Transient analysis of electrical circuits

Circuit transient analysis

1. The generation of transient process and the rule of switching routes

What is a transient process

1. Steady state: a stable working state, not necessarily voltage and currentkeep constant, You can also pressPeriodic change。

2. Transient state: from a steady stateConversionortransitionTo another stable state.

   Transition example:
   
   Transition example:
   

Causes of transient processes

3. Internal reason: Contains energy storage components
4. External cause: change of routes occurred

The significance of studying transient processes

1. Utilization: Generate the required signal
2. Prevention: excessive voltage or excessive current may occur

Tasks for studying transient processes

1. Overall task: solve the differential equation of the relationship between excitation and response
2. Specific tasks:
3. Initial value $f(0+)$
4. Steady state value $f(\infty )$
5. Time constant τ
6. Differential equation solution or curve

Rule of change

The essential reason for the non-mutation:
The electric field energy W _c and the magnetic field energy W _L stored by the energy storage element do not change at the moment of switching.

Loop rule: At the moment of switching, the capacitor voltage u _c and the inductor current i _L cannot undergo mutations.
C ： u _c（0 +） = u _c（0-）
L ： i _L（0+） = i _L（0-）
When C and L are constants, the amount of charge and magnetic flux chain are conserved

Determination of initial value and steady state value

1. Determination of initial value

• Find the 0- of u _c and i _L
• Get 0+ of u _c and i _L
• Find the initial values ​​of other currents and voltages
  [Note]
  1.The current in C and the voltage in L, as well as the voltage and current in the resistance can change abruptly
  2. The initial energy storage is 0: C short circuit, L open circuit
  Initial energy storage: C is equivalent to a constant voltage source, L is equivalent to a constant current source

2. Determination of steady state value

• C is equivalent to an open circuit, and L is equivalent to a short circuit, ie $ic（\infty ）=0$ ,$andL(\infty )=0$
• Find the steady state values ​​of other currents and voltages
  

• The voltage displayed by the voltmeter is actually the voltage flowing through itself
• At the moment of switching, a high reverse voltage will appear at both ends of the voltmeter, which will damage the voltmeter.
• Solution: 1. Before the switch is disconnected, take the voltmeter; 2. Connect a diversion diode

2. Response of RC circuit

Zero input response

Definition: After switching the circuit, there is no power excitation, the circuit response is generated by the initial value of the capacitive element

u _c (0+)=u _c (0-)=U ₀
to obtain the first-order linear homogeneous differential equation: $RC\frac{d{u}_c}{\mathrm{d}t}+u_c=0$
general solution:
$u_c\left(t\right)=U_0{e}^{\frac{-t}{\tau }}=u_c\left(0+\right)e^{\frac{-t}{\tau }}$（t≥0）

Speed ​​of discharge:

• Time constant $\tau$ =RC (unit s) determines howfastu_cdecays (generally 3~5$\tau$ , that the transient process is basically over)
• U _{0 is} certain, the larger the C, the more Q, the more energy storage, the slower the discharge
• The larger the R, the smaller the i, the slower the discharge

The energy obtained on the resistance is equal to the energy consumed during the discharge

Zero state response

Definition: After switching, there is power excitation, and the response of the circuit is generated by the external power excitation

u _c (0+)=u _c (0-)=0 (equivalent to a short circuit)
to obtain the first-order linear inhomogeneous differential equation:
$RC\frac{d{u}_c}{\mathrm{d}t}+u_c=U$
general solution:
$u_c\left(t\right)=U\left(1-e^{\frac{-t}{\tau }}\right)=u_c\left(\infty \right)\left(1-e^{\frac{-t}{\tau }}\right)$（t≥0）
(Steady state component + transient component)

Full response

Definition: With power supply excitation, the initial value of the capacitor is not 0

u _c ( 0+ ) = u _c (0-) = U ₀
to obtain the first-order linear inhomogeneous differential equation:
$RC\frac{d{u}_c}{\mathrm{d}t}+u_c=U$

3. Response of RL circuit

4. Three-element analysis method of first-order linear circuit