8/3/2019 Transient 1st

1/3

RC and RL Transient Analysis Basics

In these notes, we briefly review the basics of RC and RL circuits and their response

(transient) to a step change in an input voltage or current. The RC and RL circuits

discussed lead to 1st order differential equations in time, and have basic solutions typicalof 1st order differential equations.

A. GENERAL SOLUTION

The general solution for the four basic circuits is

X(t) = 1 + 2/

= (= ) + (= 0+

) (= )[ ] /

Here, X(t) is the transient output voltage or transient output current, depending on the

specific circuit being considered. The time constant is either RC or L/R, depending on

whether one has an RC or an RL circuit.

To obtain the values of X(t = 0+) and X(t = infinity), we use DC analysis. Before t = 0,

the input is constant (current source or voltage source depending on circuit). Under DC

conditions, the CAPACITOR becomes an OPEN CIRCUIT while an INDUCTOR

becomes a CLOSED CIRCUIT. Use these replacements to find the DC currents and

voltages at t < 0, before the input switches.

CAPACITORS: Use the fact that "voltages stick on capacitors" to translate a

capacitor voltage just before t = 0 into a capacitor voltage just after t = 0.

INDUCTORS: Use the fact that "currents stick through inductors" to translate an

inductor current just before t = 0 into an inductor current just after t = 0.

B. RC NETWORKS (time constant is RC)

Figure 1a below illustrates the case of series combination of a voltage source, resistorand capacitor. The voltage source switches between voltage levels at t=0 as shown in

Figure 1b. In this case, the relevant variable (voltage or current) is the common

current I(t) flowing around the loop, through the voltage source, resistor, and

capacitor.

The variable X(t) above is therefore the common current I(t).

Knowing the common current, the voltages across the resistor and

capacitor can be readily calculated.

8/3/2019 Transient 1st

2/3

Figure 2a below illustrates a parallel combination of a current source, resistor and

capacitor. The current source switches between current levels at t=0 as shown in

Figure 2b. In this case, the relevant variable would be the common voltage V(t)

across the current source, resistor and capacitor.

The variable X(t) above is therefore the common voltage V(t). Knowing the common voltage V(t), the current through the resistor or

capacitor can be readily calculated.

Vin(t)

Vout(t)R

C

(a)

+

Vin(t)

time tt=0

V0

(b)

V1Current

I

Figure 1

Iin(t)

Vout(t)R

C

(a)

Iin(t)

time tt=0

I0

(b)

I1

VoltageV

Figure 2

C. RL NETWORKS (time constant is L/R)

Figure 3a below illustrates the case of series combination of a voltage source, resistor

and inductor. The voltage source switches between voltage levels at t=0 as shown in

Figure 3b. In this case, the relevant variable (voltage or current) is the common

current I(t) flowing around the loop, through the voltage source, resistor, and

capacitor.

The variable X(t) above is therefore the common current I(t).

8/3/2019 Transient 1st

3/3

Knowing the common current, the voltages across the resistor and

capacitor can be readily calculated.

Figure 4a below illustrates a parallel combination of a current source, resistor and

inductor. The current source switches between current levels at t=0 as shown in

Figure 4b. In this case, the relevant variable would be the common voltage V(t)across the current source, resistor and capacitor.

The variable X(t) above is therefore the common voltage V(t).

Knowing the common voltage V(t), the current through the resistor or

capacitor can be readily calculated.

For RL circuits, the value of X(t = 0+) is based on the value of the current through the

inductor just before the input switches. Since the current through an inductor can not

change instantaneously, the current just after switching is the same as just before

switching. X(t = 0+) = X(t = 0-).

Vin(t)

Vout(t)R

L

(a)

+

Vin(t)

time tt=0

V0

(b)

V1

CurrentI

Figure 3

Iin(t)

Vout(t)R

L

(a)

Iin(t)

time tt=0

I0

(b)

I1

VoltageV

Figure 4