1. ECE201 Lect-201 First-Order Circuits Cont’d Dr. Holbert April 17, 2006

2. ECE201 Lect-202 Introduction In a circuit with energy storage elements, voltages and currents are the solutions to linear, constant coefficient differential equations. Real engineers almost never solve the differential equations directly. It is important to have a qualitative understanding of the solutions.

3. ECE201 Lect-203 Important Concepts The differential equation for the circuit Forced (particular) and natural (complementary) solutions Transient and steady-state responses 1st order circuits: the time constant (  )

4. ECE201 Lect-204 The Differential Equation Every voltage and current is the solution to a differential equation. In a circuit of order n, these differential equations have order n. The number and configuration of the energy storage elements determines the order of the circuit. n  # of energy storage elements

5. ECE201 Lect-205 The Differential Equation Equations are linear, constant coefficient: The variable x(t) could be voltage or current. The coefficients a n through a 0 depend on the component values of circuit elements. The function f(t) depends on the circuit elements and on the sources in the circuit.

6. ECE201 Lect-206 Building Intuition Even though there are an infinite number of differential equations, they all share common characteristics that allow intuition to be developed: –Particular and complementary solutions –Effects of initial conditions

7. ECE201 Lect-207 Differential Equation Solution The total solution to any differential equation consists of two parts: x(t) = x p (t) + x c (t) Particular (forced) solution is x p (t) –Response particular to a given source Complementary (natural) solution is x c (t) –Response common to all sources, that is, due to the “passive” circuit elements

8. ECE201 Lect-208 The Forced Solution The forced (particular) solution is the solution to the non-homogeneous equation: The particular solution is usually has the form of a sum of f(t) and its derivatives. –If f(t) is constant, then v p (t) is constant

9. ECE201 Lect-209 The Natural Solution The natural (or complementary) solution is the solution to the homogeneous equation: Different “look” for 1 st and 2 nd order ODEs

10. ECE201 Lect-2010 First-Order Natural Solution The first-order ODE has a form of The natural solution is Tau (  ) is the time constant For an RC circuit,  = RC For an RL circuit,  = L/R

11. ECE201 Lect-2011 Initial Conditions The particular and complementary solutions have constants that cannot be determined without knowledge of the initial conditions. The initial conditions are the initial value of the solution and the initial value of one or more of its derivatives. Initial conditions are determined by initial capacitor voltages, initial inductor currents, and initial source values.

12. ECE201 Lect-2012 Transients and Steady State The steady-state response of a circuit is the waveform after a long time has passed, and depends on the source(s) in the circuit. –Constant sources give DC steady-state responses DC SS if response approaches a constant –Sinusoidal sources give AC steady-state responses AC SS if response approaches a sinusoid The transient response is the circuit response minus the steady-state response.

13. ECE201 Lect-2013 Step-by-Step Approach 1.Assume solution (only dc sources allowed): x(t) = K 1 + K 2 e -t/  2.At t=0 –, draw circuit with C as open circuit and L as short circuit; find I L (0 – ) or V C (0 – ) 3.At t=0 +, redraw circuit and replace C or L with appropriate source of value obtained in step #2, and find x(0)=K 1 +K 2 4.At t= , repeat step #2 to find x(  )=K 1

14. ECE201 Lect-2014 Step-by-Step Approach 5.Find time constant (  ) Looking across the terminals of the C or L element, form Thevenin equivalent circuit;  =R Th C or  =L/R Th 6.Finish up Simply put the answer together.

15. ECE201 Lect-2015 Class Examples Learning Extension E7.3 Learning Extension E7.4 Learning Extension E7.5