First-Order Circuits Cont’d

1. First-Order Circuits Cont’d Dr. Holbert April 17, 2006 ECE201 Lect-20

2. 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. ECE201 Lect-20

3. 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 () ECE201 Lect-20

4. 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 ECE201 Lect-20

5. The Differential Equation • Equations are linear, constant coefficient: • The variable x(t) could be voltage or current. • The coefficients an through a0 depend on the component values of circuit elements. • The function f(t) depends on the circuit elements and on the sources in the circuit. ECE201 Lect-20

6. 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 ECE201 Lect-20

7. Differential Equation Solution • The total solution to any differential equation consists of two parts: x(t) = xp(t) + xc(t) • Particular (forced) solution is xp(t) • Response particular to a given source • Complementary (natural) solution is xc(t) • Response common to all sources, that is, due to the “passive” circuit elements ECE201 Lect-20

8. 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 vp(t) is constant ECE201 Lect-20

9. The Natural Solution • The natural (or complementary) solution is the solution to the homogeneous equation: • Different “look” for 1st and 2nd order ODEs ECE201 Lect-20

10. First-Order Natural Solution • The first-order ODE has a form of • The natural solution is • Tau (t) is the time constant • For an RC circuit, t = RC • For an RL circuit, t = L/R ECE201 Lect-20

11. 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. ECE201 Lect-20

12. 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. ECE201 Lect-20

13. Step-by-Step Approach • Assume solution (only dc sources allowed): x(t) = K1 + K2 e-t/ • At t=0–, draw circuit with C as open circuit and L as short circuit; find IL(0–) or VC(0–) • At t=0+, redraw circuit and replace C or L with appropriate source of value obtained in step #2, and find x(0)=K1+K2 • At t=, repeat step #2 to find x()=K1 ECE201 Lect-20

14. Step-by-Step Approach • Find time constant () Looking across the terminals of the C or L element, form Thevenin equivalent circuit; =RThC or =L/RTh • Finish up Simply put the answer together. ECE201 Lect-20

15. Class Examples • Learning Extension E7.3 • Learning Extension E7.4 • Learning Extension E7.5 ECE201 Lect-20