There are many elements of a house's design that make it net zero, or at least qualify it as a 'passiv haus'. What people need to realize is that a house built to code is the absolutely worst house you are allowed to build by law. Over the course of its life, a house built to code will be the most expensive to operate and maintain.
Roof
The typical ceiling rating is R-38. In an efficient home, the value rises to R-80 or more. This is accomplished by using more insulation in the attic space or ceiling joists and roof itself and by making sure that any venting does not allow the insulation to cool off, just the roof itself.
Then there's the slope and face. If you're going to mount solar panels on the roof, it should face the south, or at least south-east (statistically less cloud in the morning). Any roof that doesn't have panels on it should not be sloped toward the sun. This results in interesting shapes.
Walls
The typical thickness of walls in a standard house are only thick enough to fill with a limited amount of insulation. Most exterior walls are built with 2 x 4 studs which only allows for enough insulation to achieve an 'R' value of R-12.
Inside the home, any interior wall that will be exposed to sunlight in winter could be made of thermal mass, such as concrete or stone. I'll explain why in the next section on floors.
Moisture and air leakage are issues too. An efficient home must have air tight walls (and windows, when closed). It also must have a layer underneath the exterior cladding to allow for any water to get completely down and out of the wall.
Floors
Concrete acts as a heat sink. It absorbs heat slowly and releases it just as slowly as the air cools. A well designed house will situate its windows to allow sunlight to come through and shine and heat the concrete floor in winter most of the day, which lessens the load on other heat sources. Then at night, the heat is slowly released back into the air to lessen the load on other heat sources. Ordinarily, this heat sink wouldn't be enough to heat the house much, but because an efficient home has much better insulation, it makes a huge difference. Also, the overhangs only let this thermal warming of the floor happen when the sun is low in the sky, as in winter.
Incidentally, this only works to maximum advantage if the home is oriented east-west with the south exposure having a good deal of windows.
Radiant heat. Since the most efficient house has concrete floors, your floors are already a great heat sink. If that concrete is embedded with tubing, you can maintain a level of warmth beyond what the winter sun can provide by pumping warm water through the floor for radiant heat. An alternative is electric elements in the floor and using electricity instead. Radiant heating is so much more efficient than forced air heating, because the warmth rises out of the floor, walking on a warm floor gives the illusion of warmth even when the surrounding air is a bit cooler. It also sucks less humidity out of the air than would a forced air furnace or radiator system.
Foundation
You wouldn't think that much heat is lost in the basement through the foundation, but when your furnace starts operating less often in the spring, you notice your basement starts getting significantly cooler. That's because the heat stored in the concrete walls and floor release the heat into the air, both inside and outside and that heat isn't getting replaced if the furnace isn't running as often. In order to build a more energy efficient home, you need to insulate the concrete foundation inside and out. The designs I saw had slabs of foam insulation under the concrete basement floor and on both sides of the foundation walls. This improves the slab's insulation rating from near zero to R-16, and the basement walls from R-8 to R-36.
Windows
An east-west orientation ensures that the low morning and evening sun doesn't get much of an opportunity to enter the home through the windows, creating too much summer solar heating. The high midday sun shouldn't enter the south facing windows, especially if awnings, etc. are used as discussed in the next section. But the windows should allow the low midday winter sun to enter through as much window as possible to maximize solar heating of any thermal mass inside, typically concrete floor and possibly walls.
The north, west and east walls can have windows, but they should be as small as possible and higher up on the wall to minimize solar heating. Get the kind that open if possible to allow for cross ventilation in the warmer months.
Awnings and overhangs
In the high sun angle of summer, awnings or overhangs are needed over south facing windows to prevent direct sunlight from shining in the windows. You want to minimize solar heating. Some folks try to prevent the solar heating by using blinds instead, but the heat is still getting through the window itself. You could pick windows that have the blinds built in. That can help. But shading the window is the best. The awnings should not prevent the sun from coming in once summer is over, as this is going to help heat the thermal mass inside.
HVAC
The newest ultra high efficiency air-to-air heat exchangers can draw heat out of the outside air right down to -25C or lower and work as air conditioning in summer. When you've got the insulation of a passiv haus, this is pretty much all you need for heating and cooling. Most high efficiency homes even forego the furnace entirely. But some homes install an electric furnace as a backup.
Air exchange
Because a passiv haus is air tight, it doesn't get any fresh air when the windows are closed. These homes need a heat recovery ventilation system. This device brings fresh air from outside and using a very efficient heat transfer device, pre-heats the fresh air by extracting heat from the air that will be exhausted. It usually works out to be 75% efficient, but newer models are as much as 90% efficient. It makes for a healthy home with lots of air circulation and filtered air too, if that's necessary. The best systems get rid of excess moisture, VOCs, carbon dioxide and other toxins. They also make it unnecessary to exhaust bathroom air directly outside.
Your dryer needs to be considered here as well, because a normal dryer exhausts air outside. There are now condensing dryers that don't need to exhaust anything outside, but remove the water and dump it down the drain.
Hot water
You can use a few of the kind of solar panels that heat water to provide pre-heated or hot water, but these systems are too expensive now compared to the newest alternatives. Modern heat pump water heaters extract heat from the ambient air of your house to heat the water in the tank, and if that's not enough, then electric heating elements kick in. Some of these units can even exhaust cool air into the rest of the house in summer. Think about it, a hot water heater that also acts as an air conditioner. Smart.
Solar panels
The finishing touch of a passiv haus is generating your own electricity. This part of the project requires a lot of planning and consultation with the utility company. This is because some utilities will only allow you to generate a certain amount of power, so that you don't become a regular producer all year long. They want you to produce just enough power to provide for your home. Surpluses are fine, as long as they're not constant. The way around this is to also have battery storage for the excess. That way anything you collect in excess can be stored for later, when it's night or the sun isn't as strong.
One of the smartest things I saw was a house design where the panels were mounted on the overhangs shading the south facing windows. This prevented the entire roof from having to be covered in panels.
By the way, solar panels have really dropped in price these days, which is why they're becoming more viable even in less expensive, non net zero designs.
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