ECEN 301Discussion #15 – 1 st Order Transient Response1 Good, Better, Best Elder Oaks (October 2007): As we consider various choices, we should remember that it is not enough that something is good. Other choices are better, and still others are best. Even though a particular choice is more costly, its far greater value may make it the best choice of all. I have never known of a man who looked back on his working life and said, I just didn't spend enough time with my job.

ECEN 301Discussion #15 – 1 st Order Transient Response2 Lecture 15 – Transient Response of 1 st Order Circuits DC Steady-State Transient Response

ECEN 301Discussion #15 – 1 st Order Transient Response3 1 st Order Circuits Electric circuit 1 st order system: any circuit containing a single energy storing element (either a capacitor or an inductor) and any number of sources and resistors R s R1R1 vsvs +–+– L2L2 R2R2 L1L1 1 st order 2 nd order R 1 R2R2 C vsvs +–+–

ECEN 301Discussion #15 – 1 st Order Transient Response4 Capacitor/Inductor Voltages/Currents Review of capacitor/inductor currents and voltages Exponential growth/decay Capacitor voltage v C (t) Inductor current i L (t) NB: neither can change instantaneously Capacitor current i C (t) Inductor voltage v L (t) NB: both can change instantaneously NB: note the duality between inductors and capacitors

ECEN 301Discussion #15 – 1 st Order Transient Response5 Transient Response Transient response of a circuit consists of 3 parts: 1.Steady-state response prior to the switching on/off of a DC source 2.Transient response – the circuit adjusts to the DC source 3.Steady-state response following the transient response R 1 R2R2 C vsvs +–+– t = 0 DC Source Switch Energy element

ECEN 301Discussion #15 – 1 st Order Transient Response6 1. DC Steady State 1 st and 3 rd Step in Transient Response

ECEN 301Discussion #15 – 1 st Order Transient Response7 DC Steady-State DC steady-state: the stable voltages and currents in a circuit connected to a DC source Capacitors act like open circuits at DC steady-state Inductors act like short circuits at DC steady-state

ECEN 301Discussion #15 – 1 st Order Transient Response8 DC Steady-State Initial condition x(0): DC steady state before a switch is first activated x(0 – ): right before the switch is closed x(0 + ): right after the switch is closed Final condition x(): DC steady state a long time after a switch is activated R 1 R2R2 C vsvs +–+– t = 0 R3R3 R 1 R2R2 C vsvs +–+– t R3R3 Initial condition Final condition

ECEN 301Discussion #15 – 1 st Order Transient Response9 DC Steady-State Example 1: determine the final condition capacitor voltage v s = 12V, R 1 = 100Ω, R 2 = 75Ω, R 3 = 250Ω, C = 1uF R 1 R2R2 C vsvs +–+– t = 0 R3R3

ECEN 301Discussion #15 – 1 st Order Transient Response10 DC Steady-State Example 1: determine the final condition capacitor voltage v s = 12V, R 1 = 100Ω, R 2 = 75Ω, R 3 = 250Ω, C = 1uF R 1 R2R2 C vsvs +–+– t 0 + R3R3 1.Close the switch and find initial conditions to the capacitor NB: Initially (t = 0 + ) current across the capacitor changes instantly but voltage cannot change instantly, thus it acts as a short circuit

ECEN 301Discussion #15 – 1 st Order Transient Response11 DC Steady-State Example 1: determine the final condition capacitor voltage v s = 12V, R 1 = 100Ω, R 2 = 75Ω, R 3 = 250Ω, C = 1uF R 1 R2R2 C vsvs +–+– t R3R3 2.Close the switch and apply finial conditions to the capacitor NB: since we have an open circuit no current flows through R 2

ECEN 301Discussion #15 – 1 st Order Transient Response12 DC Steady-State Remember – capacitor voltages and inductor currents cannot change instantaneously Capacitor voltages and inductor currents dont change right before closing and right after closing a switch

ECEN 301Discussion #15 – 1 st Order Transient Response13 DC Steady-State Example 2: find the initial and final current conditions at the inductor i s = 10mA isis t = 0 R L iLiL

ECEN 301Discussion #15 – 1 st Order Transient Response14 DC Steady-State Example 2: find the initial and final current conditions at the inductor i s = 10mA isis t = 0 R L iLiL 1.Initial conditions – assume the current across the inductor is in steady-state. isis iLiL NB: in DC steady state inductors act like short circuits, thus no current flows through R

ECEN 301Discussion #15 – 1 st Order Transient Response15 DC Steady-State Example 2: find the initial and final current conditions at the inductor i s = 10mA 1.Initial conditions – assume the current across the inductor is in steady-state. isis iLiL

ECEN 301Discussion #15 – 1 st Order Transient Response16 DC Steady-State Example 2: find the initial and final current conditions at the inductor i s = 10mA isis t = 0 R L iLiL 1.Initial conditions – assume the current across the inductor is in steady-state. 2.Throw the switch NB: inductor current cannot change instantaneously R L iLiL

ECEN 301Discussion #15 – 1 st Order Transient Response17 DC Steady-State Example 2: find the initial and final current conditions at the inductor i s = 10mA isis t = 0 R L iLiL 1.Initial conditions – assume the current across the inductor is in steady-state. 2.Throw the switch 3.Find initial conditions again (non-steady state) NB: inductor current cannot change instantaneously –R+–R+ L iLiL NB: polarity of R

ECEN 301Discussion #15 – 1 st Order Transient Response18 DC Steady-State Example 2: find the initial and final current conditions at the inductor i s = 10mA isis t = 0 R L iLiL 1.Initial conditions – assume the current across the inductor is in steady-state. 2.Throw the switch 3.Find initial conditions again (non-steady state) 4.Final conditions (steady-state) NB: since there is no source attached to the inductor, its current is drained by the resistor R

ECEN 301Discussion #15 – 1 st Order Transient Response19 2. Adjusting to Switch 2 nd Step in Transient Response

ECEN 301Discussion #15 – 1 st Order Transient Response20 General Solution of 1 st Order Circuits Expressions for voltage and current of a 1 st order circuit are a 1 st order differential equation + R – +C–+C– iCiC vsvs +–+– NB: Review lecture 11 for derivation of these equations

ECEN 301Discussion #15 – 1 st Order Transient Response21 General Solution of 1 st Order Circuits Expressions for voltage and current of a 1 st order circuit are a 1 st order differential equation

ECEN 301Discussion #15 – 1 st Order Transient Response22 General Solution of 1 st Order Circuits Expressions for voltage and current of a 1 st order circuit are a 1 st order differential equation NB: Constants

ECEN 301Discussion #15 – 1 st Order Transient Response23 General Solution of 1 st Order Circuits Expressions for voltage and current of a 1 st order circuit are a 1 st order differential equation NB: similarities

ECEN 301Discussion #15 – 1 st Order Transient Response24 General Solution of 1 st Order Circuits Expressions for voltage and current of a 1 st order circuit are a 1 st order differential equation

ECEN 301Discussion #15 – 1 st Order Transient Response25 General Solution of 1 st Order Circuits Expressions for voltage and current of a 1 st order circuit are a 1 st order differential equation Capacitor/inductor voltage/current

ECEN 301Discussion #15 – 1 st Order Transient Response26 General Solution of 1 st Order Circuits Expressions for voltage and current of a 1 st order circuit are a 1 st order differential equation Forcing function (F – for DC source)

ECEN 301Discussion #15 – 1 st Order Transient Response27 General Solution of 1 st Order Circuits Expressions for voltage and current of a 1 st order circuit are a 1 st order differential equation Combinations of circuit element parameters (Constants)

ECEN 301Discussion #15 – 1 st Order Transient Response28 General Solution of 1 st Order Circuits Expressions for voltage and current of a 1 st order circuit are a 1 st order differential equation Forcing Function F

ECEN 301Discussion #15 – 1 st Order Transient Response29 General Solution of 1 st Order Circuits Expressions for voltage and current of a 1 st order circuit are a 1 st order differential equation DC gain

ECEN 301Discussion #15 – 1 st Order Transient Response30 General Solution of 1 st Order Circuits Expressions for voltage and current of a 1 st order circuit are a 1 st order differential equation Time constant DC gain

ECEN 301Discussion #15 – 1 st Order Transient Response31 General Solution of 1 st Order Circuits The solution to this equation (the complete response) consists of two parts: Natural response (homogeneous solution) Forcing function equal to zero Forced response (particular solution)

ECEN 301Discussion #15 – 1 st Order Transient Response32 General Solution of 1 st Order Circuits Natural response (homogeneous or natural solution) Forcing function equal to zero Has known solution of the form:

ECEN 301Discussion #15 – 1 st Order Transient Response33 General Solution of 1 st Order Circuits Forced response (particular or forced solution) F is constant for DC sources, thus derivative is zero NB: This is the DC steady-state solution

ECEN 301Discussion #15 – 1 st Order Transient Response34 General Solution of 1 st Order Circuits Complete response (natural + forced) Solve for α by solving x(t) at t = 0 Initial condition Final condition Time constant

ECEN 301Discussion #15 – 1 st Order Transient Response35 General Solution of 1 st Order Circuits Complete response (natural + forced) Transient ResponseSteady-State Response

ECEN 301Discussion #15 – 1 st Order Transient Response36 3. DC Steady-State + Transient Response Full Transient Response

ECEN 301Discussion #15 – 1 st Order Transient Response37 Transient Response Transient response of a circuit consists of 3 parts: 1.Steady-state response prior to the switching on/off of a DC source 2.Transient response – the circuit adjusts to the DC source 3.Steady-state response following the transient response R 1 R2R2 C vsvs +–+– t = 0 DC Source Switch Energy element

ECEN 301Discussion #15 – 1 st Order Transient Response38 Transient Response Solving 1 st order transient response: 1.Solve the DC steady-state circuit: Initial condition x(0 – ): before switching (on/off) Final condition x(): After any transients have died out (t ) 2.Identify x(0 + ): the circuit initial conditions Capacitors: v C (0 + ) = v C (0 – ) Inductors: i L (0 + ) = i L (0 – ) 3.Write a differential equation for the circuit at time t = 0 + Reduce the circuit to its Thévenin or Norton equivalent The energy storage element (capacitor or inductor) is the load The differential equation will be either in terms of v C (t) or i L (t) Reduce this equation to standard form 4.Solve for the time constant Capacitive circuits: τ = R T C Inductive circuits: τ = L/R T 5.Write the complete response in the form: x(t) = x() + [x(0) - x()]e -t/τ

ECEN 301Discussion #15 – 1 st Order Transient Response39 Transient Response Example 3: find v c (t) for all t v s = 12V, v C (0 – ) = 5V, R = 1000Ω, C = 470uF R C vsvs +–+– t = 0 i(t) + v C (t) –

ECEN 301Discussion #15 – 1 st Order Transient Response40 Transient Response Example 3: find v c (t) for all t v s = 12V, v C (0 – ) = 5V, R = 1000Ω, C = 470uF R C vsvs +–+– t = 0 i(t) + v C (t) – 1.DC steady-state a)Initial condition: v C (0) b)Final condition: v C () NB: as t the capacitor acts like an open circuit thus v C () = v S

ECEN 301Discussion #15 – 1 st Order Transient Response41 Transient Response Example 3: find v c (t) for all t v s = 12V, v C (0 – ) = 5V, R = 1000Ω, C = 470uF R C vsvs +–+– t = 0 i(t) + v C (t) – 2.Circuit initial conditions: v C (0 + )

ECEN 301Discussion #15 – 1 st Order Transient Response42 Transient Response Example 3: find v c (t) for all t v s = 12V, v C (0 – ) = 5V, R = 1000Ω, C = 470uF R C vsvs +–+– t = 0 i(t) + v C (t) – 3.Write differential equation (already in Thévenin equivalent) at t = 0 Recall:

ECEN 301Discussion #15 – 1 st Order Transient Response43 Transient Response Example 3: find v c (t) for all t v s = 12V, v C (0 – ) = 5V, R = 1000Ω, C = 470uF R C vsvs +–+– t = 0 i(t) + v C (t) – 3.Write differential equation (already in Thévenin equivalent) at t = 0 NB: solution is of the form:

ECEN 301Discussion #15 – 1 st Order Transient Response44 Transient Response Example 3: find v c (t) for all t v s = 12V, v C (0 – ) = 5V, R = 1000Ω, C = 470uF R C vsvs +–+– t = 0 i(t) + v C (t) – 4.Find the time constant τ

ECEN 301Discussion #15 – 1 st Order Transient Response45 Transient Response Example 3: find v c (t) for all t v s = 12V, v C (0 – ) = 5V, R = 1000Ω, C = 470uF R C vsvs +–+– t = 0 i(t) + v C (t) – 5.Write the complete response x(t) = x() + [x(0) - x()]e -t/τ

ECEN 301Discussion #15 – 1 st Order Transient Response46 Transient Response Example 4: find v c (t) for all t v a = 12V, v b = 5V, R 1 = 10Ω, R 2 = 5Ω, R 3 = 10Ω, C = 1uF R 3 C vava +–+– t = 0 + v C (t) – vbvb +–+– R 1 R 2

ECEN 301Discussion #15 – 1 st Order Transient Response47 Transient Response Example 4: find v c (t) for all t v a = 12V, v b = 5V, R 1 = 10Ω, R 2 = 5Ω, R 3 = 10Ω, C = 1uF R 3 C vava +–+– t = 0 + v C (t) – vbvb +–+– R 1 R 2 1.DC steady-state a)Initial condition: v C (0) b)Final condition: v C () For t < 0 the capacitor has been charged by v b thus v C (0 – ) = v b For t v c () is not so easily determined – it is equal to v T (the open circuited Thévenin equivalent)

ECEN 301Discussion #15 – 1 st Order Transient Response48 Transient Response Example 4: find v c (t) for all t v a = 12V, v b = 5V, R 1 = 10Ω, R 2 = 5Ω, R 3 = 10Ω, C = 1uF R 3 C vava +–+– t = 0 + v C (t) – vbvb +–+– R 1 R 2 2.Circuit initial conditions: v C (0 + )

ECEN 301Discussion #15 – 1 st Order Transient Response49 Transient Response Example 4: find v c (t) for all t v a = 12V, v b = 5V, R 1 = 10Ω, R 2 = 5Ω, R 3 = 10Ω, C = 1uF R 3 C vava +–+– t = 0 + v C (t) – vbvb +–+– R 1 R 2 3.Write differential equation at t = 0 a)Find Thévenin equivalent

ECEN 301Discussion #15 – 1 st Order Transient Response50 Transient Response Example 4: find v c (t) for all t v a = 12V, v b = 5V, R 1 = 10Ω, R 2 = 5Ω, R 3 = 10Ω, C = 1uF 3.Write differential equation at t = 0 a)Find Thévenin equivalent C + v C (t) – R 1 R 2 iaia ibib R 3

ECEN 301Discussion #15 – 1 st Order Transient Response51 Transient Response Example 4: find v c (t) for all t v a = 12V, v b = 5V, R 1 = 10Ω, R 2 = 5Ω, R 3 = 10Ω, C = 1uF 3.Write differential equation at t = 0 a)Find Thévenin equivalent C + v C (t) – R 1 R 2 iTiT R 3

ECEN 301Discussion #15 – 1 st Order Transient Response52 Transient Response Example 4: find v c (t) for all t v a = 12V, v b = 5V, R 1 = 10Ω, R 2 = 5Ω, R 3 = 10Ω, C = 1uF 3.Write differential equation at t = 0 a)Find Thévenin equivalent C + v C (t) – R T iTiT

ECEN 301Discussion #15 – 1 st Order Transient Response53 Transient Response Example 4: find v c (t) for all t v a = 12V, v b = 5V, R 1 = 10Ω, R 2 = 5Ω, R 3 = 10Ω, C = 1uF 3.Write differential equation at t = 0 a)Find Thévenin equivalent b)Reduce equation to standard form C + v C (t) – R T vTvT +–+–

ECEN 301Discussion #15 – 1 st Order Transient Response54 Transient Response Example 4: find v c (t) for all t v a = 12V, v b = 5V, R 1 = 10Ω, R 2 = 5Ω, R 3 = 10Ω, C = 1uF C + v C (t) – R T vTvT +–+– 4.Find the time constant τ

ECEN 301Discussion #15 – 1 st Order Transient Response55 Transient Response Example 4: find v c (t) for all t v a = 12V, v b = 5V, R 1 = 10Ω, R 2 = 5Ω, R 3 = 10Ω, C = 1uF C + v C (t) – R T vTvT +–+– 5.Write the complete response x(t) = x() + [x(0) - x()]e -t/τ

ECEN 301Discussion #15 – 1 st Order Transient Response56 Transient Response Example 4: find v c (t) for all t v a = 12V, v b = 5V, R 1 = 10Ω, R 2 = 5Ω, R 3 = 10Ω, C = 1uF 5.Write the complete response x(t) = x() + [x(0) - x()]e -t/τ R 3 C vava +–+– t = 0 + v C (t) – vbvb +–+– R 1 R 2