Scientists and engineers find it useful to consider the physical world from two points of view:

1. The steady state operation of a system, and
2. The temporary transient behavior that occurs in transition between various steady-state conditions.

While this bifurcation doesn't apply to everything (for example, a random signal might never reach a "steady state"), it does apply to an enormous number of practical systems and makes their design and analysis more tractable.

Consider a turbofan jet engine used on a commercial airliner. There are a number of different modes of steady state operation. For example:

• Off and cold is a steady state.
• On but idle on the ground is a second steady state.
• On at full takeoff power is a third steady state.
• Cruising at 30% power is a fourth steady state.

There are possibly infinitely many steady states. They may not along a single dimension (percent thrust, for example), but may also be multidimensional. For the jet engine, environmental effects are significant: for example, the "full takeoff power" condition when the air is $-20^{\circ}\text{C}$ outside may be very different from when it's $+40^{\circ}\text{C}$.

Each of these steady states has different values of fuel consumption, thrust produced, turbine RPMs and pressures -- and hundreds of other variables.

The advantage of steady state analysis is that we can use a much simpler set of equations to make a workable model of steady state operation because we only need model what happens when that long-term steady state is reached, rather than the full dynamics of achieving it.

### Transient Behavior

In contrast, the transient behavior (non-steady state) happens when we are transitioning from one steady-state to another.

To continue our jet engine example: when the pilot increases throttle from idle to takeoff power, it takes several seconds for the new steady state to be reached.

At a physical level, that's because, for example the turbine itself has significant angular mass (moment of inertia), and other closely-related systems like fuel pumps, oil pumps, etc. have their own sources of delay. It might take several seconds to go from idle to full power, and then many more seconds later for that power to accelerate the significant mass of the airplane.

Without getting too technical, we may call this behavior delay or frequency response, two very closely related concepts.

Sometimes, this delay or frequency response is imposed intentionally by the designer. For example, in the jet engine, perhaps the engineers know that if they spin up the turbine too rapidly, it can put too much stress on the brittle turbine blades, and may cause them to crack and fail. In this case, the system designer might artifically limit the acceleration or deceleration of the turbine.

Or perhaps the limitation is one level higher up: perhaps they found that if the left and right engines of a plane spin up to full power at significantly different rates, that there's a net yaw torque on the plane which causes the pilots to lose control. In this case, the system designer might artifically limit the two engines to be synchronized closely enough, by artifically slowing one via a control system somewhere.

### Multi-Modal Systems

The concept of steady state doesn't have to be along just a single axis, either. Many systems have multiple modes of operation which are quite different from each other. This is important because different modes have different functions and can often be analyzed separately with different models.

Here are two quick examples of multiple modes of operation from the jet engine:

#### Auxiliary Power Unit

When a gasoline-powered car is started, it must draw energy from the battery to turn the starter motor and to power the engine's fuel injection and control electronics. But once the engine is started and running, the alternator converts mechanical energy into electrical energy both to power the electronics and recharge the battery.

Similarly, when an airplane's jet engine is first started, it needs to be supplied power from elsewhere on the plane for the startup process. This power comes from an Auxiliary Power Unit (APU), an additional, tiny jet engine (that is itself small enough to be jump-started from batteries). This APU engine is not designed to generate thrust. It only powers an electrical generator, which is only powerful enough to get the main engines started, and possibly some critical hydraulic or air compressor equipment. But, after the main jet engines are running, the main turbines have built-in generators which supply electrical power to the rest of the plane, so the APU can be turned off after main engines are running. Operating with and without the APU are two quite different steady states.

#### Thrust Reverser

Another example of a different axis of operation is that, immediately after landing, many jet engines deploy a thrust reverser. This actually reconfigures the engine exhaust mechanically so that instead of pushing air backwards, it pushes air forwards, slowing the plane faster than the plane's wheel brakes could alone. This thrust reverser operation mode is another entirely different steady state.

### Physical Stresses of Mode Transition

Transitioning between modes and/or between different steady states can impose loads on your design (whether electronic, mechanical, thermal, or other) that normal steady state operation does not.

An electrical example is the concept of inrush current. Many components such as transformers, motors, switching power supplies, and capacitors may draw substantially more current when they're first turned on than when they've reached steady state a few seconds later. This can significantly change your worst case design requirements for what each component has to be designed to handle. For example, fuses, circuit breakers, wires, PCB traces, and more must be redesigned to accomodate this higher-than-normal transition load.

For a quick example, check out this simple circuit:

Interactive Exercise Click the circuit, click "Simulate," then "Run Time-Domain Simulation." Without worrying yet about what this circuit does precisely, observe that the steady state on-current is only about 2 amps. However, the spike when the switch is first flipped is over 100 amps -- over 50 times greater current than the steady state! As we'll discuss in Chapter 2's sections on Resistance and Ohm's Law and Practical Resistors: Power Rating (Wattage), higher current means higher power dissipation, which can lead to resistor failure. If our fuse, wiring, printed circuit board, and other components are not properly designed to handle the brief spike of 100 amps, they can be permanently damaged or destroyed, even if they would have handled the steady state on and steady state off conditions perfectly fine.

This is not a purely academic example: many professional electronics designers have made some version of this mistake, with destructive and expensive consequences. For many components and electronic systems, the most likely time for a failure to occur is during startup, because of inrush current or similar effects.

In any case, it is often useful to model both the steady state (a simplified model that is more useful in higher-level system design), and the transient or non-steady state behavior.

Just as in the Linear & Nonlinear section, there are a few different definitions of steady state, each of which builds on the previous layer, and all can be useful from an engineering perspective.

### Level 1: Everything at Rest

The idea that a system is at rest is one that is often used in analyzing physical systems, but it's not really true at any non-absolute-zero temperature. Thermal motion of atoms and electrons keeps them vibrating and in constant motion.

This is not purely a theoretical consideration: in fact, thermal motion and collisions between atoms and electrons is exactly why Ohm's Law is a linear relationship, as we'll discuss later.

Electrons at Rest wouldn't be moving, so there'd never be a nonzero current if everything was at rest.

Semiconductors also don't work at absolute zero. Thermal processes are necessary to move electrons to a conduction band where they are free to move through the lattice.

While we can sometimes use the term "at rest" as our steady state criteria, it's usually qualified with a number of unspoken assumptions or is too simplistic. Let's look at other ways to define steady state below.

### Level 2: d/dt = 0

We can consider steady state from a mathematical perspective, rather than a physical one. We can simply say that at steady state, the time derivative of all our variables of interest is zero:

$$\frac {d} {dt} = 0$$

This definition allows for nonzero DC (direct current) currents and voltages. We can have some macroscopic current running through a resistor at some nominally constant flow rate, even though when we zoom in to look at individual electrons in motion, their individual behavior is chaotic and dominated by random collisions. (See the later section on Thermodynamics, Energy, and Equilibrium for more.)

This zero derivative assumption is often called operating point analysis or a DC solution, and it's the technique that circuit simulation software like CircuitLab uses when asked to find a DC solution of a circuit: it writes out all the differential equations of the circuit, and then discards any derivative terms.

(We'll look more at solving linear DC circuits in Chapter 2, with Kirchhoff's Voltage Law and Kirchhoff's Current Law and Solving Circuit Systems.)

This type of DC analysis is very useful, and is the type of analysis generally made before any deeper analysis (such as time-domain or frequency-domain analysis) is done.

Often, this assumption is good enough even when there's a hidden non-steady variable. For example, we might not be concerned about a battery's decreasing state of charge, as long as the circuit it's powering is under steady state operation -- until the battery runs out!

### Level 3: Linear Sinusoidal AC Steady State

Beyond the DC-only steady states, there is another very useful class of steady state.

Imagine a steady tone, such as a single musical note played from a speaker or musical instrument. While the air is in constant motion vibrating back and forth to carry the tone, we can consider this to be a periodic steady state. If all variables have the same values at the start of cycle #2 as they had at the start of cycle #1, then we can cyclically continue that situation indefinitely, and only analyze a single cycle in greater detail.

In the special case where the system is linear and the input is sinusoidal, we have something even more powerful.

Because the system is linear, a sinusoid at the input will mean that we'll find sine waves at any output we consider, at the same frequency as the input, but possibly at different amplitudes and phases.

Furthermore, even if our input is periodic but nonsinusoidal, as long as the system is linear we can actually decompose our input into a bunch of different sine waves and consider them independently, and then reassemble, so the same technique applies. We'll discuss more in the Laplace and Fourier Transforms section.

This linear sinusoidal AC steady state is where the Complex Number notation for phasors is incredibly helpful, and we'll discuss this more in the Linear Time-Invariant (LTI) Systems and Frequency Domain sections.

### Level 4: Nonlinear Periodic Steady State

The analysis discussed in Level 3 relates only to linear systems.

However, the analysis of periodic steady state is useful even in highly nonlinear systems, like switching power supplies.

While we won't go into more detail mathematically about nonlinear periodic steady state analysis, the key assumption in solving for a periodic steady state is to set the values of interest (such as a capacitor's level of charge, or an inductor's current) equal at the start and end of each period so that the overall period remains a cycle.

### Level 5: Active Feedback Steady State

Many systems have elements that look approximately steady state at a high level, but when we look more closely, they may operate with some form of active feedback control to maintain that approximate state.

This activity may involve active changes to reject disturbances and respond to noise. It may involve substantial changes in some underlying state variables in order to make other variables approximately steady.

Consider an airplane suddenly encountering gust of wind. The engines may have to briefly throttle up or down (possibly by action of the human pilot, or possibly by electronic autopilot) just to maintain the same airspeed. The engine speed, fuel flow, temperature, and other variables may experience a major transient as they work to maintain a steady airspeed.

Or for a domestic example, consider a thermostat turning a furnace on and off to maintain a comfortable temperature in a house. To the person designing the thermostat and heater, it's important to consider that the system is cycling between on and off to maintain temperature. However, to the person in the house, hopefully it appears that the house is simply maintaining an approximately steady state temperature. It all depends on at what level we're looking at the system. If the house's thermostat and heater system is designed well and working well, we can often treat the indoor temperature as steady state. But, if for some reason the feedback system is not working well, we often must break the steady state assumption to diagnose the issue.

Active feedback steady state is all around us in our modern lives, and we often don't notice it until it breaks in some way.

### Macroscopic Steady State and Microscopic Chaos

What's the pressure within an unopened soda can? It seems like it's a system at rest. We can come up with a number -- perhaps it's 200 kPa. That's a pressure exerted on the entire surface area of the interior of the can: 200,000 Newtons of force per square meter, or 20 Newtons per square centimeter of wall area.

(By the way, did you know that internally pressurized containers like cans and bottles get more strength due to pressurization compared than their non-pressurized counterparts? That's why, for example, soda can be in a thin aluminum can while unpressurized canned soup has to be in a thicker, heavier steel one.)

For our purposes the number doesn't matter, but if we were able to zoom in closely enough, we wouldn't see a steady state 200kPa. Instead, we'd see billions and trillions of individual random collisions between gas molecules and the aluminum wall every second, with each collision contributing a tiny impulse. When these individual molecular impulses are summed up, on average, there's some average pressure which we can measure. We'd also see collisions between pairs of gas molecules.

Furthermore, we wouldn't see a static interface between liquid and gas: instead, we'd see gas molecules becoming dissolved into the liquid, and others coming out of solution and into the gas space at the top of the container. And trillions of other interactions within the container.

All that activity doesn't sound like much of a steady state, but that's happening in every unopened soda can on the planet right now!

We'd go crazy trying to keep track of all those interactions, so we don't. We make assumptions and we basically do accounting to track averages over reasonable periods of time.

Because of the law of large numbers, a concept from probability and statistics, when we average over billions and trillions of random interactions, it looks like the individual randomness largely disappears into a smooth result.

This lets us say, "For a soda can at rest, the pressure inside is about 200kPa and there's about 10mL of gas volume at the top." This is a good enough steady-state approximation that we can design everything else around it. In particular, if the dynamics of this equilibrium are much faster than any time scales of interest, we can basically ignore the dynamics even as the situation changes to a new steady state.

Engineering is a hierarchial process, and making the assumption of steady state at any of these levels can be a helpful simplification. Learn to appreciate both the underlying micro-scale interactions and the aggregate-level simplification: both have their place and both contribute valuable intuition for analysis and design.

### What's Next

In the next section, Lumped Element Model, we'll talk more about how and when we can treat complex systems as simpler building blocks.