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**Response of First-Order Circuits**

RL Circuits RC Circuits ECE 201 Circuit Theory I

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**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

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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

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**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

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**General Configurations for RL**

If the independent sources are equal to zero, the circuits simplify to ECE 201 Circuit Theory I

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**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

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**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

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**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

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Just after t = 0 The current source and its parallel resistor R0 are disconnected from the rest of the circuit, and the inductor begins to release energy. ECE 201 Circuit Theory I

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**The expression for the current**

ECE 201 Circuit Theory I

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**A first-order ordinary differential equation with constant coefficients.**

How do we solve it? ECE 201 Circuit Theory I

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ECE 201 Circuit Theory I

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**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

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The Complete Solution ECE 201 Circuit Theory I

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**The voltage drop across the resistor**

ECE 201 Circuit Theory I

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**The Power Dissipated in the Resistor**

ECE 201 Circuit Theory I

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**The Energy Delivered to the Resistor**

ECE 201 Circuit Theory I

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Time Constant The rate at which the current or voltage approaches zero. ECE 201 Circuit Theory I

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**Rewriting in terms of Time Constant**

ECE 201 Circuit Theory I

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Table 7.1 page 233 of the text ECE 201 Circuit Theory I

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**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

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**Graphical Interpretation**

Tangent at t = 0 intersects the time axis at the time constant ECE 201 Circuit Theory I

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**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 I0 and τ using ECE 201 Circuit Theory I

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