECE 201 Circuit Theory I 14 The Complete Solution ECE 201 Circuit Theory I 14](https://slidetodoc.com/presentation_image_h/adfca17fa27cc2b99b95d23904b82f5e/image-14.jpg)
The Complete Solution ECE 201 Circuit Theory I 14
![The voltage drop across the resistor ECE 201 Circuit Theory I 15 The voltage drop across the resistor ECE 201 Circuit Theory I 15](https://slidetodoc.com/presentation_image_h/adfca17fa27cc2b99b95d23904b82f5e/image-15.jpg)
The voltage drop across the resistor ECE 201 Circuit Theory I 15
![The Power Dissipated in the Resistor ECE 201 Circuit Theory I 16 The Power Dissipated in the Resistor ECE 201 Circuit Theory I 16](https://slidetodoc.com/presentation_image_h/adfca17fa27cc2b99b95d23904b82f5e/image-16.jpg)
The Power Dissipated in the Resistor ECE 201 Circuit Theory I 16
![The Energy Delivered to the Resistor ECE 201 Circuit Theory I 17 The Energy Delivered to the Resistor ECE 201 Circuit Theory I 17](https://slidetodoc.com/presentation_image_h/adfca17fa27cc2b99b95d23904b82f5e/image-17.jpg)
The Energy Delivered to the Resistor ECE 201 Circuit Theory I 17
![Time Constant The rate at which the current or voltage approaches zero ECE Time Constant • The rate at which the current or voltage approaches zero. ECE](https://slidetodoc.com/presentation_image_h/adfca17fa27cc2b99b95d23904b82f5e/image-18.jpg)
Time Constant • The rate at which the current or voltage approaches zero. ECE 201 Circuit Theory I 18
![Rewriting in terms of Time Constant ECE 201 Circuit Theory I 19 Rewriting in terms of Time Constant ECE 201 Circuit Theory I 19](https://slidetodoc.com/presentation_image_h/adfca17fa27cc2b99b95d23904b82f5e/image-19.jpg)
Rewriting in terms of Time Constant ECE 201 Circuit Theory I 19
![Table 7 1 page 217 of the text ECE 201 Circuit Theory I 20 Table 7. 1 page 217 of the text ECE 201 Circuit Theory I 20](https://slidetodoc.com/presentation_image_h/adfca17fa27cc2b99b95d23904b82f5e/image-20.jpg)
Table 7. 1 page 217 of the text ECE 201 Circuit Theory I 20
![Graphical Interpretation of Time Constant Determine the time constant from the plot of Graphical Interpretation of Time Constant • Determine the time constant from the plot of](https://slidetodoc.com/presentation_image_h/adfca17fa27cc2b99b95d23904b82f5e/image-21.jpg)
Graphical Interpretation of Time Constant • Determine the time constant from the plot of the circuit’s natural response. Straight Line Approximation ECE 201 Circuit Theory I 21
![Graphical Interpretation Tangent at t 0 intersects the time axis at the time Graphical Interpretation Tangent at t = 0 intersects the time axis at the time](https://slidetodoc.com/presentation_image_h/adfca17fa27cc2b99b95d23904b82f5e/image-22.jpg)
Graphical Interpretation Tangent at t = 0 intersects the time axis at the time constant ECE 201 Circuit Theory I 22
![Procedure to Determine the Natural Response of an RL Circuit Find the initial Procedure to Determine the Natural Response of an RL Circuit • Find the initial](https://slidetodoc.com/presentation_image_h/adfca17fa27cc2b99b95d23904b82f5e/image-23.jpg)
Procedure to Determine the Natural Response of an RL Circuit • Find the initial current through the inductor. • Find the time constant, τ, of the circuit (L/R). • Generate i(t) from I 0 and τ using ECE 201 Circuit Theory I 23
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