Cellulosic ethanol

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Cellulosic_ethanol

Cellulosic ethanol is ethanol (ethyl alcohol) produced from cellulose (the stringy fiber of a plant) rather than from the plant's seeds or fruit. It can be produced from grasses, wood, algae, or other plants. It is generally discussed for use as a biofuel. The carbon dioxide that plants absorb as they grow offsets some of the carbon dioxide emitted when ethanol made from them is burned, so cellulosic ethanol fuel has the potential to have a lower carbon footprint than fossil fuels.

Interest in cellulosic ethanol is driven by its potential to replace ethanol made from corn or sugarcane. Since these plants are also used for food products, diverting them for ethanol production can cause food prices to rise; cellulose-based sources, on the other hand, generally do not compete with food, since the fibrous parts of plants are mostly inedible to humans. Another potential advantage is the high diversity and abundance of cellulose sources; grasses, trees and algae are found in almost every environment on Earth. Even municipal solid waste components like paper could conceivably be made into ethanol. The main current disadvantage of cellulosic ethanol is its high cost of production, which is more complex and requires more steps than corn-based or sugarcane-based ethanol.

Cellulosic ethanol received significant attention in the 2000s and early 2010s. The United States government in particular funded research into its commercialization and set targets for the proportion of cellulosic ethanol added to vehicle fuel. A large number of new companies specializing in cellulosic ethanol, in addition to many existing companies, invested in pilot-scale production plants. However, the much cheaper manufacturing of grain-based ethanol, along with the low price of oil in the 2010s, meant that cellulosic ethanol was not competitive with these established fuels. As a result, most of the new refineries were closed by the mid-2010s and many of the newly founded companies became insolvent. A few still exist, but are mainly used for demonstration or research purposes; as of 2021, none produces cellulosic ethanol at scale.

Overview

Cellulosic ethanol is a type of biofuel produced from lignocellulose, a structural material that comprises much of the mass of plants and is composed mainly of cellulose, hemicellulose and lignin. Popular sources of lignocellulose include both agricultural waste products (e.g. corn stover or wood chips) and grasses like switchgrass and miscanthus species. These raw materials for ethanol production have the advantage of being abundant and diverse and would not compete with food production, unlike the more commonly used corn and cane sugars. However, they also require more processing to make the sugar monomers available to the microorganisms typically used to produce ethanol by fermentation, which drives up the price of cellulos-derived ethanol.

Cellulosic ethanol can reduce greenhouse gas emissions by 85% over reformulated gasoline. By contrast, starch ethanol (e.g., from corn), which most frequently uses natural gas to provide energy for the process, may not reduce greenhouse gas emissions at all depending on how the starch-based feedstock is produced. According to the National Academy of Sciences in 2011, there is no commercially viable bio-refinery in existence to convert lignocellulosic biomass to fuel. Absence of production of cellulosic ethanol in the quantities required by the regulation was the basis of a United States Court of Appeals for the District of Columbia decision announced January 25, 2013, voiding a requirement imposed on car and truck fuel producers in the United States by the Environmental Protection Agency requiring addition of cellulosic biofuels to their products. These issues, along with many other difficult production challenges, led George Washington University policy researchers to state that "in the short term, [cellulosic] ethanol cannot meet the energy security and environmental goals of a gasoline alternative."

History

The French chemist, Henri Braconnot, was the first to discover that cellulose could be hydrolyzed into sugars by treatment with sulfuric acid in 1819. The hydrolyzed sugar could then be processed to form ethanol through fermentation. The first commercialized ethanol production began in Germany in 1898, where acid was used to hydrolyze cellulose. In the United States, the Standard Alcohol Company opened the first cellulosic ethanol production plant in South Carolina in 1910. Later, a second plant was opened in Louisiana. However, both plants were closed after World War I due to economic reasons.

The first attempt at commercializing a process for ethanol from wood was done in Germany in 1898. It involved the use of dilute acid to hydrolyze the cellulose to glucose, and was able to produce 7.6 liters of ethanol per 100 kg of wood waste (18 US gal (68 L) per ton). The Germans soon developed an industrial process optimized for yields of around 50 US gallons (190 L) per ton of biomass. This process soon found its way to the US, culminating in two commercial plants operating in the southeast during World War I. These plants used what was called "the American Process" — a one-stage dilute sulfuric acid hydrolysis. Though the yields were half that of the original German process (25 US gallons (95 L) of ethanol per ton versus 50), the throughput of the American process was much higher. A drop in lumber production forced the plants to close shortly after the end of World War I. In the meantime, a small but steady amount of research on dilute acid hydrolysis continued at the USFS's Forest Products Laboratory. During World War II, the US again turned to cellulosic ethanol, this time for conversion to butadiene to produce synthetic rubber. The Vulcan Copper and Supply Company was contracted to construct and operate a plant to convert sawdust into ethanol. The plant was based on modifications to the original German Scholler process as developed by the Forest Products Laboratory. This plant achieved an ethanol yield of 50 US gal (190 L) per dry ton, but was still not profitable and was closed after the war.

With the rapid development of enzyme technologies in the last two decades, the acid hydrolysis process has gradually been replaced by enzymatic hydrolysis. Chemical pretreatment of the feedstock is required to hydrolyze (separate) hemicellulose, so it can be more effectively converted into sugars. The dilute acid pretreatment is developed based on the early work on acid hydrolysis of wood at the USFS's Forest Products Laboratory. Recently, the Forest Products Laboratory together with the University of Wisconsin–Madison developed a sulfite pretreatment to overcome the recalcitrance of lignocellulose for robust enzymatic hydrolysis of wood cellulose.

In his 2007 State of the Union Address on January 23, 2007, US President George W. Bush announced a proposed mandate for 35 billion US gallons (130×10⁹ L) of ethanol by 2017. Later that year, the US Department of Energy awarded $385 million in grants aimed at jump-starting ethanol production from nontraditional sources like wood chips, switchgrass, and citrus peels.

Production methods

Bioreactor for cellulosic ethanol research.

The stages to produce ethanol using a biological approach are:

1. A "pretreatment" phase to make the lignocellulosic material such as wood or straw amenable to hydrolysis
2. Cellulose hydrolysis (cellulolysis) to break down the molecules into sugars
3. Microbial fermentation of the sugar solution
4. Distillation and dehydration to produce pure alcohol

In 2010, a genetically engineered yeast strain was developed to produce its own cellulose-digesting enzymes. Assuming this technology can be scaled to industrial levels, it would eliminate one or more steps of cellulolysis, reducing both the time required and costs of production.

Although lignocellulose is the most abundant plant material resource, its usability is curtailed by its rigid structure. As a result, an effective pretreatment is needed to liberate the cellulose from the lignin seal and its crystalline structure so as to render it accessible for a subsequent hydrolysis step. By far, most pretreatments are done through physical or chemical means. To achieve higher efficiency, both physical and chemical pretreatments are required. Physical pretreatment involves reducing biomass particle size by mechanical processing methods such as milling or extrusion. Chemical pretreatment partially depolymerizes the lignocellulose so enzymes can access the cellulose for microbial reactions.

Chemical pretreatment techniques include acid hydrolysis, steam explosion, ammonia fiber expansion, organosolv, sulfite pretreatment, SO2-ethanol-water fractionation, alkaline wet oxidation and ozone pretreatment. Besides effective cellulose liberation, an ideal pretreatment has to minimize the formation of degradation products because they can inhibit the subsequent hydrolysis and fermentation steps. The presence of inhibitors further complicates and increases the cost of ethanol production due to required detoxification steps. For instance, even though acid hydrolysis is probably the oldest and most-studied pretreatment technique, it produces several potent inhibitors including furfural and hydroxymethylfurfural. Ammonia Fiber Expansion (AFEX) is an example of a promising pretreatment that produces no inhibitors.

Most pretreatment processes are not effective when applied to feedstocks with high lignin content, such as forest biomass. These require alternative or specialized approaches. Organosolv, SPORL ('sulfite pretreatment to overcome recalcitrance of lignocellulose') and SO2-ethanol-water (AVAP®) processes are the three processes that can achieve over 90% cellulose conversion for forest biomass, especially those of softwood species. SPORL is the most energy efficient (sugar production per unit energy consumption in pretreatment) and robust process for pretreatment of forest biomass with very low production of fermentation inhibitors. Organosolv pulping is particularly effective for hardwoods and offers easy recovery of a hydrophobic lignin product by dilution and precipitation. AVAP® process effectively fractionates all types of lignocellulosics into clean highly digestible cellulose, undegraded hemicellulose sugars, reactive lignin and lignosulfonates, and is characterized by efficient recovery of chemicals.

Cellulolytic processes

The hydrolysis of cellulose (cellulolysis) produces simple sugars that can be fermented into alcohol. There are two major cellulolysis processes: chemical processes using acids, or enzymatic reactions using cellulases.

Chemical hydrolysis

In the traditional methods developed in the 19th century and at the beginning of the 20th century, hydrolysis is performed by attacking the cellulose with an acid. Dilute acid may be used under high heat and high pressure, or more concentrated acid can be used at lower temperatures and atmospheric pressure. A decrystallized cellulosic mixture of acid and sugars reacts in the presence of water to complete individual sugar molecules (hydrolysis). The product from this hydrolysis is then neutralized and yeast fermentation is used to produce ethanol. As mentioned, a significant obstacle to the dilute acid process is that the hydrolysis is so harsh that toxic degradation products are produced that can interfere with fermentation. BlueFire Renewables uses concentrated acid because it does not produce nearly as many fermentation inhibitors, but must be separated from the sugar stream for recycle [simulated moving bed chromatographic separation, for example] to be commercially attractive.

Agricultural Research Service scientists found they can access and ferment almost all of the remaining sugars in wheat straw. The sugars are located in the plant's cell walls, which are notoriously difficult to break down. To access these sugars, scientists pretreated the wheat straw with alkaline peroxide, and then used specialized enzymes to break down the cell walls. This method produced 93 US gallons (350 L) of ethanol per ton of wheat straw.

Enzymatic hydrolysis

Cellulose chains can be broken into glucose molecules by cellulase enzymes. This reaction occurs at body temperature in the stomachs of ruminants such as cattle and sheep, where the enzymes are produced by microbes. This process uses several enzymes at various stages of this conversion. Using a similar enzymatic system, lignocellulosic materials can be enzymatically hydrolyzed at a relatively mild condition (50 °C and pH 5), thus enabling effective cellulose breakdown without the formation of byproducts that would otherwise inhibit enzyme activity. All major pretreatment methods, including dilute acid, require an enzymatic hydrolysis step to achieve high sugar yield for ethanol fermentation.

Fungal enzymes can be used to hydrolyze cellulose. The raw material (often wood or straw) still has to be pre-treated to make it amenable to hydrolysis. In 2005, Iogen Corporation announced it was developing a process using the fungus Trichoderma reesei to secrete "specially engineered enzymes" for an enzymatic hydrolysis process.

Another Canadian company, SunOpta, uses steam explosion pretreatment, providing its technology to Verenium (formerly Celunol Corporation)'s facility in Jennings, Louisiana, Abengoa's facility in Salamanca, Spain, and a China Resources Alcohol Corporation in Zhaodong. The CRAC production facility uses corn stover as raw material.

Microbial fermentation

Main article: Ethanol fermentation

Traditionally, baker's yeast (Saccharomyces cerevisiae), has long been used in the brewery industry to produce ethanol from hexoses (six-carbon sugars). Due to the complex nature of the carbohydrates present in lignocellulosic biomass, a significant amount of xylose and arabinose (five-carbon sugars derived from the hemicellulose portion of the lignocellulose) is also present in the hydrolysate. For example, in the hydrolysate of corn stover, approximately 30% of the total fermentable sugars is xylose. As a result, the ability of the fermenting microorganisms to use the whole range of sugars available from the hydrolysate is vital to increase the economic competitiveness of cellulosic ethanol and potentially biobased proteins.

In recent years, metabolic engineering for microorganisms used in fuel ethanol production has shown significant progress. Besides Saccharomyces cerevisiae, microorganisms such as Zymomonas mobilis and Escherichia coli have been targeted through metabolic engineering for cellulosic ethanol production. An attraction towards alternative fermentation organism is its ability to ferment five carbon sugars improving the yield of the feed stock. This ability is often found in bacteria based organisms.

Recently, engineered yeasts have been described efficiently fermenting xylose, and arabinose, and even both together. Yeast cells are especially attractive for cellulosic ethanol processes because they have been used in biotechnology for hundreds of years, are tolerant to high ethanol and inhibitor concentrations and can grow at low pH values to reduce bacterial contamination.

Combined hydrolysis and fermentation

Some species of bacteria have been found capable of direct conversion of a cellulose substrate into ethanol. One example is Clostridium thermocellum, which uses a complex cellulosome to break down cellulose and synthesize ethanol. However, C. thermocellum also produces other products during cellulose metabolism, including acetate and lactate, in addition to ethanol, lowering the efficiency of the process. Some research efforts are directed to optimizing ethanol production by genetically engineering bacteria that focus on the ethanol-producing pathway.

Gasification process (thermochemical approach)

Fluidized Bed Gasifier in Güssing Burgenland Austria

The gasification process does not rely on chemical decomposition of the cellulose chain (cellulolysis). Instead of breaking the cellulose into sugar molecules, the carbon in the raw material is converted into synthesis gas, using what amounts to partial combustion. The carbon monoxide, carbon dioxide and hydrogen may then be fed into a special kind of fermenter. Instead of sugar fermentation with yeast, this process uses Clostridium ljungdahlii bacteria. This microorganism will ingest carbon monoxide, carbon dioxide and hydrogen and produce ethanol and water. The process can thus be broken into three steps:

1. Gasification — Complex carbon-based molecules are broken apart to access the carbon as carbon monoxide, carbon dioxide and hydrogen
2. Fermentation — Convert the carbon monoxide, carbon dioxide and hydrogen into ethanol using the Clostridium ljungdahlii organism
3. Distillation — Ethanol is separated from water

A recent study has found another Clostridium bacterium that seems to be twice as efficient in making ethanol from carbon monoxide as the one mentioned above.

Alternatively, the synthesis gas from gasification may be fed to a catalytic reactor where it is used to produce ethanol and other higher alcohols through a thermochemical process. This process can also generate other types of liquid fuels, an alternative concept successfully demonstrated by the Montreal-based company Enerkem at their facility in Westbury, Quebec.

Hemicellulose to ethanol

Studies are intensively conducted to develop economic methods to convert both cellulose and hemicellulose to ethanol. Fermentation of glucose, the main product of cellulose hydrolyzate, to ethanol is an already established and efficient technique. However, conversion of xylose, the pentose sugar of hemicellulose hydrolyzate, is a limiting factor, especially in the presence of glucose. Moreover, it cannot be disregarded as hemicellulose will increase the efficiency and cost-effectiveness of cellulosic ethanol production.

Sakamoto (2012) et al. show the potential of genetic engineering microbes to express hemicellulase enzymes. The researchers created a recombinant Saccharomyces cerevisiae strain that was able to:

1. hydrolyze hemicellulase through codisplaying endoxylanase on its cell surface,
2. assimilate xylose by expression of xylose reductase and xylitol dehydrogenase.

The strain was able to convert rice straw hydrolyzate to ethanol, which contains hemicellulosic components. Moreover, it was able to produce 2.5x more ethanol than the control strain, showing the highly effective process of cell surface-engineering to produce ethanol.

Advantages

General advantages of ethanol fuel

Ethanol burns more cleanly and more efficiently than gasoline. Because plants consume carbon dioxide as they grow, bioethanol has an overall lower carbon footprint than fossil fuels. Substituting ethanol for oil can also reduce a country's dependence on oil imports.

Advantages of cellulosic ethanol over corn or sugar-based ethanol

See also: Environmental and social impacts of ethanol fuel in the U.S., Indirect land use change impacts of biofuels, and Low-carbon fuel standard

U.S. Environmental Protection Agency Draft life cycle GHG emissions reduction results for different time horizon and discount rate approaches (includes indirect land use change effects)
Fuel Pathway	100 years + 2% discount rate	30 years + 0% discount rate
Corn ethanol (natural gas dry mill)(1)	-16%	+5%
Corn ethanol (Best case NG DM)(2)	-39%	-18%
Corn ethanol (coal dry mill)	+13%	+34%
Corn ethanol (biomass dry mill)	-39%	-18%
Corn ethanol (biomass dry mill with combined heat and power)	-47%	-26%
Brazilian sugarcane ethanol	-44%	-26%
Cellulosic ethanol from switchgrass	-128%	-124%
Cellulosic ethanol from corn stover	-115%	-116%
Notes: (1) Dry mill (DM) plants grind the entire kernel and generally produce only one primary co-product: distillers grains with solubles (DGS). (2) Best case plants produce wet distillers grains co-product.

Commercial production of cellulosic ethanol, which unlike corn and sugarcane would not compete with food production, would be highly attractive since it would alleviate pressure on these foodcrops.

Although its processing costs are higher, the price of cellulose biomass is much cheaper than that of grains or fruits. Moreover, since cellulose is the main component of plants, the whole plant can be harvested, rather than just the fruit or seeds. This results in much better yields; for instance, switchgrass yields twice as much ethanol per acre as corn. Biomass materials for cellulose production require fewer inputs, such as fertilizer, herbicides, and their extensive roots improve soil quality, reduce erosion, and increase nutrient capture. The overall carbon footprint and global warming potential of cellulosic ethanol are considerably lower (see chart) and the net energy output is several times higher than that of corn-based ethanol.

The potential raw material is also plentiful. Around 44% of household waste generated worldwide consists of food and greens. An estimated 323 million tons of cellulose-containing raw materials which could be used to create ethanol are thrown away each year in US alone. This includes 36.8 million dry tons of urban wood wastes, 90.5 million dry tons of primary mill residues, 45 million dry tons of forest residues, and 150.7 million dry tons of corn stover and wheat straw. Moreover, even land marginal for agriculture could be planted with cellulose-producing crops, such as switchgrass, resulting in enough production to substitute for all the current oil imports into the United States.

Paper, cardboard, and packaging comprise around 17% of global household waste; although some of this is recycled. As these products contain cellulose, they are transformable into cellulosic ethanol, which would avoid the production of methane, a potent greenhouse gas, during decomposition.

Disadvantages

General disadvantages

The main overall drawback of ethanol fuel is its lower fuel economy compared to gasoline when using ethanol in an engine designed for gasoline with a lower compression ratio.

Disadvantages of cellulosic ethanol over corn or sugar-based ethanol

The main disadvantage of cellulosic ethanol is its high cost and complexity of production, which has been the main impediment to its commercialization.

Economics

Although the global bioethanol market is sizable (around 110 billion liters in 2019), the vast majority is made from corn or sugarcane, not cellulose. In 2007, the cost of producing ethanol from cellulosic sources was estimated ca. USD 2.65 per gallon (€0.58 per liter), which is around 2–3 times more expensive than ethanol made from corn. However, the cellulosic ethanol market remains relatively small and reliant on government subsidies. The US government originally set cellulosic ethanol targets gradually ramping up from 1 billion liters in 2011 to 60 billion liters in 2022. However, these annual goals have almost always been waived after it became clear there was no chance of meeting them. Most of the plants to produce cellulosic ethanol were canceled or abandoned in the early 2010s. Plants built or financed by DuPont, General Motors and BP, among many others, were closed or sold. As of 2018, only one major plant remains in the US.

In order for it to be grown on a large-scale production, cellulose biomass must compete with existing uses of agricultural land, mainly for the production of crop commodities. Of the United States' 2.26 billion acres (9.1 million km²) of unsubmerged land, 33% are forestland, 26% pastureland and grassland, and 20% crop land. A study by the U.S. Departments of Energy and Agriculture in 2005 suggested that 1.3 billion dry tons of biomass is theoretically available for ethanol use while maintaining an acceptable impact on forestry, agriculture.

Comparison with corn-based ethanol

Currently, cellulose is more difficult and more expensive to process into ethanol than corn or sugarcane. The US Department of Energy estimated in 2007 that it costs about $2.20 per gallon to produce cellulosic ethanol, which is 2–3 times much as ethanol from corn. Enzymes that destroy plant cell wall tissue cost US$0.40 per gallon of ethanol compared to US$0.03 for corn. However, cellulosic biomass is cheaper to produce than corn, because it requires fewer inputs, such as energy, fertilizer, herbicide, and is accompanied by less soil erosion and improved soil fertility. Additionally, nonfermentable and unconverted solids left after making ethanol can be burned to provide the fuel needed to operate the conversion plant and produce electricity. Energy used to run corn-based ethanol plants is derived from coal and natural gas. The Institute for Local Self-Reliance estimates the cost of cellulosic ethanol from the first generation of commercial plants will be in the $1.90–$2.25 per gallon range, excluding incentives. This compares to the current cost of $1.20–$1.50 per gallon for ethanol from corn and the current retail price of over $4.00 per gallon for regular gasoline (which is subsidized and taxed).

Enzyme-cost barrier

Cellulases and hemicellulases used in the production of cellulosic ethanol are more expensive compared to their first generation counterparts. Enzymes required for maize grain ethanol production cost 2.64-5.28 US dollars per cubic meter of ethanol produced. Enzymes for cellulosic ethanol production are projected to cost 79.25 US dollars, meaning they are 20-40 times more expensive. The cost differences are attributed to quantity required. The cellulase family of enzymes have a one to two order smaller magnitude of efficiency. Therefore, it requires 40 to 100 times more of the enzyme to be present in its production. For each ton of biomass it requires 15-25 kilograms of enzyme. More recent estimates are lower, suggesting 1 kg of enzyme per dry tonne of biomass feedstock. There is also relatively high capital costs associated with the long incubation times for the vessel that perform enzymatic hydrolysis. Altogether, enzymes comprise a significant portion of 20-40% for cellulosic ethanol production. A recent paper estimates the range at 13-36% of cash costs, with a key factor being how the cellulase enzyme is produced. For cellulase produced offsite, enzyme production amounts to 36% of cash cost. For enzyme produced onsite in a separate plant, the fraction is 29%; for integrated enzyme production, the fraction is 13%. One of the key benefits of integrated production is that biomass instead of glucose is the enzyme growth medium. Biomass costs less, and it makes the resulting cellulosic ethanol a 100% second-generation biofuel, i.e., it uses no ‘food for fuel’.

Feedstocks

In general there are two types of feedstocks: forest (woody) Biomass and agricultural biomass. In the US, about 1.4 billion dry tons of biomass can be sustainably produced annually. About 370 million tons or 30% are forest biomass. Forest biomass has higher cellulose and lignin content and lower hemicellulose and ash content than agricultural biomass. Because of the difficulties and low ethanol yield in fermenting pretreatment hydrolysate, especially those with very high 5 carbon hemicellulose sugars such as xylose, forest biomass has significant advantages over agricultural biomass. Forest biomass also has high density which significantly reduces transportation cost. It can be harvested year around which eliminates long-term storage. The close to zero ash content of forest biomass significantly reduces dead load in transportation and processing. To meet the needs for biodiversity, forest biomass will be an important biomass feedstock supply mix in the future biobased economy. However, forest biomass is much more recalcitrant than agricultural biomass. Recently, the USDA Forest Products Laboratory together with the University of Wisconsin–Madison developed efficient technologies that can overcome the strong recalcitrance of forest (woody) biomass including those of softwood species that have low xylan content. Short-rotation intensive culture or tree farming can offer an almost unlimited opportunity for forest biomass production.

Woodchips from slashes and tree tops and saw dust from saw mills, and waste paper pulp are forest biomass feedstocks for cellulosic ethanol production.

Switchgrass (Panicum virgatum) is a native tallgrass prairie grass. Known for its hardiness and rapid growth, this perennial grows during the warm months to heights of 2–6 feet. Switchgrass can be grown in most parts of the United States, including swamplands, plains, streams, and along the shores & interstate highways. It is self-seeding (no tractor for sowing, only for mowing), resistant to many diseases and pests, & can produce high yields with low applications of fertilizer and other chemicals. It is also tolerant to poor soils, flooding, & drought; improves soil quality and prevents erosion due its type of root system.

Switchgrass is an approved cover crop for land protected under the federal Conservation Reserve Program (CRP). CRP is a government program that pays producers a fee for not growing crops on land on which crops recently grew. This program reduces soil erosion, enhances water quality, and increases wildlife habitat. CRP land serves as a habitat for upland game, such as pheasants and ducks, and a number of insects. Switchgrass for biofuel production has been considered for use on Conservation Reserve Program (CRP) land, which could increase ecological sustainability and lower the cost of the CRP program. However, CRP rules would have to be modified to allow this economic use of the CRP land.

Miscanthus × giganteus is another viable feedstock for cellulosic ethanol production. This species of grass is native to Asia and is a sterile hybrid of Miscanthus sinensis and Miscanthus sacchariflorus. It has high crop yields, is cheap to grow, and thrives in a variety of climates. However, because it is sterile, it also requires vegetative propagation, making it more expensive.

It has been suggested that Kudzu may become a valuable source of biomass.

Cellulosic ethanol commercialization

Fueled by subsidies and grants, a boom in cellulosic ethanol research and pilot plants occurred in the early 2000s. Companies such as Iogen, POET, and Abengoa built refineries that can process biomass and turn it into ethanol, while companies such as DuPont, Diversa, Novozymes, and Dyadic invested in enzyme research. However, most of these plants were canceled or closed in the early 2010s as technical obstacles proved too difficult to overcome. As of 2018, only one cellulosic ethanol plant remained operational.

In the later 2010s, various companies occasionally attempted smaller-scale efforts at commercializing cellulosic ethanol, although such ventures generally remain at experimental scales and often dependent on subsidies. The companies Granbio, Raízen and the Centro de Tecnologia Canavieira each run a pilot-scale facility operate in Brazil, which together produce around 30 million liters in 2019. Iogen, which started as an enzyme maker in 1991 and re-oriented itself to focus primarily on cellulosic ethanol in 2013, owns many patents for cellulosic ethanol production and provided the technology for the Raízen plant. Other companies developing cellulosic ethanol technology as of 2021 are Inbicon (Denmark); companies operating or planning pilot production plants include New Energy Blue (US), Sekab (Sweden) and Clariant (in Romania). Abengoa, a Spanish company with cellulosic ethanol assets, became insolvent in 2021.

The Australian Renewable Energy Agency, along with state and local governments, partially funded a pilot plant in 2017 and 2020 in New South Wales as part of efforts to diversify the regional economy away from coal mining.

US Government support

From 2006, the US Federal government began promoting the development of ethanol from cellulosic feedstocks. In May 2008, Congress passed a new farm bill that contained funding for the commercialization of second-generation biofuels, including cellulosic ethanol. The Food, Conservation, and Energy Act of 2008 provided for grants covering up to 30% of the cost of developing and building demonstration-scale biorefineries for producing "advanced biofuels," which effectively included all fuels not produced from corn kernel starch. It also allowed for loan guarantees of up to $250 million for building commercial-scale biorefineries.

In January 2011, the USDA approved $405 million in loan guarantees through the 2008 Farm Bill to support the commercialization of cellulosic ethanol at three facilities owned by Coskata, Enerkem and INEOS New Planet BioEnergy. The projects represent a combined 73 million US gallons (280,000 m³) per year production capacity and will begin producing cellulosic ethanol in 2012. The USDA also released a list of advanced biofuel producers who will receive payments to expand the production of advanced biofuels. In July 2011, the US Department of Energy gave in $105 million in loan guarantees to POET for a commercial-scale plant to be built Emmetsburg, Iowa.

Electrical impedance

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Electrical_impedance

In electrical engineering, impedance is the opposition to alternating current presented by the combined effect of resistance and reactance in a circuit.

Quantitatively, the impedance of a two-terminal circuit element is the ratio of the complex representation of the sinusoidal voltage between its terminals, to the complex representation of the current flowing through it. In general, it depends upon the frequency of the sinusoidal voltage.

Impedance extends the concept of resistance to alternating current (AC) circuits, and possesses both magnitude and phase, unlike resistance, which has only magnitude.

Impedance can be represented as a complex number, with the same units as resistance, for which the SI unit is the ohm (Ω). Its symbol is usually Z, and it may be represented by writing its magnitude and phase in the polar form |Z|∠θ. However, Cartesian complex number representation is often more powerful for circuit analysis purposes.

The notion of impedance is useful for performing AC analysis of electrical networks, because it allows relating sinusoidal voltages and currents by a simple linear law. In multiple port networks, the two-terminal definition of impedance is inadequate, but the complex voltages at the ports and the currents flowing through them are still linearly related by the impedance matrix.

The reciprocal of impedance is admittance, whose SI unit is the siemens, formerly called mho.

Instruments used to measure the electrical impedance are called impedance analyzers.

History

Perhaps the earliest use of complex numbers in circuit analysis was by Johann Victor Wietlisbach in 1879 in analysing the Maxwell bridge. Wietlisbach avoided using differential equations by expressing AC currents and voltages as exponential functions with imaginary exponents (see § Validity of complex representation). Wietlisbach found the required voltage was given by multiplying the current by a complex number (impedance), although he did not identify this as a general parameter in its own right.

The term impedance was coined by Oliver Heaviside in July 1886. Heaviside recognised that the "resistance operator" (impedance) in his operational calculus was a complex number. In 1887 he showed that there was an AC equivalent to Ohm's law.

Arthur Kennelly published an influential paper on impedance in 1893. Kennelly arrived at a complex number representation in a rather more direct way than using imaginary exponential functions. Kennelly followed the graphical representation of impedance (showing resistance, reactance, and impedance as the lengths of the sides of a right angle triangle) developed by John Ambrose Fleming in 1889. Impedances could thus be added vectorially. Kennelly realised that this graphical representation of impedance was directly analogous to graphical representation of complex numbers (Argand diagram). Problems in impedance calculation could thus be approached algebraically with a complex number representation. Later that same year, Kennelly's work was generalised to all AC circuits by Charles Proteus Steinmetz. Steinmetz not only represented impedances by complex numbers but also voltages and currents. Unlike Kennelly, Steinmetz was thus able to express AC equivalents of DC laws such as Ohm's and Kirchhoff's laws. Steinmetz's work was highly influential in spreading the technique amongst engineers.

Introduction

In addition to resistance as seen in DC circuits, impedance in AC circuits includes the effects of the induction of voltages in conductors by the magnetic fields (inductance), and the electrostatic storage of charge induced by voltages between conductors (capacitance). The impedance caused by these two effects is collectively referred to as reactance and forms the imaginary part of complex impedance whereas resistance forms the real part.

Complex impedance

A graphical representation of the complex impedance plane

The impedance of a two-terminal circuit element is represented as a complex quantity ${\displaystyle Z}$. The polar form conveniently captures both magnitude and phase characteristics as

${\displaystyle Z=|Z|e^{j\arg (Z)}}$

where the magnitude ${\displaystyle |Z|}$ represents the ratio of the voltage difference amplitude to the current amplitude, while the argument ${\displaystyle \arg (Z)}$ (commonly given the symbol ${\displaystyle \theta }$) gives the phase difference between voltage and current. ${\displaystyle j}$ is the imaginary unit, and is used instead of ${\displaystyle i}$ in this context to avoid confusion with the symbol for electric current.

In Cartesian form, impedance is defined as

${\displaystyle Z=R+jX}$

where the real part of impedance is the resistance R and the imaginary part is the reactance X.

Where it is needed to add or subtract impedances, the cartesian form is more convenient; but when quantities are multiplied or divided, the calculation becomes simpler if the polar form is used. A circuit calculation, such as finding the total impedance of two impedances in parallel, may require conversion between forms several times during the calculation. Conversion between the forms follows the normal conversion rules of complex numbers.

Complex voltage and current

Generalized impedances in a circuit can be drawn with the same symbol as a resistor (US ANSI or DIN Euro) or with a labeled box.

To simplify calculations, sinusoidal voltage and current waves are commonly represented as complex-valued functions of time denoted as ${\displaystyle V}$ and ${\displaystyle I}$.

${\displaystyle \begin{array}{rl}V& =|V|e^{j(\omega t+{\varphi }_V)},\\ I& =|I|e^{j(\omega t+{\varphi }_I)}.\end{array}}$

The impedance of a bipolar circuit is defined as the ratio of these quantities:

${\displaystyle Z=\frac{V}{I}=\frac{|V|}{|I|}{e}^{j({\varphi }_V-{\varphi }_I)}.}$

Hence, denoting ${\displaystyle \theta ={\varphi }_V-{\varphi }_I}$, we have

${\displaystyle \begin{array}{rl}|V|& =|I||Z|,\\ {\varphi }_V& ={\varphi }_I+\theta .\end{array}}$

The magnitude equation is the familiar Ohm's law applied to the voltage and current amplitudes, while the second equation defines the phase relationship.

Validity of complex representation

This representation using complex exponentials may be justified by noting that (by Euler's formula):

${\displaystyle \cos (\omega t+\varphi )=\frac{1}{2}[e^{j(\omega t+\varphi )}+e^{-j(\omega t+\varphi )}]}$

The real-valued sinusoidal function representing either voltage or current may be broken into two complex-valued functions. By the principle of superposition, we may analyse the behaviour of the sinusoid on the left-hand side by analysing the behaviour of the two complex terms on the right-hand side. Given the symmetry, we only need to perform the analysis for one right-hand term. The results are identical for the other. At the end of any calculation, we may return to real-valued sinusoids by further noting that

${\displaystyle \cos (\omega t+\varphi )={\mathcal{R}}_{\mathcal{e}}\{e^{j(\omega t+\varphi )}\}}$

Ohm's law

An AC supply applying a voltage ${\displaystyle V}$, across a load ${\displaystyle Z}$, driving a current

 

Main article: Ohm's law

The meaning of electrical impedance can be understood by substituting it into Ohm's law. Assuming a two-terminal circuit element with impedance ${\displaystyle Z}$ is driven by a sinusoidal voltage or current as above, there holds

${\displaystyle V=IZ=I|Z|e^{j\arg (Z)}}$

The magnitude of the impedance ${\displaystyle |Z|}$ acts just like resistance, giving the drop in voltage amplitude across an impedance ${\displaystyle Z}$ for a given current ${\displaystyle I}$. The phase factor tells us that the current lags the voltage by a phase of ${\displaystyle \theta =\arg (Z)}$ (i.e., in the time domain, the current signal is shifted $\frac{\theta }{2\pi }T$ later with respect to the voltage signal).

Just as impedance extends Ohm's law to cover AC circuits, other results from DC circuit analysis, such as voltage division, current division, Thévenin's theorem and Norton's theorem, can also be extended to AC circuits by replacing resistance with impedance.

Phasors

Main article: Phasor

Further information: Analytic representation

A phasor is represented by a constant complex number, usually expressed in exponential form, representing the complex amplitude (magnitude and phase) of a sinusoidal function of time. Phasors are used by electrical engineers to simplify computations involving sinusoids (such as in AC circuits), where they can often reduce a differential equation problem to an algebraic one.

The impedance of a circuit element can be defined as the ratio of the phasor voltage across the element to the phasor current through the element, as determined by the relative amplitudes and phases of the voltage and current. This is identical to the definition from Ohm's law given above, recognising that the factors of ${\displaystyle e^{j\omega t}}$ cancel.

Device examples

Resistor

The phase angles in the equations for the impedance of capacitors and inductors indicate that the voltage across a capacitor lags the current through it by a phase of ${\displaystyle \pi /2}$, while the voltage across an inductor leads the current through it by ${\displaystyle \pi /2}$. The identical voltage and current amplitudes indicate that the magnitude of the impedance is equal to one.

The impedance of an ideal resistor is purely real and is called resistive impedance:

${\displaystyle Z_R=R}$

In this case, the voltage and current waveforms are proportional and in phase.

Inductor and capacitor

Ideal inductors and capacitors have a purely imaginary reactive impedance:

the impedance of inductors increases as frequency increases;

${\displaystyle Z_L=j\omega L}$

the impedance of capacitors decreases as frequency increases;

${\displaystyle Z_C=\frac{1}{j\omega C}}$

In both cases, for an applied sinusoidal voltage, the resulting current is also sinusoidal, but in quadrature, 90 degrees out of phase with the voltage. However, the phases have opposite signs: in an inductor, the current is lagging; in a capacitor the current is leading.

Note the following identities for the imaginary unit and its reciprocal:

${\displaystyle \begin{array}{rl}j& \equiv \cos \left(\frac{\pi }{2}\right)+j\sin \left(\frac{\pi }{2}\right)\equiv e^{j\frac{\pi }{2}}\\ \frac{1}{j}\equiv -j& \equiv \cos \left(-\frac{\pi }{2}\right)+j\sin \left(-\frac{\pi }{2}\right)\equiv e^{j\left(-\frac{\pi }{2}\right)}\end{array}}$

Thus the inductor and capacitor impedance equations can be rewritten in polar form:

${\displaystyle \begin{array}{rl}{Z}_L& =\omega L{e}^{j\frac{\pi }{2}}\\ Z_C& =\frac{1}{\omega C}{e}^{j\left(-\frac{\pi }{2}\right)}\end{array}}$

The magnitude gives the change in voltage amplitude for a given current amplitude through the impedance, while the exponential factors give the phase relationship.

Deriving the device-specific impedances

What follows below is a derivation of impedance for each of the three basic circuit elements: the resistor, the capacitor, and the inductor. Although the idea can be extended to define the relationship between the voltage and current of any arbitrary signal, these derivations assume sinusoidal signals. In fact, this applies to any arbitrary periodic signals, because these can be approximated as a sum of sinusoids through Fourier analysis.

Resistor

For a resistor, there is the relation

${\displaystyle v_{\text{R}}\left(t\right)=i_{\text{R}}\left(t\right)R}$

which is Ohm's law.

Considering the voltage signal to be

${\displaystyle v_{\text{R}}(t)=V_p\sin (\omega t)}$

it follows that

${\displaystyle \frac{v_{\text{R}}\left(t\right)}{i_{\text{R}}\left(t\right)}=\frac{V_p\sin (\omega t)}{I_p\sin \left(\omega t\right)}=R}$

This says that the ratio of AC voltage amplitude to alternating current (AC) amplitude across a resistor is ${\displaystyle R}$, and that the AC voltage leads the current across a resistor by 0 degrees.

This result is commonly expressed as

${\displaystyle Z_{\text{resistor}}=R}$

Capacitor

For a capacitor, there is the relation:

${\displaystyle i_{\text{C}}(t)=C\frac{\mathrm{d}{v}_{\text{C}}(t)}{\mathrm{d}t}}$

Considering the voltage signal to be

${\displaystyle v_{\text{C}}(t)=V_p{e}^{j\omega t}}$

it follows that

${\displaystyle \frac{\mathrm{d}{v}_{\text{C}}(t)}{\mathrm{d}t}=j\omega V_p{e}^{j\omega t}}$

and thus, as previously,

${\displaystyle Z_{\text{capacitor}}=\frac{v_{\text{C}}\left(t\right)}{i_{\text{C}}\left(t\right)}=\frac{1}{j\omega C}.}$

Conversely, if the current through the circuit is assumed to be sinusoidal, its complex representation being

${\displaystyle i_{\text{C}}(t)=I_p{e}^{j\omega t}}$

then integrating the differential equation

${\displaystyle i_{\text{C}}(t)=C\frac{\mathrm{d}{v}_{\text{C}}(t)}{\mathrm{d}t}}$

leads to

${\displaystyle v_C(t)=\frac{1}{j\omega C}{I}_p{e}^{j\omega t}+\text{Const.}=\frac{1}{j\omega C}{i}_C(t)+\text{Const.}}$

The Const term represents a fixed potential bias superimposed to the AC sinusoidal potential, that plays no role in AC analysis. For this purpose, this term can be assumed to be 0, hence again the impedance

${\displaystyle Z_{\text{capacitor}}=\frac{1}{j\omega C}.}$

Inductor

For the inductor, we have the relation (from Faraday's law):

${\displaystyle v_{\text{L}}(t)=L\frac{\mathrm{d}{i}_{\text{L}}(t)}{\mathrm{d}t}}$

This time, considering the current signal to be:

${\displaystyle i_{\text{L}}(t)=I_p\sin (\omega t)}$

it follows that:

${\displaystyle \frac{\mathrm{d}{i}_{\text{L}}(t)}{\mathrm{d}t}=\omega I_p\cos \left(\omega t\right)}$

This result is commonly expressed in polar form as

${\displaystyle Z_{\text{inductor}}=\omega L{e}^{j\frac{\pi }{2}}}$

or, using Euler's formula, as

${\displaystyle Z_{\text{inductor}}=j\omega L}$

As in the case of capacitors, it is also possible to derive this formula directly from the complex representations of the voltages and currents, or by assuming a sinusoidal voltage between the two poles of the inductor. In the latter case, integrating the differential equation above leads to a constant term for the current, that represents a fixed DC bias flowing through the inductor. This is set to zero because AC analysis using frequency domain impedance considers one frequency at a time and DC represents a separate frequency of zero hertz in this context.

Generalised s-plane impedance

Impedance defined in terms of jω can strictly be applied only to circuits that are driven with a steady-state AC signal. The concept of impedance can be extended to a circuit energised with any arbitrary signal by using complex frequency instead of jω. Complex frequency is given the symbol s and is, in general, a complex number. Signals are expressed in terms of complex frequency by taking the Laplace transform of the time domain expression of the signal. The impedance of the basic circuit elements in this more general notation is as follows:

Element	Impedance expression
Resistor	${\displaystyle R}$
Inductor	${\displaystyle sL}$
Capacitor	${\displaystyle \frac{1}{sC}}$

For a DC circuit, this simplifies to s = 0. For a steady-state sinusoidal AC signal s = jω.

Formal derivation

The impedance ${\displaystyle Z}$ of an electrical component is defined as the ratio between the Laplace transforms of the voltage over it and the current through it, i.e.

${\displaystyle Z(s)=\frac{\mathcal{L}\{v(t)\}}{\mathcal{L}\{i(t)\}}=\frac{V(s)}{I(s)}\text{(general impedance)}}$

where ${\displaystyle s=\sigma +j\omega }$ is the complex Laplace parameter. As an example, according to the I-V-law of a capacitor, ${\displaystyle \mathcal{L}\{i(t)\}=\mathcal{L}\{C\mathrm{d}v(t)/\mathrm{d}t\}=sC\mathcal{L}\{v(t)\}}$, from which it follows that ${\displaystyle Z_C(s)=1/sC}$.

In the phasor regime (steady-state AC, meaning all signals are represented mathematically as simple complex exponentials ${\displaystyle v(t)=\hat{V}{e}^{j\omega t}}$ and ${\displaystyle i(t)=\hat{I}{e}^{j\omega t}}$ oscillating at a common frequency ${\displaystyle \omega }$), impedance can simply be calculated as the voltage-to-current ratio, in which the common time-dependent factor cancels out:

${\displaystyle Z(\omega )=\frac{v(t)}{i(t)}=\frac{\hat{V}{e}^{j\omega t}}{\hat{I}{e}^{j\omega t}}=\frac{\hat{V}}{\hat{I}}\text{(phasor-regime impedance)}}$

Again, for a capacitor, one gets that ${\displaystyle i(t)=C\mathrm{d}v(t)/\mathrm{d}t=j\omega Cv(t)}$, and hence ${\displaystyle Z_C(\omega )=1/j\omega C}$. The phasor domain is sometimes dubbed the frequency domain, although it lacks one of the dimensions of the Laplace parameter. For steady-state AC, the polar form of the complex impedance relates the amplitude and phase of the voltage and current. In particular:

• The magnitude of the complex impedance is the ratio of the voltage amplitude to the current amplitude;
• The phase of the complex impedance is the phase shift by which the current lags the voltage.

These two relationships hold even after taking the real part of the complex exponentials (see phasors), which is the part of the signal one actually measures in real-life circuits.

Resistance vs reactance

Resistance and reactance together determine the magnitude and phase of the impedance through the following relations:

${\displaystyle \begin{array}{rl}|Z|& =\sqrt{Z{Z}^{\ast }}=\sqrt{R^2+X^2}\\ \theta & =\arctan \left(\frac{X}{R}\right)\end{array}}$

In many applications, the relative phase of the voltage and current is not critical so only the magnitude of the impedance is significant.

Resistance

Main article: Electrical resistance

Resistance ${\displaystyle R}$ is the real part of impedance; a device with a purely resistive impedance exhibits no phase shift between the voltage and current.

${\displaystyle R=|Z|\cos \theta }$

Reactance

Main article: Electrical reactance

Reactance ${\displaystyle X}$ is the imaginary part of the impedance; a component with a finite reactance induces a phase shift ${\displaystyle \theta }$ between the voltage across it and the current through it.

${\displaystyle X=|Z|\sin \theta }$

A purely reactive component is distinguished by the sinusoidal voltage across the component being in quadrature with the sinusoidal current through the component. This implies that the component alternately absorbs energy from the circuit and then returns energy to the circuit. A pure reactance does not dissipate any power.

Capacitive reactance

Main article: Capacitance

A capacitor has a purely reactive impedance that is inversely proportional to the signal frequency. A capacitor consists of two conductors separated by an insulator, also known as a dielectric.

${\displaystyle X_{\mathsf{C}}=\frac{-1}{\omega C}=\frac{-1}{2\pi fC}.}$

The minus sign indicates that the imaginary part of the impedance is negative.

At low frequencies, a capacitor approaches an open circuit so no current flows through it.

A DC voltage applied across a capacitor causes charge to accumulate on one side; the electric field due to the accumulated charge is the source of the opposition to the current. When the potential associated with the charge exactly balances the applied voltage, the current goes to zero.

Driven by an AC supply, a capacitor accumulates only a limited charge before the potential difference changes sign and the charge dissipates. The higher the frequency, the less charge accumulates and the smaller the opposition to the current.

Inductive reactance

Main article: Inductance

Inductive reactance ${\displaystyle X_L}$ is proportional to the signal frequency ${\displaystyle f}$ and the inductance ${\displaystyle L}$.

${\displaystyle X_L=\omega L=2\pi fL}$

An inductor consists of a coiled conductor. Faraday's law of electromagnetic induction gives the back emf ${\displaystyle \mathcal{E}}$ (voltage opposing current) due to a rate-of-change of magnetic flux density ${\displaystyle B}$ through a current loop.

${\displaystyle \mathcal{E}=-\frac{d{\mathrm{\Phi }}_B}{dt}}$

For an inductor consisting of a coil with ${\displaystyle N}$ loops this gives:

${\displaystyle \mathcal{E}=-N\frac{d{\mathrm{\Phi }}_B}{dt}}$

The back-emf is the source of the opposition to current flow. A constant direct current has a zero rate-of-change, and sees an inductor as a short-circuit (it is typically made from a material with a low resistivity). An alternating current has a time-averaged rate-of-change that is proportional to frequency, this causes the increase in inductive reactance with frequency.

Total reactance

The total reactance is given by

${\displaystyle X=X_L+X_C}$ (note that ${\displaystyle X_C}$ is negative)

so that the total impedance is

${\displaystyle Z=R+jX}$

Combining impedances

Main article: Series and parallel circuits

The total impedance of many simple networks of components can be calculated using the rules for combining impedances in series and parallel. The rules are identical to those for combining resistances, except that the numbers in general are complex numbers. The general case, however, requires equivalent impedance transforms in addition to series and parallel.

Series combination

For components connected in series, the current through each circuit element is the same; the total impedance is the sum of the component impedances.

${\displaystyle Z_{\text{eq}}=Z_1+Z_2+\cdots +Z_n}$

Or explicitly in real and imaginary terms:

${\displaystyle Z_{\text{eq}}=R+jX=(R_1+R_2+\cdots +R_n)+j(X_1+X_2+\cdots +X_n)}$

Parallel combination

For components connected in parallel, the voltage across each circuit element is the same; the ratio of currents through any two elements is the inverse ratio of their impedances.

Hence the inverse total impedance is the sum of the inverses of the component impedances:

${\displaystyle \frac{1}{Z_{\text{eq}}}=\frac{1}{Z_1}+\frac{1}{Z_2}+\cdots +\frac{1}{Z_n}}$

or, when n = 2:

${\displaystyle \frac{1}{Z_{\text{eq}}}=\frac{1}{Z_1}+\frac{1}{Z_2}=\frac{Z_1+Z_2}{Z_1{Z}_2}}$

${\displaystyle Z_{\text{eq}}=\frac{Z_1{Z}_2}{Z_1+Z_2}}$

The equivalent impedance ${\displaystyle Z_{\text{eq}}}$ can be calculated in terms of the equivalent series resistance ${\displaystyle R_{\text{eq}}}$ and reactance ${\displaystyle X_{\text{eq}}}$.

${\displaystyle \begin{array}{rl}{Z}_{\text{eq}}& =R_{\text{eq}}+j{X}_{\text{eq}}\\ R_{\text{eq}}& =\frac{(X_1{R}_2+X_2{R}_1)(X_1+X_2)+(R_1{R}_2-X_1{X}_2)(R_1+R_2)}{(R_1+R_2{)}^2+(X_1+X_2{)}^2}\\ X_{\text{eq}}& =\frac{(X_1{R}_2+X_2{R}_1)(R_1+R_2)-(R_1{R}_2-X_1{X}_2)(X_1+X_2)}{(R_1+R_2{)}^2+(X_1+X_2{)}^2}\end{array}}$

Measurement

The measurement of the impedance of devices and transmission lines is a practical problem in radio technology and other fields. Measurements of impedance may be carried out at one frequency, or the variation of device impedance over a range of frequencies may be of interest. The impedance may be measured or displayed directly in ohms, or other values related to impedance may be displayed; for example, in a radio antenna, the standing wave ratio or reflection coefficient may be more useful than the impedance alone. The measurement of impedance requires the measurement of the magnitude of voltage and current, and the phase difference between them. Impedance is often measured by "bridge" methods, similar to the direct-current Wheatstone bridge; a calibrated reference impedance is adjusted to balance off the effect of the impedance of the device under test. Impedance measurement in power electronic devices may require simultaneous measurement and provision of power to the operating device.

The impedance of a device can be calculated by complex division of the voltage and current. The impedance of the device can be calculated by applying a sinusoidal voltage to the device in series with a resistor, and measuring the voltage across the resistor and across the device. Performing this measurement by sweeping the frequencies of the applied signal provides the impedance phase and magnitude.

The use of an impulse response may be used in combination with the fast Fourier transform (FFT) to rapidly measure the electrical impedance of various electrical devices.

The LCR meter (Inductance (L), Capacitance (C), and Resistance (R)) is a device commonly used to measure the inductance, resistance and capacitance of a component; from these values, the impedance at any frequency can be calculated.

Example

Consider an LC tank circuit. The complex impedance of the circuit is

${\displaystyle Z(\omega )=\frac{j\omega L}{1-{\omega }^{2}LC}.}$

It is immediately seen that the value of $\frac{1}{|Z|}$ is minimal (actually equal to 0 in this case) whenever

${\displaystyle {\omega }^{2}LC=1.}$

Therefore, the fundamental resonance angular frequency is

${\displaystyle \omega =\frac{1}{\sqrt{LC}}.}$

Variable impedance

See also: Network analysis (electrical circuits) § Time-varying components

In general, neither impedance nor admittance can vary with time, since they are defined for complex exponentials in which −∞ < t < +∞. If the complex exponential voltage to current ratio changes over time or amplitude, the circuit element cannot be described using the frequency domain. However, many components and systems (e.g., varicaps that are used in radio tuners) may exhibit non-linear or time-varying voltage to current ratios that seem to be linear time-invariant (LTI) for small signals and over small observation windows, so they can be roughly described as-if they had a time-varying impedance. This description is an approximation: Over large signal swings or wide observation windows, the voltage to current relationship will not be LTI and cannot be described by impedance