Derivation of Models for Electrical Systems(1) Start from the basic circuit component principle:BasicVoltage-CurrentRelationsv(t) = Ri(t)Resistor of R (ohms):V(s) = R I(s)cdv() =i(t)= C(sV(s) - v(0) = I(s)Capacitor of C (farads):dtI(s) + v(0) -[1(s) +Cv(0)]V(s) =SCsI di(t)Inductor ofL (henries):v(t)dt(0)V(s) = LsI(s) - Li(O) = Ls I(s)1S
Derivation of Models for Electrical Systems dt C dv(t) = i(t) C(sV (s) − v(0)) = I (s) Resistor of R (ohms): Capacitor of C (farads): Cs s Cs Inductor of L(henries): V(s) = 1 I(s) + v(0) = 1 I(s) +Cv(0) s V (s) = LsI(s) − Li(0) = Ls I (s) − i(0) v(t) = Ri(t) V(s) = R I(s) dt v(t) = L di(t) (1) Start from the basic circuit component principle: Basic Voltage-CurrentRelations:
V=-RIVRV= RIVR1dtdtdlVdtdt
+V I -+VI -+V I R V = RI CL dt I = C dV dt V = L dI +V I -+VI -+V I R V = −RI CL dt I = − C dV dt V = − L dI
V=RIVRV=-RI1dvdtdtdlVdlVdtdt
-V I +-V+ I +-V I R V = −RI CL dV I = −C dt dt V = − L dI I -V+-V+ I +-V I R V = RI CL dt I = C dV dt V = L dI
(2)BuildUp Circuit InterconnectionsExample: The RC-Branch Model in a BatteryRpVocvVpR,dtpdvdv2PRdtdtRD
Rp R v i vocv vp + - Cp ip - + p p p p p p p p p dvp , vp R dvp dvp vp C dt R dt R C C i =C = i −i dt = i − vp = − + 1 i (2) Build Up Circuit Interconnections Example: The RC-Branch Model in a Battery
Initial Condition Response (Zero-Input Response)i = O, the initial conditionis v, (O)dyVRpCp v(t)=v(0)e二dtR,C.pDT,= R,C,= Time ConstantIf C, is small, then the time constant is small= The initial consdition response will go down to zero relativelyfast= The RC branch will reach the steady state fastSteady State of theRC Branch(after the initial condition response diminishes)dyP= O and v(o) is now a constant.The steady state meansdtdyvFromwehave i, =0dt(8)VFromi-i, =i, we have v,(o)= R, iR,p
p p RpCp p p p p dv v dt R C i = 0, the initial condition is vp (0) − t = − v (t) = v (0)e Initial Condition Response (Zero-Input Response) Tp = RpCp = Time Constant If Cp is small, then the time constant is small. The initial consdition response will go down to zero relativelyfast The RC branch will reach the steady state fast. Steady State of the RC Branch (after the initial condition response diminishes) p p p p p p p p dt dt R dv The steady state means p = 0 and v () is now a constant. Fromi = C dvp , wehave i = 0 v () From p = i −i = i, we have v () = R i