time,is called time constant.It is an important physical quantity to characterize the speed of transient process.Another characteristic number related with time constant r is so called half-value time Ti,which means half of time that Uc(t)decrease to its initial value or increase to a terminal value.It is also used to describe the speed of transient process.The relationship between Tin and r is: T12=ln2=0.693(ort=1.443T1n) (9) (2)series RL circuits The analysis is similar to RC circuits.Through which we can obtain the time constant r and half-value time Ti: ÷上，T2=0.693x=0.6931 (10) (3)series RLC circuit. Firstly,discuss the condition that a power supply is turned on suddenly,the equation of capacitor voltage Uc is LCU+RC Uc+Ve=E dt2 dt (11) Divide by LC on both sides,and let B=R/2L and 00= LC (12) Then the Eq.11 changes into: dU+260+U。=ogE dr2 dt (13) Eq.13 is a typical equation of damped oscillation.B is damping coefficient,@o is the natural frequency of the circuit.Together with the two initial condition of this process: dUc =0 =0:dr o (14) Therefore,the solution to Eq.13 depends on the ratio ofB and @o. 44 time, is called time constant. It is an important physical quantity to characterize the speed of transient process. Another characteristic number related with time constant τ is so called half-value time T1/2, which means half of time that UC（t）decrease to its initial value or increase to a terminal value. It is also used to describe the speed of transient process. The relationship between T1/2 and τ is: T1/2 =τln2 = 0.693（or τ = 1.443T1/2） （9） （2）series RL circuits The analysis is similar to RC circuits. Through which we can obtain the time constant τ and half-value time T1/2: R L T R L τ = , 1 / 2 = 0.693τ = 0.693 （10） （3）series RLC circuit. Firstly, discuss the condition that a power supply is turned on suddenly, the equation of capacitor voltage UC is U E t U RC t U LC C C C + + = d d d d 2 2 （11） Divide by LC on both sides，and let β = R 2L and LC 1 ω0 = （12） Then the Eq.11 changes into: U E t U t U C C C 2 0 2 2 0 2 d d 2 d d + β + ω =ω （13） Eq. 13 is a typical equation of damped oscillation. β is damping coefficient，ω0 is the natural frequency of the circuit．Together with the two initial condition of this process: 0 0 = C t= U ； 0 d d 0 = t= C t U （14） Therefore, the solution to Eq. 13 depends on the ratio of β and ω0．