EGR 2201 Unit 9 First-Order Circuits  Read Alexander & Sadiku, Chapter 7.  Homework #9 and Lab #9 due next week.  Quiz next week.

Review: DC Conditions in a Circuit with Inductors or Capacitors  Recall that when power is first applied to a dc circuit with inductors or capacitors, voltages and currents change briefly as the inductors and capacitors become energized.  But once they are fully energized (i.e., “under dc conditions”), all voltages and currents in the circuit have constant values.  To analyze a circuit under dc conditions, replace all capacitors with open circuits and replace all inductors with short circuits.

What About the Time Before DC Conditions?  We also want to be able to analyze such circuits during the time while the voltages and currents are changing, before dc conditions have been reached.  This is sometimes called transient analysis, because the behavior that we’re looking at is short-lived. It’s the focus of Chapters 7 and 8 in the textbook.

 The circuits we’ll study in this unit are called first-order circuits because they are described mathematically by first-order differential equations.  We’ll study four kinds of first-order circuits: Source-free RC circuits Source-free RL circuits RC circuits with sources RL circuits with sources Four Kinds of First-Order Circuits

 We use the term natural response to refer to the behavior of source-free circuits.  And we use the term step response to refer to the behavior of circuits in which a source is applied at some time.  So our goal in this unit is to understand the natural response of source-free RC and RL circuits, and to understand the step response of RC and RL circuits with sources. Natural Response and Step Response

 Consider the circuit shown. Assume that at time t=0, the capacitor is charged and has an initial voltage, V 0.  As time passes, the initial charge on the capacitor will flow through the resistor, gradually discharging the capacitor.  This results in changing voltage v(t) and currents i C (t) and i R (t), which we wish to calculate. Natural Response of Source-Free RC Circuit (1 of 2)

Natural Response of Source-Free RC Circuit (2 of 2)  This equation is an example of a first- order differential equation. How do we solve it for v(t)?

Math Detour: Differential Equations  Differential equations arise frequently in science and engineering. Some examples:  The equations above are all called linear ordinary differential equations with constant coefficients. A first-order diff. eq. A second-order diff. eq. A fourth-order diff. eq.

Solving Differential Equations  The differential equations on the previous slide are quite easy to solve. The ones shown below are more difficult.  In a later math course you’ll learn many techniques for solving such equations. Non-constant coefficient Non-linear A partial differential equation

A Closer Look at Our Differential Equation

Solving Our Differential Equation where A is a constant

Apply the Initial Condition

The Bottom Line

Graph of Voltage Versus Time  Note that at first the voltage falls steeply from its initial value (V 0 ). But as time passes, the descent becomes less steep.

The Time Constant

Don’t Confuse t and 

Rules of Thumb  After one time constant (i.e., when t =  ), the voltage has fallen to about 36.8% of its initial value.  After five time constants (i.e., when t = 5  ), the voltage has fallen to about 0.7% of its initial value. For most practical purposes we say that the capacitor is completely discharged and v = 0 after five time constants.

Comparing Different Values of   The greater  is, the more slowly the voltage descends, as shown below for a few values of .

Finding Values of Other Quantities

The Keys to Working with a Source-Free RC Circuit

More Complicated Cases  A circuit that looks more complicated at first might be reducible to a simple source- free RC circuit by combining resistors. Example: Here we can combine the three resistors into a single equivalent resistor, as seen from the capacitor’s terminals.

Where Did V 0 Come From?  In previous examples you’ve simply been given the capacitor’s initial voltage, V 0.  More realistically, you have to find V 0 by considering what happened before t = 0. Example: Suppose you’re told that the switch in this circuit has been closed for a long time before it’s opened at t = 0. Can you find the capacitor’s voltage V 0 at time t = 0?

 Consider the circuit shown. Assume that at time t=0, the inductor is energized and has an initial current, I 0.  As time passes, the inductor’s energy will gradually dissipate as current flows through the resistor.  This results in changing current i(t) and voltages v L (t) and v R (t), which we wish to calculate. Natural Response of Source-Free RL Circuit (1 of 2)

Natural Response of Source-Free RL Circuit (2 of 2)  This first-order differential equation is similar to the equation we had for source-free RC circuits.

Solving Our Differential Equation

Apply the Initial Condition

The Bottom Line

The Time Constant

Don’t Confuse t and 

Graph of Current Versus Time  It’s a decaying exponential curve, with the current falling steeply from its initial value ( I 0 ). But as time passes, the descent becomes less steep.

Rules of Thumb  After one time constant (i.e., when t =  ), the current has fallen to about 36.8% of its initial value.  After five time constants (i.e., when t = 5  ), the current has fallen to about 0.7% of its initial value. For most practical purposes we say that the inductor is completely de- energized and i = 0 after five time constants.

Finding Values of Other Quantities

The Keys to Working with a Source-Free RL Circuit

More Complicated Cases  A circuit that looks more complicated at first might be reducible to a simple source- free RL circuit by combining resistors. Example: Here we can combine the resistors into a single equivalent resistor, as seen from the inductor’s terminals.

Where Did I 0 Come From?  In previous examples you’ve simply been given the inductor’s initial current, I 0.  More realistically, you have to find I 0 by considering what happened before t = 0. Example: Suppose you’re told that the switch in this circuit has been closed for a long time before it’s opened at t = 0. Can you find the inductor’s current I 0 at time t = 0?

Where We Are  We’ve looked at: Source-free RC circuits Source-free RL circuits  We still need to look at: RC circuits with sources RL circuits with sources  Before doing this, we’ll look at some mathematical functions called singularity functions (or switching functions), which are widely used to model electrical signals that arise during switching operations.

Three Singularity Functions  The three singularity functions that we’ll study are: The unit step function The unit impulse function The unit ramp function

The Unit Step Function  The unit step function u(t) is 0 for negative values of t and 1 for positive values of t:

 We can obtain other step functions by shifting the unit step function to the left or right…  …or by multiplying the unit step function by a scaling constant: Shifting and Scaling the Unit Step Function

 As with any mathematical function, we can “flip” the unit step function horizontally by replacing t with t: Flipping the Unit Step Function Horizontally

 By adding two or more step functions we can obtain more complex “step- like” functions, such as the one shown below from the book’s Practice Problem 7.6. Adding Step Functions

 Step functions are useful for modeling sources that are switched on (or off) at some time: Using Step Functions to Model Switched Sources

The Unit Impulse Function  The unit step function’s derivative is the unit impulse function (t), also called the delta function.  The unit impulse function is 0 everywhere except at t =0, where it is undefined:  It’s useful for modeling “spikes” that can occur during switching operations.

 We can obtain other impulse functions by shifting the unit impulse function to the left or right, or by multiplying the unit impulse function by a scaling constant:  We won’t often use impulse functions in this course. Shifting and Scaling the Unit Impulse Function

The Unit Ramp Function  The unit step function’s integral is the unit ramp function r(t).  The unit ramp function is 0 for negative values of t and has a slope of 1 for positive values of t :

 We can obtain other ramp functions by shifting the unit ramp function to the left or right…  …or by multiplying the unit ramp function by a scaling constant: Shifting and Scaling the Unit Ramp Function

 By adding two or more step functions or ramp functions we can obtain more complex functions, such as the one shown below from the book’s Example 7.7. Adding Step and Ramp Functions

 A circuit’s step response is the circuit’s behavior due to a sudden application of a dc voltage or current source.  We can use a step function to model this sudden application. Step Response of a Circuit

 We distinguish the following two times: t = 0  (the instant just before the switch closes) t = 0 + (the instant just after the switch closes)  Since a capacitor’s voltage cannot change abruptly, we know that v(0  ) = v(0 + ) in this circuit.  But on the other hand, i(0  )  i(0 + ) in this circuit. t = 0  versus t = 0 +

 Assume that in the circuit shown, the capacitor’s initial voltage is V 0 (which may equal 0 V).  As time passes after the switch closes, the capacitor’s voltage will gradually approach the source voltage V S.  This results in changing voltage v(t) and current i(t), which we wish to calculate. Step Response of RC Circuit (1 of 2)

Step Response of RC Circuit (2 of 2)

Solution of Our Differential Equation

The Bottom Line  I won’t expect you to be able to reproduce the derivation on the previous slides.  The important point is to realize that for a circuit like this: the solution for v(t) is:

Graph of Voltage Versus Time  Here’s a graph of (assuming V 0 < V S ).  This is a saturating exponential curve.  Note that the voltage at first rises steeply from its initial value (V 0 ), and then gradually approaches its final value (V S ).

Rules of Thumb, etc.  We can repeat many of the same remarks as for source-free circuits, such as: The greater  is, the more slowly v(t) approaches its final value. For most practical purposes, v(t) reaches its final value after 5 . Knowing v(t), we can use Ohm’s law to find current, and we can use other familiar formulas to find power, energy, etc.

Another Way of Looking At It

The Keys to Finding an RC Circuit’s Step Response

More Complicated Cases  A circuit that looks more complicated at first might be reducible to a simple RC circuit by combining resistors. Example: Here, for t > 0 we can combine the resistors into a single equivalent resistor, as seen from the capacitor’s terminals.

Two Ways of Breaking It Down

Natural Response Versus Forced Response

Transient Response Versus Steady-State Response

“Under DC Conditions” = Steady- State  Recall that earlier we used the term “under dc conditions” to refer to the time after an RC or RL circuit’s currents and voltages have “settled down” to their final values.  This is just another way of referring to what we’re now calling steady-state values. So way can say that in the steady state, capacitors look like open circuits and inductors look like short circuits.

 Assume that in the circuit shown, the inductor’s initial current is I 0 (which may equal 0 A).  As time passes, the inductor’s current will gradually approach a steady-state value.  This results in changing current i(t) and voltage v(t), which we wish to calculate. Step Response of RL Circuit (1 of 2)

 Using the same sort of math we used previously for RC circuits, we find where  = L/R, just as for source-free RL circuits. Step Response of RL Circuit (2 of 2)

Another Way of Looking At It

Graph of Voltage Versus Time

Rules of Thumb, etc.  We can repeat many of the same remarks as for previous circuits, such as: The greater  is, the more slowly i(t) approaches its final value. For most practical purposes, i(t) reaches its final value after 5 . Knowing i(t), we can use Ohm’s law to find voltage, and we can use other familiar formulas to find power, energy, etc.

The Keys to Finding an RL Circuit’s Step Response

More Complicated Cases  A circuit that looks more complicated at first might be reducible to a simple RL circuit by combining resistors. Example: Here we can combine the resistors into a single equivalent resistor, as seen from the inductor’s terminals.

A General Approach for First- Order Circuits (1 of 3)

A General Approach for First- Order Circuits (2 of 3)

A General Approach for First- Order Circuits (3 of 3)