# Transient Analysis

Quick Review of Theory

## Example 1

Find the voltage, Vc(t), over the capacitor for all time t>0.

## Example 2

The voltage source Vs in the circuit shown below is nominally 5V. A power surge at time t=0 causes it to instantaneously jump to 10V, and then at time t=5ms, it instantaneously drops back to 5V. Express the current flowing down through the 30 ohm resistor (as a function of time).
a) What is the 30 ohm resistor current at t=0+?
b) What would the final 30 ohm resistor current be if Vs stayed at 10V?
c) What is the 30 ohm resistor current at t=5ms+?
d) What is the final 30 ohm resistor current?
e ) Write the full expression(s) for the 30 ohm resistor current as a function of time.

## Example 3

The switch in the circuit shown below closes at t=0 and then opens at time t=5ms. Solve for the voltage Vx over the 3K resistor (as a function of time).

## Example 4

Design a circuit that has a critically damped response using a 10mH inductor, a 1mF capacitor, a 10 V source with a 5 O internal source resistance, and a load resistance that you choose.

Solution

## Example 5

In the circuit shown above, the switch is controlled by the voltage Vx:

Vx > 2V, the switch opens
Vx < 0.1V, the switch closes

A sketch of the voltage Vx over time is shown below (the scale of each axis is not shown). Give a mathematical expression to describe Vx during the first period. You will need to describe it in two different portions of time: as Vx rises and as Vx falls, so be sure to note the time range during which your formula is valid.

Solution

## Example 6

You and your lab partner wire the circuit shon below on a protoboard.  However, the 2 KW resistor is inserted in the board poorly and when your partner touches the resistor it becomes disconnected momentarily from the circuit at time 0 and then reconnected after 100ms.

(a) Describe the current i(t) flowing right through the 8KΩ resistor by providing the mathematical formula(s) for t>0 (split into multiple intervals if needed, but clearly label the time range of each interval).

(b) Sketch a graph of i(t) for t>0.

Hint

Solution

## Example 7

A 12-V power supply is connected to a 1K ohm resistor and a 3mF capacitor which are in series. The power supply has an internal source resistance of approximately 50 ohms.

(a) Draw a diagram to model this circuit

(b) How long will it take the circuit to reach steady-state from the time the power supply is switched on?

(c) Sketch the current through the 1K ohm resistor as a function of time.

## Example 8

Given a course voltage of 10V, a source resistance of 10 ohms, a 10mH inductor, and a 5nF capacitor, choose a load resistance and draw a circuit that will have an underdamped (decaying sine wave) second order transient response.

## Example 9

You and your lab partner wire the circuit shown below on a protoboard. However, the voltage source is poorly connected to the board and when your partner jiggles the cable, it becomes disconnected momentarily from the circuit at time 0 and then reconnected after 100ms.

a) Use transient analysis to describe the current i(t) flowing right through the 8 Kohm resistor by providing the mathematical formula(s) for t>0 (split into multiple intervals if needed, but clearly label the time range of each interval).

b) Sketch a graph of i(t) for t>0.

c) What is the current at time 50ms and at time 150ms?

Solution

## Example 10

In lab you try to charge up a large capacitor (50 mF) using the circuit shown below. However, when you turn on the supply, the 10 ohm resistor makes a loud pop, turns black, and smoke rises! The other students in the lab give you a dirty look as a bad smell permeates the room.
• What went wrong? Quantify.
• What resistor value should you have used?
• How long will it take the capacitor to charge up using the new resistor?

Solution

## Example 11

In the circuit below, resistor R3 is suddenly removed, leaving a gap (an open circuit). Sketch the current through R2 from this moment on.
• Label the graph scales to show the quantified maximum and quantified minimum current that flows.
• At what time is the current in R2 slightly more than half way between the maximum and minimum?
• At what time is the current in R2 at exactly 20.0 mA?

Solution

## Example 12

The circuit shown below starts out without the 4 Kohm resistor connected. At time t=0, the 4 Kohm resistor is connected in parallel to the 1 nF capacitor. After 10us, the 4 Kohm resistor is removed. Compute the voltage across the capacitor for all time. Give your answer as one or more formulas (clearly indicating the time range for each formula) and a sketch of VC versus time.

Solution

## Example 13

The voltage supply in the circuit shown below is switched on at time 0. Compute the voltage over the 3K ohm resistor for all time.

Solution

(c) 2014, Steven H. VanderLeest