Response of First-Order Circuits RL Circuits RC Circuits ECE 201 Circuit Theory I

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

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

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

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

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

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

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

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

The expression for the current ECE 201 Circuit Theory I

A first-order ordinary differential equation with constant coefficients. How do we solve it? ECE 201 Circuit Theory I

ECE 201 Circuit Theory I

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

The Complete Solution ECE 201 Circuit Theory I

The voltage drop across the resistor ECE 201 Circuit Theory I

The Power Dissipated in the Resistor ECE 201 Circuit Theory I

The Energy Delivered to the Resistor ECE 201 Circuit Theory I

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

Rewriting in terms of Time Constant ECE 201 Circuit Theory I

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

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

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

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