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Lecture 14 •Introduction to dynamic systems •Energy storage •Basic time-varying signals •Related educational modules: –Sections 2.0, 2.1 Review and Background • Our circuits have not contained any energy storage elements • Resistors dissipate energy • Governing equations are algebraic, the system responds instantaneously to changes Example: Inverting voltage amplifier VOUT Rf Vin Rin • The system output at some time depends only on the input at that time • Example: If the input changes suddenly, the output changes suddenly Inverting voltage amplifier – switched response • Input and response: Dynamic Systems • We now consider circuits containing energy storage elements • Capacitors and inductors store energy • The circuits are dynamic systems • They are governed by differential equations • Physically, they are performing integrations • If we apply a time-varying input to the system, the output may not have the same “shape” as the input • The system output depends upon the state of the system at previous times Dynamic System – example • Heating a frying pan Body with: mass m, specific heat cP, temperature TB Heat Input, qin Heat Dissipation, qout Ambient Temperature, T0 Dynamic System Example – continued • The rate at which the temperature can respond is dictated by the body’s mass and material properties dT mc p B qin qout dt • The heat out of the mass is governed by the difference in temperature between the body and the surroundings: qout R( TB T0 ) • The mass is storing heat as temperature Dynamic System Example – continued TB(t) qin(t) Final Temperature Initial Temperature t=0 t t=0 t Time-varying signals • We now have to account for changes in the system response with time • Previously, our analyses could be viewed as being independent of time • The system inputs and outputs will become functions of time • Generically referred to a signals • We need to introduce the basic time-varying signals we will be using Basic Time-Varying Signals • In this class, we will restrict our attention to a few basic types of signals: • Step functions • Exponential functions • Sinusoidal functions • Sinusoidal functions will be used extensively later; we will introduce them at that time Step Functions • The unit step function is defined as: 0 , t 0 u0 ( t ) 1, t 0 • Circuit to generate the signal: Scaled and shifted step functions • Scaling • Multiply by a constant 0,t 0 K u0 ( t ) K , t 0 • Shifting • Moving in time 0 , t a u0 ( t a ) 1, t a Example 1 • Sketch 5u0(t-3) Example 2 • Represent v(t) in the circuit below in terms of step functions t = 3 sec t = 1 sec Example 3 cos( t ), 0 t 2 • Represent the function f ( t ) as a single 0 , otherwise • function defined over -<t<. Exponential Functions • An exponential function is defined by f ( t ) Ae t • is the time constant • >0 Exponential Functions – continued • Our exponential functions will generally be limited to t≥0: f ( t ) Ae t , t 0 0.368A • or: f ( t ) Ae t u0 ( t ) • Note: f(t) decreases by 63.2% every seconds Effect of varying Exponential Functions – continued • Why are exponential functions important? • They are the form of the solutions to ordinary, linear differential equations with constant coefficients