EGR 2201 Unit 10 Second-Order Circuits Read Alexander & Sadiku, Sections 8.1 through 8.4. Homework #10 and Lab #10 due next week. Quiz next week. Review: Four Kinds of First-Order Circuits

The circuits we studied last week are called first-order circuits because they are described mathematically by first-order differential equations. We studied four kinds of first-order circuits: Source-free RC circuits Source-free RL circuits RC circuits with sources

RL circuits with sources Review: A General Approach for First-Order Circuits (1 of 3) 1. 2. 3. 4. General approach for most of the problems we studied last week: Find the quantitys initial value . Find the quantitys final value . Find the time constant: for an RC circuit.

for an RL circuit. Once you know these items, solution is : Review: A General Approach for First-Order Circuits (2 of 3) The equation from the previous slide, graphs as either: A decaying exponential curve if the initial value x(0) is greater than the final value x(). Or a saturating exponential curve if the initial value x(0)

is less than the final value x(). Review: A General Approach for First-Order Circuits (3 of 3) Recall also that we can think of the complete response as being the sum of a transient response that dies away with time and a steady-state response that is constant and remains after the transient has died away: Steady-state response Transient response

Transient Analysis with Multisim The textbooks Sections 7.8 and 8.9 discuss using PSpice simulation software to perform transient analysis of first-order and secondorder circuits. We can also do this with Multisim, as shown here. The steps are summarized in Lab 10.

Our Goal: A General Approach for Second-Order Circuits Next we will develop a general approach for analyzing more complicated circuits called secondorder circuits. Unfortunately the general approach for second-order circuits is quite a bit more complicated than the one for first-order circuits. Second-Order Circuits The circuits well study are called second-order circuits because they

are described mathematically by second-order differential equations. Whereas first-order circuits contain a single energy-storing element (capacitor or inductor), second-order circuits contain two energy-storing elements. These two elements could both be capacitors or both be inductors, but well focus on circuits containing one capacitor and one inductor. Four Kinds of Second-Order Circuits The book treats four kinds of second-order circuits:

Source-free series RLC circuits Source-free parallel RLC circuits Series RLC circuits with sources (We wont cover these.) Parallel RLC circuits with sources Natural Response and Step Response

Recall that the term natural response refers to the behavior of source-free circuits. And the term step response refers to the behavior of circuits in which a source is applied at some time. So the goal of this chapter in the book is to understand the natural response of source-free RLC circuits, and to

understand the step response of RLC circuits with sources. Redraw, Redraw, Redraw! Our procedure will usually require us to find values of voltages or currents at the following three times: At t = 0, just before a switch is opened or closed. At t = 0+, just after a switch is opened or closed. As t , a long time after a switch is opened or closed.

Usually the circuit looks different at these three times, so youll want to redraw the circuit for each of these times. Finding Initial Values To completely solve a first-order differential equation, you need one initial condition, usually either: An initial inductor current i(0+), or An initial capacitor voltage v(0+).

To completely solve a second-order differential equation, you need two initial conditions, usually either: An initial inductor current i(0+) and its derivative di(0+)/dt, or An initial capacitor voltage v(0+) and its derivative dv(0+)/dt. Finding Initial Derivative Values To find initial derivative values such as dv(0+)/dt, well rely on the basic relationships for capacitors and inductors:

= = For example, if we know a capacitors initial current i(0+), then we can use the left-hand equation above to find the initial derivative of that capacitors voltage, dv(0+)/dt. Quantities that Cannot Change Abruptly

Well also rely on the fact that a capacitors voltage and an inductors current cannot change abruptly. Example: In this circuit, i(0+) must be equal to i(0), and v(0+) must be equal to v(0). Since these values must be equal, we dont really need to distinguish between their values at time t = 0 and at time t = 0+. So we could just write i(0) instead of i(0+) and i(0). Caution: Some Quantities Can Change Abruptly

Dont assume that every quantity has the same value at times t = 0 and t = 0+. Example: In the same circuit, iC(t) changes abruptly from 0 A to 2 A at time t = 0. So we must distinguish between iC(0) and iC(0+): i (0) = 0 A C i (0+) = 2 A

C i (0) is undefined. C Finding Final Values Our procedure will sometimes also require us to find final or steady-state values, such as: A final inductor current i() A final capacitor voltage v(). Usually these final values are easier to find than initial values, because:

1. 2. We dont have to worry about abrupt changes as t , so we never need to distinguish between t and t +. We dont have to find derivatives of currents or voltages as t . Natural Response of Source-Free Series RLC Circuit (1 of 2)

Consider the circuit shown. Assume that at time t=0, the inductor has initial current I0, and the capacitor has initial voltage V0. As time passes, the initial energy in the capacitor and inductor will dissipate as current flows through the resistor. This results in changing current i(t), which we wish to calculate. Natural Response of Source-Free Series RLC Circuit (2 of 2)

Applying KVL, A standard trick for such integro-differential equations is to take the derivative of both sides: Now divide by L and rearrange terms: A Closer Look at Our Differential Equation

Our equation, , is a second-order linear differential equation with constant coefficients. Such equations have been studied extensively. The following slides outline the standard solution. Solving Our Differential Equation (1 of 4) To solve , first assume that the solution is of the form where A and s are constants to be found. Plugging this into the equation yields:

Factoring out the common term, Solving Our Differential Equation (2 of 4) From it follows that: This is called the characteristic equation of our original differential equation. Note that it is purely algebraic with one variable (s). It

has no derivatives, no integrals, no exponentials. Now we can use the quadratic formula to solve for s. But first. Solving Our Differential Equation (3 of 4) For convenience we introduce two new symbols: and

We call the neper frequency, and we call 0 the undamped natural frequency. Then we can rewrite as: Now use the quadratic formula. Solving Our Differential Equation (4 of 4) Applying the quadratic formula to gives two solutions for s, which we call the natural frequencies: , Because of the square root, we must distinguish three cases:

> 0, called the overdamped case. = 0, called the critically damped case. < 0, called the underdamped case. The Overdamped Case ( > 0)

If > 0, our two solutions for s are distinct negative real numbers: Real number , In this case, the solution to our differential equation is The initial conditions determine A1 and A2: , The Critically Damped Case ( = 0) If = 0, our two solutions for s are equal to each other and to : , Zero

In this case, the solution to our differential equation is The initial conditions determine A1 and A2: , The Underdamped Case ( < 0) If < 0, our two solutions for s are complex numbers: Imaginary number

Here j is the imaginary unit, , and d is called the damped natural frequency, In this case, the solution to our differential equation is The initial conditions determine B1 and B2: , Graphs of the Three Cases

Details will differ based on initial conditions and element values, but the shapes shown here are typical. Note the oscillation in the underdamped case. Typing Equations in Word 2013 1. Select Insert > Equation on Words menu bar. 2. Use the toolbars Structures section to create fractions, exponents, square roots, and more. 3. Use the toolbars Symbols section to insert basic math symbols, Greek letters, special operators, and

more. Oscilloscope Looking ahead, well use an oscilloscope to display and measure fast-changing voltages, including transients. Oscilloscope Challenge Game The oscilloscope is a complex instrument that you must learn to use.

To learn the basics, play my Oscilloscope Challenge game at http://nreeder.com/flashgames.htm. Natural Response of Source-Free Parallel RLC Circuit (1 of 2) Preview: The math for a source-free parallel circuit is almost the same as the math on the previous slides, except that: 1. Now the variable in our differential equation is v(t) instead of i(t).

2. We define (the neper frequency) to be equal to instead of . Natural Response of Source-Free Parallel RLC Circuit (2 of 2) By applying KCL and some algebra, we get: By assuming a solution of the form we can derive the characteristic equation

Solving Our Differential Equation For convenience we define: and Note that 0 (the undamped natural frequency) is the same as for series RLC circuits, but (the neper frequency) is not. Therefore, just as for series circuits, We get the same three cases as before (overdamped, critically damped, and

underdamped). The Overdamped Case ( > 0) If > 0, our two solutions for s are distinct negative real numbers , In this case, the solution to our differential equation is The initial conditions determine A1 and A2: , The Critically Damped Case ( = 0)

If = 0, our two solutions for s are equal to each other and to : , In this case, the solution to our differential equation is The initial conditions determine A1 and A2: , The Underdamped Case ( < 0)

If < 0, our two solutions for s are complex numbers: Here j is the imaginary unit, , and d is called the damped natural frequency, In this case, the solution to our differential equation is The initial conditions determine B1 and B2: , Graphs of the Three Cases

Details will differ based on initial conditions and element values, but the shapes shown here are typical. Note the oscillation in the underdamped case. A General Approach for Source-Free Series or Parallel RLC Circuits (1 of 3) 1. 2. 3.

4. 5. To find x(t) in a source-free series or parallel RLC circuit, where x could be any current or voltage: Find the quantitys initial value . Find the quantitys initial derivative . Find the neper frequency: for a series RLC circuit. for a parallel RLC circuit. Find the undamped natural frequency: Compare to 0 to see whether circuit is overdamped, critically damped, or underdamped

A General Approach for Source-Free Series or Parallel RLC Circuits (2 of 3) 6. If its overdamped ( > 0), then: , Solve for A1 and A2: 7. , If its critically damped ( = 0), then: , A General Approach for Source-Free Series or Parallel RLC Circuits (3 of 3)

8. If its underdamped ( < 0), then: ,