What are Transients? - Transients describe the behavior of a circuit before it reaches a steady-state condition. This means that some new event takes place like a voltage source being turned on or a breaker being closed in, and, before the currents and voltages in the circuit can stabilize, there will be some short-lived (transient) timeframe where the old circuit parameters are transitioning to the new ones.
Transient behavior is caused by inductance and capacitance in circuits. Inductors do not permit the current flowing through them to change instantaneously. An instantaneous change in current (say from 0A to 1A in 0 seconds) would require an infinite amount of power. For the same reason, capacitors do not permit an instantaneous change in the voltage across them.
All real-world circuits have inductance and capacitance. Inductance always exists when current flows because of magnetic fields and capacitance always exists when voltages are present because of electric fields. Even for bare conductors, like on an overhead transmission line, there can be considerable amounts of inductance and/or capacitance present.
First Order Transients - In many circuits, either capacitance or inductance is dominant and the effect of the other can be neglected. A good example is a resistive load like a heater, fed by a transformer, being switched into service. The transformer is inductive and the heater is resistive.
These types of situation are considered first order transients. We can model the circuit's behavior under these conditions with a single resistance R and a single inductance L (or capacitance C, if applicable). The images below show the two different first order transient circuits.
An RL Circuit with a DC Voltage Source
An RC Circuit with a DC Voltage Source
Let's assume both the circuits above are completely de-energized before we turn on the voltage V. For the RL circuit, initially no current will flow. Over time though, the current will build its way up to a steady-state value equal to V/R. For the RC circuit, a reversed situation occurs. Current will flow initially at a magnitude of V/R, but it will taper off over time until no current flows at all.
Now, consider turning the same circuits' voltage sources off. For the RL circuit, current will continue to flow through the circuit because of the inductor. Eventually the resistor will burn off all of the stored energy in the inductor and the current will stop. For the RC circuit, the capacitor will act as the new voltage source using its stored energy and begin to discharge a current at a value of Vc/R. In other words, both circuits behave the same when discharging their stored energy.
How long do Transients Last? - The time it takes for a circuit to charge or discharge in the circuits above is given by a value known as the time constant 𝜏. 𝜏 is a measure of time that applies to both RL and RC circuits and describes the decay of the transient effect. Within three time constants, the transient effect will be about 95% gone and within 5 time constants, the transient effect is all but absent from the circuit. Time constants can be computed as follows:
RL Circuit: 𝜏 = L / R
RC Circuit: 𝜏 = R C
Where:
𝜏 is the time constant in seconds
R is the resistance in Ohms
L is the inductance in Henries
C is the capacitance in Farads
Time constants can be converted to power frequency cycles instead of seconds by multiplying the time constants by the frequency (60 Hertz in the United States).
Higher Order Transients - Some circuits have capacitances and inductances that both contribute to the circuit's behavior significantly. This could be described by a situation where a capacitor bank is switched into service near a transformer. The transformer has a large inductance, the capacitor bank has a large capacitance, and both of them have some resistance. Neglecting either value could lead to incorrect results. Below is an image of an RLC circuit, a simplified model of the transformer & capacitor bank example discussed here.
A Series RLC Circuit, One of the Simplest Higher-Order Systems
Higher order transient behavior is different from first order systems. The characterization of these systems requires us to know how "damped" the system is. Systems that are highly damped will behave like first order systems. Systems that are undamped will have ringing in their transient response. Instead of transitioning from a starting value to an ending value in a steadily increasing or decreasing way, they will oscillate.
The series RLC circuit above is a simple model, but is useful for approximating results in many instances. The damping coefficient of this circuit is given by:
d = R / 2 √(C / L)
Where d is the damping coefficient, a unitless measure of how damped the system is. When d > 1, the system is damped and will not exhibit any oscillations in its transient response. Notice that increasing the resistance and capacitance increases the damping coefficient, making the response more stable.
For real systems with many elements, there is no simple formula that can be applied to determine how the system will be damped. We have to perform a detailed time-domain analysis using the full device equations (current-voltage relationships) for inductors, capacitors, and resistors to make a determination. This same analysis is also used to determine the duration of transient effects.
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