21.1: Structure and Functions of the Skin
By the end of this section, you will be able to:
- Review the anatomy of the integumentary system
- Identify the many functions of skin
- Recognize how assessment findings may vary in special populations
The integumentary system is the body’s first line of defense, serving as a physical barrier between the external and internal environments. The external environment includes microorganisms, radiation, temperature, or other threats of physical trauma or injury. The internal environment is the body that the integumentary system serves to protect and maintain homeostasis. The integumentary system includes the skin, hair, nails, as well as glands (e.g., sebaceous, sweat, mammary) (Kim & Dao, 2023).
Nurses play a vital role in maintaining skin integrity and promoting wound healing . To effectively protect and preserve a patient’s skin, the nurse must understand the anatomy and physiology of the integumentary system, factors affecting skin integrity, and how to appropriately manage wounds.
Anatomy of the Skin
The skin, which covers the body entirely, is the largest organ of the body. It serves as a protective barrier against heat, light, infection , and injury and performs additional essential functions. The skin varies in thickness, color, and texture across the body. For example, the palms of the hands and soles of the feet are thicker than skin on the abdomen. The skin is made up of three distinctive layers: the epidermis (the outermost layer of the skin); dermis (the layer below the epidermis); and hypodermis or subcutaneous tissue (the deepest layer) (Figure 21.2). Each layer varies in its anatomy and function (Yousef et al., 2022).
Epidermis
The epidermis is the thin outer layer of the skin and consists of epithelial cells. The main function of the epidermis is to protect deeper tissue layers from water, mechanical and chemical trauma, exposure to microorganisms, and damage from ultraviolet (UV) light. The epidermis has four or five layers depending on its location (Figure 21.3). There are five layers over the palms of the hands and soles of the feet, while the rest of the body has four layers. Starting from the bottom, the layers are the stratum basale , stratum spinosum , stratum granulosum , stratum lucidum , and stratum corneum.
The stratum basale (or stratum germinativum) is the deepest layer of the epidermis and contains a cell known as a melanocyte that produce melanin and a keratinocyte that produces keratin . Melanin forms a protective shield to protect the keratinocytes and nerve endings from damage caused by ultraviolet light. Melanin is also responsible for the color of the skin. Keratin is a fibrous, water-repellent protein that gives the epidermis its strong, protective quality. The keratinocytes move upward through the layers as they mature and become dead cells on the outermost layer of the skin (Osseiran et al., 2018). The Merkel cell is also found in the stratum basale and serves as a sensory receptor for light touch.
The next layer above the stratum basale is the stratum spinosum . This layer is several layers thick and contains cells that arise from the bone marrow and move up to the epidermis. The cells are “irregular, polyhedral cells with cytoplasmic processes and are often called ‘spines’ that extend outward and contact neighboring cells by desmosomes” (Yousef et al., 2022). The Langerhans cell is primarily located in the stratum spinosum. These cells are first-line defenders and are antigen-producing cells that determine the appropriate response to (e.g., inflammation) or tolerance of foreign substances.
The next layer, the stratum granulosum , is only two to three cell layers thick. This layer contains a glycolipid that slows water loss across the epidermis. The process of keratinization , a thickening of plasma membranes of a cell, begins in this layer. The next layer, the stratum lucidum , is only found in the thicker areas of the skin (e.g., palms and soles of feet).  The outermost layer of the epidermis, the stratum corneum , is the thickest layer and makes up about 75 percent of the total thickness. This layer contains dead keratinocytes that secrete defensin (a class of host-defense peptides found in neutrophils that have antimicrobial and/or cytotoxic properties) and is part of the body’s first immune defense.
Dermis
The second, deeper layer of the skin is the dermis , which is connected to the epidermis by dermal papillae. Its main function is regulating temperature and supporting, protecting, and nourishing the epidermis. This layer consists of the papillary layer and reticular layer . The upper papillary layer is the thinner of the two and is composed of loose connective tissue and contacts the epidermis. It also contains capillaries as well as touch and pain receptors. The deeper layer, the reticular layer, is the thicker of the two. This layer consists of dense connective tissue, collagen bundles, and houses most of the sweat glands, sebaceous glands, hair follicles, hair, muscles, blood vessels, and deep sensory neurons (Yousuf et al., 2022).
The sebaceous gland is attached to the hair follicles and is present over the body minus the palms of hands and soles of feet. The sebaceous glands secrete sebum, an oily substance that makes the hair and skin waterproof. There are two types of sweat glands: eccrine glands and apocrine glands. The eccrine gland is located over the skin entirely, secretes sweat, and aids in thermoregulation . The apocrine gland correlates with the hair follicles present in the perineum, axillae, and areolae of the breasts. These glands are nonfunctioning and small until puberty has been reached. After the onset of puberty, the glands secrete a milky sweat.
Hypodermis
The hypodermis ( subcutaneous tissue ) is the subcutaneous fatty layer beneath the dermis that separates the skin from the underlying tissue. The hypodermis also anchors the dermis to the underlying tissues of the body and insulates, protects, and stores fat for the body.  This layer provides a cushion for the body against trauma, stores fat for energy, and insulates the body from heat loss. Some skin appendages (e.g., hair follicles, blood vessels, sensory neurons) are found in the hypodermis. This layer also contains blood and lymph vessels and nerves. The blood vessels supply nutrients to the tissues, and the lymph vessels provide a pathway for the movement of waste products to and from the skin.
Functions of the Skin
The skin performs essential functions including protection, thermoregulation, sensation , absorption , elimination , and vitamin D production . The skin contributes to the psychosocial aspect of an individual because of its contribution to the external appearance of the person and plays a role in self-esteem. The skin also aids in communication with the brain via the sensory neurons located in the skin.
Protection
The skin provides protection because it covers the body entirely and serves as the physical barrier between the internal and external environments. Breaks in the skin trigger an immune response to promote healing and fight off foreign debris that may lead to infection . The many layers of the skin provide protection from injury to underlying tissues and organs. The skin also acts as a shield from microorganisms, damaging ultraviolet rays, and other substances including water. The layers of keratin and glycolipids in the skin serve to protect the body against moisture loss from both the surface and underlying structures.
Thermoregulation
Temperature regulation is another function of the skin. The skin is highly vascularized, which allows the body to regulate body temperature through vasoconstriction and vasodilation of blood vessels. When body temperature decreases, blood vessels vasoconstrict to decrease blood flow to the periphery in an effort to conserve core body heat. The arrector pili muscle triggers hair follicles on the body to flex causing the hair to rise up (goosebumps), which further prevents heat loss (Figure 21.4). When body temperature increases, blood vessels vasodilate to increase blood flow. Sweat is secreted, which evaporates from the skin contributing to heat loss and bringing the body temperature down (Kim & Dao, 2023).
During the first hours of life, newborns experience difficulties regulating their body temperature. The balance of heat production and heat loss is linked to the metabolism and oxygen consumption of the newborn. Newborns can attempt to conserve heat and increase their heat production by increasing peripheral vasoconstriction, metabolic rate, and muscular activity through moving. Newborns may also assume a fetal position to hold in heat and decrease body surface area exposure. Nursing interventions to aid in maintaining body temperature of newborns include the following (Albahrani & Hunt, 2019):
- Dry newborns completely after birth or baths to prevent heat loss through evaporation. (Baths should be performed after newborn is stable, and a radiant heat source should be used.)
- Prewarm blankets and caps to reduce heat loss through conduction.
- Place a cap on a newborn after drying thoroughly.
- Use a warm cover on the scale when weighing a newborn.
- Warm stethoscopes and hands before touching a newborn.
- Avoid placing a newborn near air vents or areas with drafts, and avoid placing cribs near cold outer walls.
- Keep infant transporter fully charged and heated.
- Avoid placing skin temperature probes over brown fat areas (found on the back, neck, and shoulders in babies to help regulate body temperature).
- Encourage parent skin-to-skin contact with stable newborns.
- Use heated and humidified oxygen.
Sensation
The skin provides sensation for the body and allows a patient to feel temperature, pressure, pain, and touch through the various sensory nerve endings. Each type of sensory receptor sends signals to the brain and spinal cord that allow a patient to respond and adjust to the environment as needed. For example, when a patient touches something that is extremely hot, they typically move their hand away quickly, which is the body’s reflex in response to the external stimuli in an effort to protect itself from harm.
Scenario: A nurse is assessing an older adult with diabetic neuropathy who is being seen in the clinic for a sore on his right foot.
Nurse: Hi. My name is Linda, and I will be your nurse today. Will you tell me your name and date of birth?
Patient: Yes. Hi, my name is Joseph Gellar. My date of birth is April 25, 1941.
Nurse: Thank you. What brings you in today?
Patient: Well, I have this sore on my foot, and I am not sure how I got it.
Nurse: Okay, let me take a look.
Nurse: [after assessing the wound] Let’s verify your health history.
Patient: Well, I have diabetes, high blood pressure, and high cholesterol.
Nurse: What medications do you currently take?
Patient: I take my insulin, metoprolol, and atorvastatin. But I forgot to check my blood sugar today and take my insulin.
Nurse: Okay, have you noticed any changes in sensation? Like being able to feel things that are painful, hot, or cold?
Patient: Yeah, I noticed that a few years ago, but I just thought it was normal.
 Because of this, it is important to be aware of your surroundings to avoid injury. Check your feet every day for any injuries like cuts, scrapes, or bruises. It is also important to try to manage your blood sugar as much as possible with your diet, exercise, and medications.
Patient: Oh, that makes sense. I think my wife and I have a friend who had something similar happen.
Nurse: Yes sir. Do you have any questions for me before I give an update to your doctor, so he can come in and see you?
Patient: No, that will be it for now.
Nurse: Okay, great. I will be in after the doctor sees you to go over any orders or treatments he prescribes for you.
Absorption
The skin is porous and therefore absorbs substances that may be on it. Substances that can be absorbed through the skin will enter the bloodstream. Certain medications may be absorbed by transdermal administration (placed on the skin and absorbed either locally or systemically), including scopolamine , estrogen , testosterone , some opioids, nicotine , and some contraceptives. It is important to note that anything on the skin may be absorbed and thus has the potential to cause harm. Certain chemicals and medications require appropriate handling to avoid the risk of harm or injury. For example, when administering fentanyl , it is important to wear gloves to avoid accidental absorption into the bloodstream. This is also why sunscreen or other lotions are not used on babies under 6 months old due to thin skin and increased absorption.
Elimination
The elimination function of the skin helps the body get rid of excess water and salts through sweat produced by the sweat glands. This perspiration or sweat may contain electrolytes, water, and nitrogenous wastes like urea, uric acid, and ammonia. In the event other organs become unable to eliminate toxins, the skin will try to push out toxins to aid in removal. In addition to removing wastes, the excretion function of the skin aids in regulating body temperature as previously described.
Vitamin D Production
The skin initiates the biochemical processes necessary for vitamin D production . Ultraviolet sun exposure leads to the conversion of 7-dehydrocholesterol to cholecalciferol (vitamin D 3 ) in the skin. Cholecalciferol is hydroxylated in the liver and then in the kidneys, which converts it into the active, metabolite form, calcitriol. The chemical process in which a hydroxl group (–OH) is introduced into an organic compound is termed hydroxylation . This metabolite leads to enhanced calcium production in the gut and is vital for bone health (Kim & Dao, 2023).
Assessment Findings for Special Populations
There are distinct considerations for special populations that the nurse must consider when assessing a patient’s skin. The assessment’s findings may vary among different cultures and ethnicities, developmental levels, and age groups. It is essential that the nurse be aware of the variations when performing a skin assessment to avoid stereotypes and to ensure accuracy when developing a plan of care to promote optimal outcomes. For example, a darker-skinned patient may not show pallor in the same way a lighter-skinned patient would. The nurse must assess the mucous membranes, such as the gums or palpebral conjunctiva, of a darker-skinned patient to adequately assess for pallor.
Name:
Bailey, BSN
Clinical setting:
Pediatric clinic
Years in practice:
3
Facility location:
Rural area of Kansas
Our pediatric clinic is the only one in a 50-mile radius so we are always swamped. One day, I was working in the well-baby side of our clinic. I was assessing a baby who was here for her six-month wellness visit. Her mother, Ali, brought her in and introduced her baby girl as Alexis. After taking her medical history, I began the physical assessment. I noticed what appeared to be bruises along Alexis’s back similar to those seen in abuse cases (Figure 21.5). I was concerned and thought that the baby may be experiencing abuse at home. The spots were bluish in color and along the baby’s backside, so the areas were not visible to others.
As I began to plan my next steps, Alexis’s mother stated, “I am ready for the spots to start disappearing!” Then, I recalled learning about these spots in nursing school. I put my fingers on the skin to blanch the area and was relieved when I observed that the spots were nonblanchable, confirming dermal melanocytosis, formerly known as Mongolian spots. The spots are actually pigmented skin lesions that may be present at birth or develop within the first few weeks of life.
I asked Ali about the first noticeable spot and said they showed up when Alexis was about 2 weeks old. We then discussed how the spots may disappear by childhood. Ali said she understood but would be happy when they were gone. It was the first time I had actually seen Mongolian spots, and I was really relieved that they were not bruises from abuse.
Cultural Considerations
While people share a similar number and distribution of eccrine gland s over the body, the glands vary in activity depending on patient-specific characteristics and environmental adaptations. For example, those born in tropical areas have more functioning eccrine glands than those who move there later in life. People who acclimate to hot environments do not excrete as much chloride in their sweat as others. People of Asian and American Indian descent have fewer functioning apocrine gland s than Whites and Blacks. The amount of sweat and body odor an individual has is genetically determined and is in relation to the functionality of the apocrine glands. Therefore, Asians and American Indians have little to no underarm sweat or body odor. People born farther from the equator usually have fairer skin and are at a greater risk of developing skin cancer s due to lower levels of the protective pigment, melanin . Additional factors for fair-skinned individuals include the presence of freckles and an increased susceptibility to sunburn. However, all people with excessive levels of sun exposure are at risk for sun damage to skin and skin cancer and should, therefore, avoid excessive exposure to UV light. Darker-skinned patients, including those of ethnicities such as Black, Latinx, Hispanic, Asian, and American Indian, typically produce more melanin in general. The higher level of melanin is a protective factor and reduces the risk of developing skin cancers. However, darker-skinned individuals are more susceptible to certain skin abnormalities (Figure 21.6), including the following:
- postinflammatory hyperpigmentation , a temporary pigmentation that occurs after an injury or inflammatory skin condition, most commonly seen in patients with darker skin types
- dermatosis papulose nigra , small dark or skin-colored bumps on the face, neck, and upper torso, common in patients with darker skin colors
- keloid , a thick fibrous scar caused by excessive collagen formation in response to trauma or a surgical incision
- pityriasis , any of various skin conditions characterized by dry scaling patches of skin
- vitiligo , a chronic autoimmune disorder that causes partial or total loss of skin color in patches
- hair loss
- dry skin
Developmental Considerations of Infants and Children
A person’s skin undergoes changes throughout the life span. Sebaceous glands do not function maturely at birth. Sebum production increases in the prepubescent and adolescent years, which makes individuals within these age ranges more prone to acne. The sweat glands of an infant function to some degree and produce sweat in response to heat or emotional stimuli.  Until the glands function fully, temperature regulation is not as efficient as it is with the fully functioning glands of older children and adults.
Infants also have a thinner epidermis than adults. Infants have less subcutaneous tissue , so the blood vessels lie closer to the skin’s surface. Therefore, an infant loses more heat through the skin’s surface than an older child or adult. In addition, substances absorb more readily through the skin of an infant than an adult because of the thinness of the skin. Infant skin contains more water than adult skin. The epidermis is loosely bound to the dermis , and this means that friction may easily separate the layers which can result in blisters or skin breakdown.
Regardless of ethnicity, infant skin is less pigmented, and this places them at higher risk of skin damage from sun exposure . As the infant ages, the skin becomes tougher and less hydrated, making them less susceptible to invasions from microorganisms. When the child reaches teenage years, the skin characteristics and thickness are at the adult level. Infant skin and mucous membranes are more easily injured and susceptible to infection ; therefore, the nurse must carefully handle infants to protect them from harm or infection.
Overheating an infant can easily happen if a caregiver fears that the infant is cold. Newborns and infants do not have a mature thermoregulatory system. This means that they are more vulnerable to overheating. Signs and symptoms of overheating include the infant
- feeling warm to the touch;
- having flushed or red skin;
- developing heat rash;
- sweating or having damp hair;
- developing tachycardia;
- developing tachypnea;
- acting fussy;
- appearing sluggish, confused, or dizzy; or
- becoming unresponsive.
Nurses should be aware of these signs and the interventions needed: administer fluids (e.g., breast milk, water, or formula depending on their age), apply cold compresses, take off any excessive clothing or bedding, and go to a well-ventilated area. If the nurse is educating a caregiver, they should also include the need to call 911 if the child is unresponsive. Interventions to prevent overheating include avoiding the use of excessive clothing and bedding, keeping the room temperature between 68°F and 72°F (20°C and 22°C), ensuring adequate room ventilation, using curtains if needed to block out excessive light and heat, and avoiding the use of heating pads.
Other developmental considerations the nurse must consider include patients with cognitive disabilities. Individuals with cognitive dysfunction may not know how to check their own skin for abnormalities or impairments, or they may not recognize any changes in their skin. These patients may also neglect skin care. This can lead to itching, scratching, and picking because of buildup of dead skin and other debris, which can lead to injury or infection. The nurse may need to establish rapport with those who have disabilities to gain their trust. This enables the nurse to perform an assessment and provide coaching to teach them how to perform a skin assessment at home. If a patient with a disability has a caregiver, the caregiver should be included in education of skin assessment as well as findings to report.
The nurse must also consider the patient’s developmental level when communicating and when describing the steps of assessment. For example, an infant may be examined in the caregiver’s lap versus the school-aged child who can sit on the examination table. The nurse should speak with terms that are appropriate for the age or developmental level. The nurse should also speak to the patient regardless of the developmental level, especially school-age children and adolescents, to help them feel included in the plan of care.
Age-Related Considerations
Much like the infant’s skin changes from birth to adulthood, the skin continues to change as a person ages (Figure 21.7). The thickness of the epidermis in older people decreases, making the skin more delicate and at risk of tears or injury. The subcutaneous tissue s also become thinner, making thermoregulation more difficult and increasing the risk of pressure ulcers. The mitotic activity , or the degree in which a population of cells proliferates, decreases, leading to delayed wound healing . As the subcutaneous and dermal tissues thin with age, the skin becomes more prone to wrinkles, and pressure and pain sensation is reduced.
As the activity levels of the sebaceous and sweat glands decrease, the skin becomes dryer, and older people may experience pruritus , itching. Melanin declines with age leading to unevenly pigmented skin and gray or white hair. Hyperplasia of the melanocytes may lead to small areas of hyperpigmentation , darkened patches of skin called liver spots. Decreased melanin in an area can cause hypopigmentation , lighter patches of skin (age spots). The elastic fibers degenerate causing the skin to lose its elasticity. There is often a redistribution of adipose tissue leading to cellulite (lumpy skin), increased abdominal fat, sagging of breasts, and reduced tenting (the skin does not snap back when pinched; instead, it maintains a tentlike shape, which may indicate dehydration ). Decreased vitamin D production , which is normal with aging, increases the risk of osteomalacia (bone demineralization) and osteoporosis (decrease in bone mass and density).


5.9: Flagella and Cilia
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- Ying Liu
- City College of San Francisco
State the difference between eukaryotic flagella and cilia.
Briefly describe and state the function of eukaryotic flagella and cilia.
Flagellar arrangement schemes
Different species of bacteria have different numbers and arrangements of flagella (Figure \(\PageIndex{7}\).7.1).
- Monotrichous bacteria have a single flagellum (e.g., Vibrio cholerae ).
- Lophotrichous bacteria have multiple flagella located at the same spot on the bacteria's surfaces which act in concert to drive the bacteria in a single direction. In many cases, the bases of multiple flagella are surrounded by a specialized region of the cell membrane, the so-called polar organelle .
- Amphitrichous bacteria have a single flagellum on each of two opposite ends (only one flagellum operates at a time, allowing the bacteria to reverse course rapidly by switching which flagellum is active).
- Peritrichous bacteria have flagella projecting in all directions (e.g., E. coli ).
Flagella and cilia consist of 9 fused pairs of protein microtubules with side arms of the motor molecule dynein that originate from a centriole. These form a ring around an inner central pair of microtubules that arise from a plate near the cell surface. The arrangement of microtubules is known as a 2X9+2 arrangement. This complex of microtubules is surrounded by a sheath continuous with the cytoplasmic membrane.
Summary
- Flagella are long and few in number whereas cilia are short and numerous.
- Both flagella and cilia consist of 9 fused pairs of protein microtubules with side arms of the motor molecule dynein that originate from a centriole. These form a ring around an inner central pair of microtubules that arise from a plate near the cell surface. This complex of microtubules is surrounded by a sheath continuous with the cytoplasmic membrane.
- Flagella and cilia function in locomotion. Cilia also function to move various materials that may surround a cell.


3.20: Power Flow on Transmission Lines
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It is often important to know the power associated with a wave on a transmission line. The power of the waves incident upon, reflected by, and absorbed by a load are each of interest. In this section we shall work out expressions for these powers and consider some implications in terms of the voltage reflection coefficient (\(\Gamma\)) and standing wave ratio (SWR).
Let’s begin by considering a lossless transmission line that is oriented along the \(z\) axis. The time-average power associated with a sinusoidal wave having potential \(v(z,t)\) and current \(i(z,t)\) is \[P_{av}(z) \triangleq \frac{1}{T}\int_{t_0}^{t_0+T}{ v(z,t)~i(z,t)~dt } \nonumber \] where \(T\triangleq 2\pi/f\) is one period of the wave and \(t_0\) is the start time for the integration. Since the time-average power of a sinusoidal signal does not change with time, \(t_0\) may be set equal to zero without loss of generality.
Let us now calculate the power of a wave incident from \(z<0\) on a load impedance \(Z_L\) at \(z=0\). We may express the associated potential and current as follows: \[v^+(z,t) = \left|V_0^+\right|\cos\left(\omega t -\beta z +\phi\right) \nonumber \] \[i^+(z,t) = \frac{\left|V_0^+\right|}{Z_0}\cos\left(\omega t -\beta z +\phi\right) \nonumber \] And so the associated time-average power is \[\begin{equation}\begin{split} P_{av}^+(z) &= \frac{1}{T}\int_0^T{ v^+(z,t)~i^+(z,t)~dt } \\ &= \frac{\left|V_0^+\right|^2}{Z_0} \cdot \frac{1}{T}\int_0^T{ \cos^2\left(\omega t -\beta z +\phi\right) ~dt } \end{split}\end{equation} \label{m0090_ePav1} \] Employing a well-known trigonometric identity: \[\cos^2\theta = \frac{1}{2} + \frac{1}{2}\cos 2\theta \nonumber \] we may rewrite the integrand as follows \[\cos^2\left(\omega t -\beta z +\phi\right) = \frac{1}{2} + \frac{1}{2}\cos\left(2\left[\omega t -\beta z +\phi\right]\right) \nonumber \] Then integrating over both sides of this quantity \[\int_0^T{ \cos^2\left(\omega t -\beta z +\phi\right) ~dt } = \frac{T}{2} + 0 \nonumber \] The second term of the integral is zero because it is the integral of cosine over two complete periods. Subsequently, we see that the position dependence (here, the dependence on \(z\)) is eliminated. In other words, the power associated with the incident wave is the same for all \(z<0\), as expected. Substituting into Equation \ref{m0090_ePav1} we obtain: \[\boxed{ P_{av}^+ = \frac{\left|V_0^+\right|^2}{2Z_0} } \label{m0090_ePav+} \] This is the time-average power associated with the incident wave, measured at any point \(z<0\) along the line.
Equation \ref{m0090_ePav+} gives the time-average power associated with a wave traveling in a single direction along a lossless transmission line.
Using precisely the same procedure, we find that the power associated with the reflected wave is \[P_{av}^- = \frac{\left|\Gamma V_0^+\right|^2}{2Z_0} = \left|\Gamma\right|^2 \frac{\left|V_0^+\right|^2}{2Z_0} \nonumber \] or simply \[\boxed{ P_{av}^- = \left|\Gamma\right|^2 P_{av}^+ } \label{m0090_ePav-} \]
Equation \ref{m0090_ePav-} gives the time-average power associated with the wave reflected from an impedance mismatch.
Now, what is the power \(P_L\) delivered to the load impedance \(Z_L\)? The simplest way to calculate this power is to use the principle of conservation of power . Applied to the present problem, this principle asserts that the power incident on the load must equal the power reflected plus the power absorbed; i.e., \[P_{av}^+ = P_{av}^- + P_L \nonumber \] Applying the previous equations we obtain: \[\boxed{ P_L = \left(1-\left|\Gamma\right|^2\right) P_{av}^+ } \label{m0090_ePL} \]
Equations \ref{m0090_ePL} gives the time-average power transferred to a load impedance, and is equal to the difference between the powers of the incident and reflected waves.
Example \(\PageIndex{1}\): How important is it to match \(50~\Omega\) to \(75~\Omega\)?
Two impedances which commonly appear in radio engineering are \(50~\Omega\) and \(75~\Omega\). It is not uncommon to find that it is necessary to connect a transmission line having a \(50~\Omega\) characteristic impedance to a device, circuit, or system having a \(75~\Omega\) input impedance, or vice-versa. If no attempt is made to match these impedances, what fraction of the power will be delivered to the termination, and what fraction of power will be reflected? What is the SWR?
Solution
The voltage reflection coefficient going from \(50~\Omega\) transmission line to a \(75~\Omega\) load is \[\Gamma = \frac{75-50}{75+50} = +0.2 \nonumber \] The fraction of power reflected is \(\left|\Gamma\right|^2 = 0.04\), which is 4%. The fraction of power transmitted is \(1-\left|\Gamma\right|^2\), which is 96%. Going from a \(50~\Omega\) transmission line to a \(75~\Omega\) termination changes only the sign of \(\Gamma\), and therefore, the fractions of reflected and transmitted power remain 4% and 96%, respectively. In either case (from Section 3.14): \[\mbox{SWR} = \frac{1+\left|\Gamma\right|}{1-\left|\Gamma\right|} = 1.5 \nonumber \] This is often acceptable, but may not be good enough in some particular applications. Suffice it to say that it is not necessarily required to use an impedance matching device to connect \(50~\Omega\) to \(75~\Omega\) devices.


5.12: Reading- Ethics for Marketing Employees
If you are hired to work in marketing at a typical company, there will likely be clear ethical standards defined in a company policy and some level of compliance among employees at all levels in the company. You will witness ethical breaches and need to decide whether to report them or not. You will see examples of outstanding ethics and have the opportunity to participate in debates about ethical disagreements and issues. No company is perfect, but most are trying to be ethical.
How can you, as a marketer, make a difference? Marketers have a specific set of responsibilities when it comes to preventing and addressing ethical issues. These are described below.
Demonstrate Respect for Your Target Customer
Marketing is not a game of manipulation. Good marketing provides compelling solutions and informs customers to help them make good selections and realize value. Recognize the customer’s need for an offering that is easy to use and includes clear instructions and appropriate warnings. Remain available to customers to hear complaints. Be humble enough to recognize that not everyone wants to hear your messages. If you demonstrate respect for the consumer, you will find new opportunities to provide value. If you treat consumers like a commodity to be manipulated, a host of ethical issues will clutter your path.
Prepare the Sales Team to Sell Effectively and Ethically
If personal sales are a part of the business plan, then marketers have an important responsibility to prepare the sales team for success. Often marketers are asked to create the message, and sales reps are asked to deliver it. When the sales rep is prepared with a strong value proposition, effective communication materials and presentations, and thorough market research, the sales rep can do her best work. When the marketing mix is not hitting the mark, the sales rep’s is much more difficult, and there is a greater risk of ethical issues. It is the marketer’s responsibility to prepare sales reps to be successful without compromising their integrity.
Demonstrate High Personal Standards in Business Relationships
Marketers often entertain and give gifts. It is not unusual for the marketing team to create the gift list for all customers. Marketers cultivate business relationships and distributor relationships, too. If marketing demonstrates a high standard for professionalism and ethics in these relationships, it sends a strong message and increases the expectation that others will, as well.
Provide Fair Value to the Target Customer
Many ethical issues result from some level of deception involving misstatement of value to the customer. Be accurate in communications to customers about the value that a product provides. Be clear in pricing and contracts. Pricing strategies that confuse customers and cost more than the customer initially believed are never a good long-term strategy.
Play Nicely in the Competitive Environment
Companies in a competitive market shift positions and introduce innovations to give them new competitive advantage. This is the very nature of a competitive marketplace. Treat competitors with respect and learn from their approaches. Do thorough competitive research to understand them better. Do not seek to gain confidential information about competitors or their products.
Be Truthful
Seek to create a relationship of trust with your target customer through honest, helpful communication. This is such a simple but important recommendation for all marketers. If customers trust the product, the company, and the brand, business results improve, and the company has greater flexibility to introduce new products or make market adjustments.
Contributors and Attributions
- Ethics for Marketing Employees. Provided by : Lumen Learning. License : CC BY: Attribution
- Integrity. Authored by : atalou. Located at : https://www.flickr.com/photos/natalieguinsler/5565750498/ . License : CC BY-NC-ND: Attribution-NonCommercial-NoDerivatives


10: Electromagnetic Induction, AC Circuits, and Electrical Technologies
Historically, it was very shortly after Oersted discovered currents cause magnetic fields that other scientists asked the following question: Can magnetic fields cause currents? The answer was soon found by experiment to be yes. In 1831, some 12 years after Oersted’s discovery, the English scientist Michael Faraday (1791–1862) and the American scientist Joseph Henry (1797–1878) independently demonstrated that magnetic fields can produce currents. The basic process of generating emfs (electromotive force) and, hence, currents with magnetic fields is known as induction ; this process is also called magnetic induction to distinguish it from charging by induction, which utilizes the Coulomb force.
-
- 10.1: Prelude to Electromagnetic Induction, AC Circuits and Electrical Technologies
- Today, currents induced by magnetic fields are essential to our technological society. The ubiquitous generator—found in automobiles, on bicycles, in nuclear power plants, and so on—uses magnetism to generate current. Other devices that use magnetism to induce currents include pickup coils in electric guitars, transformers of every size, certain microphones, airport security gates, and damping mechanisms on sensitive chemical balances.
-
- 10.3: Faraday’s Law of Induction- Lenz’s Law
- Faraday’s experiments showed that the emf induced by a change in magnetic flux depends on only a few factors. First, emf is directly proportional to the change in flux ΔΦ. Second, emf is greatest when the change in time Δt is smallest—that is, emf is inversely proportional to Δt . Finally, if a coil has N turns, an emf will be produced that is Ntimes greater than for a single coil, so that emf is directly proportional to N.
-
- 10.4: Motional Emf
- As we have seen, any change in magnetic flux induces an emf opposing that change—a process known as induction. Motion is one of the major causes of induction. For example, a magnet moved toward a coil induces an emf, and a coil moved toward a magnet produces a similar emf. In this section, we concentrate on motion in a magnetic field that is stationary relative to the Earth, producing what is loosely called motional emf.
Thumbnail: Small cheap inductor. (CC-SA-BY 3.0; FDominec).


2.5: Quadratic Equations
- Solve quadratic equations by factoring.
- Solve quadratic equations by the square root property.
- Solve quadratic equations by completing the square.
- Solve quadratic equations by using the quadratic formula.
The computer monitor on the left in Figure \(\PageIndex{1}\) is a \(23.6\)-inch model and the one on the right is a \(27\)-inch model. Proportionally, the monitors appear very similar. If there is a limited amount of space and we desire the largest monitor possible, how do we decide which one to choose? In this section, we will learn how to solve problems such as this using four different methods.
Solving Quadratic Equations by Factoring
An equation containing a second-degree polynomial is called a quadratic equation . For example, equations such as \(2x^2 +3x−1=0\) and \(x^2−4= 0\) are quadratic equations. They are used in countless ways in the fields of engineering, architecture, finance, biological science, and, of course, mathematics.
Often the easiest method of solving a quadratic equation is factoring . Factoring means finding expressions that can be multiplied together to give the expression on one side of the equation.
If a quadratic equation can be factored, it is written as a product of linear terms. Solving by factoring depends on the zero-product property, which states that if \(a⋅b=0\), then \(a = 0\) or \(b =0\), where a and b are real numbers or algebraic expressions. In other words, if the product of two numbers or two expressions equals zero, then one of the numbers or one of the expressions must equal zero because zero multiplied by anything equals zero.
Multiplying the factors expands the equation to a string of terms separated by plus or minus signs. So, in that sense, the operation of multiplication undoes the operation of factoring. For example, expand the factored expression \((x−2)(x+3)\) by multiplying the two factors together.
\[\begin{align*} (x-2)(x+3)&= x^2+3x-2x-6\\ &= x^2+x-6\\ \end{align*}\]
The product is a quadratic expression. Set equal to zero, \(x^2+x−6= 0\) is a quadratic equation. If we were to factor the equation, we would get back the factors we multiplied.
The process of factoring a quadratic equation depends on the leading coefficient, whether it is \(1\) or another integer. We will look at both situations; but first, we want to confirm that the equation is written in standard form, \(ax^2+bx+c=0\), where \(a\), \(b\), and \(c\) are real numbers, and \(a≠0\). The equation \(x^2 +x−6= 0\) is in standard form.
We can use the zero-product property to solve quadratic equations in which we first have to factor out the greatest common factor(GCF), and for equations that have special factoring formulas as well, such as the difference of squares, both of which we will see later in this section.
The zero-product property states
If \(a⋅b=0\), then \(a=0\) or \(b=0\),
where \(a\) and \(b\) are real numbers or algebraic expressions.
A quadratic equation is an equation containing a second-degree polynomial; for example
\[ax^2+bx+c=0\]
where \(a\), \(b\), and \(c\) are real numbers, and if \(a≠0\), it is in standard form.
Solving Quadratics with a Leading Coefficient of \(1\)
In the quadratic equation \(x^2 +x−6=0\), the leading coefficient, or the coefficient of \(x^2\), is \(1\). We have one method of factoring quadratic equations in this form.
- Find two numbers whose product equals \(c\) and whose sum equals \(b\).
- Use those numbers to write two factors of the form \((x+k)\) or \((x−k)\), where k is one of the numbers found in step 1. Use the numbers exactly as they are. In other words, if the two numbers are \(1\) and \(−2\), the factors are \((x+1)(x−2)\).
- Solve using the zero-product property by setting each factor equal to zero and solving for the variable.
Factor and solve the equation: \(x^2+x−6=0\).
Solution
To factor \(x^2 +x−6=0\), we look for two numbers whose product equals \(−6\) and whose sum equals \(1\). Begin by looking at the possible factors of \(−6\).
\[1⋅(−6) \nonumber \]
\[(−6)⋅1 \nonumber \]
\[2⋅(−3) \nonumber \]
\[3⋅(−2) \nonumber \]
The last pair, \(3⋅(−2)\) sums to \(1\), so these are the numbers. Note that only one pair of numbers will work. Then, write the factors.
\[(x−2)(x+3)=0 \nonumber \]
To solve this equation, we use the zero-product property. Set each factor equal to zero and solve.
\[\begin{align*} (x-2)(x+3)&= 0\\ (x-2)&= 0\\ x&= 2\\ (x+3)&= 0\\ x&= -3 \end{align*}\]
The two solutions are \(2\) and \(−3\). We can see how the solutions relate to the graph in Figure \(\PageIndex{2}\). The solutions are the x-intercepts of \(x^2 +x−6=0\).
Factor and solve the quadratic equation: \(x^2−5x−6=0\).
- Answer
-
\((x−6)(x+1)=0\), \(x=6\), \(x=−1\)
Solve the quadratic equation by factoring: \(x^2+8x+15=0\).
Solution
Find two numbers whose product equals \(15\) and whose sum equals \(8\). List the factors of \(15\).
\[1⋅15 \nonumber \]
\[3⋅5 \nonumber \]
\[(−1)⋅(−15) \nonumber \]
\[(−3)⋅(−5) \nonumber \]
The numbers that add to \(8\) are \(3\) and \(5\). Then, write the factors, set each factor equal to zero, and solve.
\[\begin{align*} (x+3)(x+5)&= 0\\ (x+3)&= 0\\ x&= -3\\ (x+5)&= 0\\ x&= -5 \end{align*}\]
The solutions are \(−3\) and \(−5\).
Solve the quadratic equation by factoring: \(x^2−4x−21=0\).
- Answer
-
\((x−7)(x+3)=0\), \(x=7\), \(x=−3\)
Solve the difference of squares equation using the zero-product property: \(x^2−9=0\).
Solution
Recognizing that the equation represents the difference of squares, we can write the two factors by taking the square root of each term, using a minus sign as the operator in one factor and a plus sign as the operator in the other. Solve using the zero-factor property.
\[\begin{align*} x^2-9&= 0\\ (x-3)(x+3)&= 0\\ x-3&= 0\\ x&= 3\\ (x+3)&= 0\\ x&= -3 \end{align*}\]
The solutions are \(3\) and \(−3\).
Solve by factoring: \(x^2−25=0\).
- Answer
-
\((x+5)(x−5)=0, x=−5, x=5\)
Factoring and Solving a Quadratic Equation of Higher Order
When the leading coefficient is not \(1\), we factor a quadratic equation using the method called grouping , which requires four terms.
With the equation in standard form, let’s review the grouping procedures
- With the quadratic in standard form, \(ax^2+bx+c=0\), multiply \(a⋅c\).
- Find two numbers whose product equals ac and whose sum equals \(b\).
- Rewrite the equation replacing the \(bx\) term with two terms using the numbers found in step \(1\) as coefficients of \(x\).
- Factor the first two terms and then factor the last two terms. The expressions in parentheses must be exactly the same to use grouping.
- Factor out the expression in parentheses.
- Set the expressions equal to zero and solve for the variable.
Use grouping to factor and solve the quadratic equation: \(4x^2+15x+9=0\).
Solution
First, multiply \(ac:4(9)=36\). Then list the factors of \(36\).
\[1⋅36 \nonumber\]
\[2⋅18 \nonumber\]
\[3⋅12 \nonumber\]
\[4⋅9 \nonumber\]
\[6⋅6 \nonumber\]
The only pair of factors that sums to \(15\) is \(3+12\). Rewrite the equation replacing the b term, \(15x\), with two terms using \(3\) and \(12\) as coefficients of \(x\). Factor the first two terms, and then factor the last two terms.
\[\begin{align*} 4x^2+3x+12x+9&= 0\\ x(4x+3)+3(4x+3)&= 0\\ (4x+3)(x+3)&= 0 \qquad \text{Solve using the zero-product property}\\ (4x+3)&= 3\\ x&= -\dfrac{3}{4}\\ (x+3)&= 0\\ x&= -3 \end{align*}\]
The solutions are \(−\dfrac{3}{4}\), and \(−3\). See Figure \(\PageIndex{3}\).
Solve using factoring by grouping: \(12x^2+11x+2=0\).
- Answer
-
\((3x+2)(4x+1)=0\), \(x=−\dfrac{2}{3}\), \(x=−\dfrac{1}{4}\)
Solve the equation by factoring: \(−3x^3−5x^2−2x=0\).
Solution
This equation does not look like a quadratic, as the highest power is \(3\), not \(2\). Recall that the first thing we want to do when solving any equation is to factor out the GCF, if one exists. And it does here. We can factor out \(−x\) from all of the terms and then proceed with grouping.
\[\begin{align*}
-3x^3-5x^2-2x&= 0\\
-x(3x^2+5x+2)&= 0\\
-x(3x^2+3x+2x+2)&= 0 \qquad \text{Use grouping on the expression in parentheses}\\
-x[3x(x+1)+2(x+1)]&= 0\\
-x(3x+2)(x+1)&= 0\\
\text{Now, we use the zero-product property. Notice that we have three factors.}\\
-x&= 0\\
x&= 0\\
3x+2&= 0\\
x&= -\dfrac{2}{3}\\
x+1&= 0\\
x&= -1
\end{align*}\]
The solutions are \(0\), \(−\dfrac{2}{3}\), and \(−1\).
Solve by factoring: \(x^3+11x^2+10x=0\).
- Answer
-
\(x=0, x=−10, x=−1\)
Using the Square Root Property
When there is no linear term in the equation, another method of solving a quadratic equation is by using the square root property , in which we isolate the \(x^2\) term and take the square root of the number on the other side of the equals sign. Keep in mind that sometimes we may have to manipulate the equation to isolate the \(x^2\) term so that the square root property can be used.
With the \(x^2\) term isolated, the square root property states that:
where \(k\) is a nonzero real number.
- Isolate the \(x^2\) term on one side of the equal sign.
- Take the square root of both sides of the equation, putting a \(±\) sign before the expression on the side opposite the squared term.
- Simplify the numbers on the side with the \(±\) sign.
Solve the quadratic using the square root property: \(x^2=8\).
Solution
Take the square root of both sides, and then simplify the radical. Remember to use a \(±\) sign before the radical symbol.
\[\begin{align*} x^2&= 8\\ x&= \pm \sqrt{8}\\ &= \pm 2\sqrt{2} \end{align*}\]
The solutions are \(2\sqrt{2}\),\(-2\sqrt{2}\)
Solve the quadratic equation: \(4x^2+1=7\).
Solution
First, isolate the \(x^2\) term. Then take the square root of both sides.
\[\begin{align*} 4x^2+1&= 7\\ 4x^2&= 6\\ x^2&= \dfrac{6}{4}\\ x&= \pm \dfrac{\sqrt{6}}{2} \end{align*}\]
The solutions are \(\dfrac{\sqrt{6}}{2}\), and \(-\dfrac{\sqrt{6}}{2}\).
Solve the quadratic equation using the square root property: \(3{(x−4)}^2=15\).
- Answer
-
\(x=4±\sqrt{5}\)
Completing the Square
Not all quadratic equations can be factored or can be solved in their original form using the square root property. In these cases, we may use a method for solving a quadratic equation known as completing the square . Using this method, we add or subtract terms to both sides of the equation until we have a perfect square trinomial on one side of the equal sign. We then apply the square root property. To complete the square, the leading coefficient, \(a\), must equal \(1\). If it does not, then divide the entire equation by \(a\). Then, we can use the following procedures to solve a quadratic equation by completing the square.
We will use the example \(x^2+4x+1=0\) to illustrate each step.
Given a quadratic equation that cannot be factored, and with \(a=1\), first add or subtract the constant term to the right sign of the equal sign.
\[\begin{align*}
x^2+4x+1&= 0\\
x^2+4x&= -1 \qquad \text{Multiply the b} \text{ term by } \dfrac{1}{2} \text{ and square it.}\\
\dfrac{1}{2}(4)&= 2 \\
2^2&= 4 \qquad \text{Add } \left ({\dfrac{1}{2}} \right )^2 \text{ to both sides of the equal sign and simplify the right side. We have}\\
x^2+4x+4&= -1+4\\
x^2+4x+4&= 3 \qquad \text{The left side of the equation can now be factored as a perfect square.}\\
{(x+2)}^2&=3\\
\sqrt{{(x+2)}^2}&= \pm \sqrt{3} \qquad \text{Use the square root property and solve.}\\
\sqrt{{(x+2)}^2}&= \pm \sqrt{3}\\
x+2&= \pm \sqrt{3}\\
x&= -2 \pm \sqrt{3}
\end{align*}\]
The solutions are \(−2+\sqrt{3}\), and \(−2−\sqrt{3}\).
Solve the quadratic equation by completing the square: \(x^2−3x−5=0\).
Solution
First, move the constant term to the right side of the equal sign.
\[\begin{align*}
x^2-3x&= 5 \qquad \text{Then, take } \dfrac{1}{2} \text{ of the b term and square it.} \\
\dfrac{1}{2}(-3)&= -\dfrac{3}{2}\\
{\left (-\dfrac{3}{2} \right )}^2=\dfrac{9}{4}\\
x^2-3x+{\left (-\dfrac{3}{2} \right )}^2&= 5+{\left (-\dfrac{3}{2} \right )}^2 \qquad \text{Add the result to both sides of the equal sign.}\\
x^2-3x+\dfrac{9}{4}&= 5+\dfrac{9}{4}\\
\text{Factor the left side as a perfect square and simplify the right side.}\\
{\left (x-\dfrac{3}{2} \right )}^2&= \dfrac{29}{4}\\
(x-\dfrac{3}{2})&= \pm \dfrac{\sqrt{29}}{2} \qquad \text{Use the square root property and solve.}\\
x&= \dfrac{3}{2} \pm \dfrac{\sqrt{29}}{2}\\
\end{align*}\]
The solutions are \(\dfrac{3}{2}+\dfrac{\sqrt{29}}{2}\), and \(\dfrac{3}{2}-\dfrac{\sqrt{29}}{2}\)
Solve by completing the square: \(x^2−6x=13\).
- Answer
-
\(x=3±\sqrt{22}\)
Using the Quadratic Formula
The fourth method of solving a quadratic equation is by using the quadratic formula , a formula that will solve all quadratic equations. Although the quadratic formula works on any quadratic equation in standard form, it is easy to make errors in substituting the values into the formula. Pay close attention when substituting, and use parentheses when inserting a negative number.
We can derive the quadratic formula by completing the square . We will assume that the leading coefficient is positive; if it is negative, we can multiply the equation by \(−1\) and obtain a positive a. Given \(ax^2+bx+c=0, a≠0\), we will complete the square as follows:
First, move the constant term to the right side of the equal sign:
\[ax^2+bx=−c \nonumber \]
As we want the leading coefficient to equal \(1\), divide through by \(a\):
\[x^2+\dfrac{b}{a}x=−\dfrac{c}{a} \nonumber \]
Then, find \(\dfrac{1}{2}\) of the middle term, and add \({(\dfrac{1}{2}\dfrac{b}{a})}^2=\dfrac{b^2}{4a^2}\) to both sides of the equal sign:
\[x^2+\dfrac{b}{a}x+\dfrac{b^2}{4a^2}=\dfrac{b^2}{4a^2}-\dfrac{c}{a} \nonumber \]
Next, write the left side as a perfect square. Find the common denominator of the right side and write it as a single fraction:
\[{(x+\dfrac{b}{2a})}^2=\dfrac{b^2-4ac}{4a^2} \nonumber \]
Now, use the square root property, which gives
\[x+\dfrac{b}{2a}=±\sqrt{\dfrac{b^2-4ac}{4a^2}} \nonumber \]
\[x+\dfrac{b}{2a}=\dfrac{±\sqrt{b^2-4ac}}{2a} \nonumber \]
Finally, add \(-\dfrac{b}{2a}\) to both sides of the equation and combine the terms on the right side. Thus,
\[x=\dfrac{-b±\sqrt{b^2-4ac}}{2a} \nonumber \]
Written in standard form, \(ax^2+bx+c=0\), any quadratic equation can be solved using the quadratic formula :
\[x=\dfrac{-b±\sqrt{b^2-4ac}}{2a}\]where \(a\), \(b\), and \(c\) are real numbers and \(a≠0\).
Given a quadratic equation, solve it using the quadratic formula
- Make sure the equation is in standard form: \(ax^2+bx+c=0\).
- Make note of the values of the coefficients and constant term, \(a\), \(b\), and \(c\).
- Carefully substitute the values noted in step 2 into the equation. To avoid needless errors, use parentheses around each number input into the formula.
- Calculate and solve.
Solve the quadratic equation: \(x^2+5x+1=0\).
Solution
Identify the coefficients: \(a=1,b=5,c=1\). Then use the quadratic formula.
\[\begin{align*} x&= \dfrac{-(5) \pm \sqrt{(5)^2-4(1)(1)}}{2(1)}\\ &= \dfrac{-5 \pm \sqrt{25-4}}{2}\\ &= \dfrac{-5 \pm \sqrt{21}}{2} \end{align*}\]
Use the quadratic formula to solve \(x^2+x+2=0\).
Solution
First, we identify the coefficients: \(a=1\),\(b=1\), and \(c=2\).
Substitute these values into the quadratic formula.
\[\begin{align*} x&= \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}\\ &= \dfrac{-(1) \pm \sqrt{(1)^2-4(1)(2)}}{2(1)}\\ &= \dfrac{-1 \pm \sqrt{1-8}}{2}\\ &= \dfrac{-1 \pm \sqrt{-7}}{2}\\ &= \dfrac{-1 \pm i\sqrt{7}}{2} \end{align*}\]
Solve the quadratic equation using the quadratic formula: \(9x^2+3x−2=0\).
- Answer
-
\(x=-\dfrac{2}{3},x=\dfrac{1}{3}\)
The Discriminant
The quadratic formula not only generates the solutions to a quadratic equation, it tells us about the nature of the solutions when we consider the discriminant , or the expression under the radical, \(b^2−4ac\). The discriminant tells us whether the solutions are real numbers or complex numbers, and how many solutions of each type to expect. Table \(\PageIndex{1}\) relates the value of the discriminant to the solutions of a quadratic equation.
| Value of Discriminant | Results |
|---|---|
| \(b^2−4ac=0\) | One rational solution (double solution) |
| \(b^2−4ac>0\), perfect square | Two rational solutions |
| \(b^2−4ac>0\), not a perfect square | Two irrational solutions |
| \(b^2−4ac<0\) | Two complex solutions |
For \(ax^2+bx+c=0\), where \(a\), \(b\), and \(c\) are real numbers, the discriminant is the expression under the radical in the quadratic formula: \(b^2−4ac\). It tells us whether the solutions are real numbers or complex numbers and how many solutions of each type to expect.
Use the discriminant to find the nature of the solutions to the following quadratic equations:
- \(x^2+4x+4=0\)
- \(8x^2+14x+3=0\)
- \(3x^2−5x−2=0\)
- \(3x^2−10x+15=0\)
Solution
Calculate the discriminant \(b^2−4ac\) for each equation and state the expected type of solutions.
a.
\(x^2+4x+4=0\)
\(b^2-4ac={(4)}^2-4(1)(4)=0\) There will be one rational double solution.
b.
\(8x^2+14x+3=0\)
\(b^2-4ac={(14)}^2-4(8)(3)=100\) As \(100\) is a perfect square, there will be two rational solutions.
c.
\(3x^2−5x−2=0\)
\(b^2-4ac={(-5)}^2-4(3)(-2)=49\) As \(49\) is a perfect square, there will be two rational solutions.
d.
\(3x^2−10x+15=0\)
\(b^2-4ac={(-10)}^2-4(3)(15)=-80\) There will be two complex solutions.
Using the Pythagorean Theorem
One of the most famous formulas in mathematics is the Pythagorean Theorem . It is based on a right triangle, and states the relationship among the lengths of the sides as \(a^2+b^2=c^2\), where \(a\) and \(b\) refer to the legs of a right triangle adjacent to the \(90°\) angle, and \(c\) refers to the hypotenuse. It has immeasurable uses in architecture, engineering, the sciences, geometry, trigonometry, and algebra, and in everyday applications.
We use the Pythagorean Theorem to solve for the length of one side of a triangle when we have the lengths of the other two. Because each of the terms is squared in the theorem, when we are solving for a side of a triangle, we have a quadratic equation. We can use the methods for solving quadratic equations that we learned in this section to solve for the missing side.
The Pythagorean Theorem is given as
\[a^2+b^2=c^2\]
where \(a\) and \(b\) refer to the legs of a right triangle adjacent to the \(90°\) angle, and \(c\) refers to the hypotenuse, as shown in .
Find the length of the missing side of the right triangle in Figure \(\PageIndex{5}\).
Solution
As we have measurements for side \(b\) and the hypotenuse, the missing side is \(a\).
\[\begin{align*} a^2+b^2&= c^2\\ a^2+{(4)}^2&= {(12)}^2\\ a^2+16&= 144\\ a^2&= 128\\ a&= \sqrt{128}\\ &= 8\sqrt{2} \end{align*}\]
Use the Pythagorean Theorem to solve the right triangle problem: Leg a measures 4 units, leg b measures 3 units. Find the length of the hypotenuse.
- Answer
-
\(5\) units
Access these online resources for additional instruction and practice with quadratic equations.
- Solving Quadratic Equations by Factoring
- The Zero-Product Property
- Completing the Square
- Quadratic Formula with Two Rational Solutions
- Length of a leg of a right triangle
Key Equations
| quadratic formula | \(x=\dfrac{−b±\sqrt{b^2-4ac}}{2a}\) |
Key Concepts
- Many quadratic equations can be solved by factoring when the equation has a leading coefficient of \(1\) or if the equation is a difference of squares. The zero-factor property is then used to find solutions. See Example , Example , and Example .
- Many quadratic equations with a leading coefficient other than \(1\) can be solved by factoring using the grouping method. See Example and Example .
- Another method for solving quadratics is the square root property. The variable is squared. We isolate the squared term and take the square root of both sides of the equation. The solution will yield a positive and negative solution. See Example and Example .
- Completing the square is a method of solving quadratic equations when the equation cannot be factored. See Example .
- A highly dependable method for solving quadratic equations is the quadratic formula, based on the coefficients and the constant term in the equation. See Example .
- The discriminant is used to indicate the nature of the roots that the quadratic equation will yield: real or complex, rational or irrational, and how many of each. See Example .
- The Pythagorean Theorem, among the most famous theorems in history, is used to solve right-triangle problems and has applications in numerous fields. Solving for the length of one side of a right triangle requires solving a quadratic equation. See Example .


<IP_ADDRESS>.3: Sample Biography - Aldous Huxley (1894-1963)
Figure 1: Aldous Huxley.
Aldous Huxley was descended from two eminent Victorian families—he was a grandson of noted biologist and writer on science, Thomas Henry Huxley, grandnephew of Matthew Arnold, and nephew of Victorian novelist Mary Augusta Ward (who wrote under her married name Mrs. Humphry Ward), Arnold’s niece. His unusual Christian name commemorates a major character, Aldous Raeburn, in the novel Marcella , which Mrs. Ward published in the year of Huxley’s birth, 1894.
Born in Godalming, Surrey, England, he received his first schooling from his mother, Julia Arnold Huxley. He then moved on to Hillside Preparatory School, Eton, and eventually, Balliol College, Oxford, taking a first in English in 1916. Two early blows—the death of his mother when he was only nine, and an attack of keratitis while he was a student at Eton, which left him nearly blind for the rest of his life—may have sharpened his tendency toward introversion. Certainly the latter affliction precluded a career in science, paving the way for a life in letters.
In the course of his long literary career, Aldous Huxley published poetry collections, plays, essays, short fiction, travel narratives, biography, and criticism, but, like George Orwell, author of Nineteen-Eighty-four , he is best remembered as the author of a hugely influential utopian satire. Both men used the genre more than once. Brave New World began, like Orwell’s Coming Up for Air , as a response to H. G. Wells’s utopia, Men Like Gods (1923), and Huxley later wrote two more utopian-dystopian novels: Ape and Essence (1948) and Island (1962).
Although Huxley felt that he was not a “born novelist, but an essayist who writes novels,” he became an innovative fictional stylist, effectively using cinematic montage technique for the purpose of exposition in the third chapter of Brave New World , thereby sparing the reader from that major defect of most utopian novels—the tedious guided tour of utopian schools, hospitals, and factories. With his use of montage and intertextuality, Huxley did for the novel what Eliot had done for modern poetry in The Waste Land 10 years earlier. Like Eliot, Huxley often uses multiple references to canonical works. In Brave New World , his 50 allusions to Shakespeare help develop theme. He also incorporates ideas of fashionable Freudians such as Ernest Jones and his interpretation of Hamlet to deepen the sense of the Savage’s sexual repression. At opportune moments, he echoes key lines from Gray’s “Elegy” and “Ode on a Distant Prospect of Eton” to underscore the ignorance-is-bliss theme and key images that recall the Utilitarian schoolroom in Dickens’s Hard Times . Nor does he hesitate to use bathos: the deliberate contrasting of high culture with popular culture—excerpts from fictional advertising jingles and popular 1920s-era romances such as Elinor Glyn’s Three Weeks satirize the contemporary lowering of musical and literary standards.
One notes a gradually emerging social conscience in Huxley at the time of his writing Brave New World . In fact, his satiric description of the reified workers in a dystopian factory was based on his visit to a Lucas automotive parts factory in Birmingham the previous spring.
In Brave New World , the novel’s most sympathetic character, Helmholtz Watson, like Wordworth on Westminster Bridge, finds the silence of the sleeping metropolis to be eloquent: the very absence of the numinous in the wholly materialistic state begins, paradoxically, to suggest a presence. Watson is one of the first in what becomes a steady string of protagonists in his future novels who advocate what Huxley called “the great central technique, which traces the art of obtaining freedom from the fundamental human disability of egotism…repeatedly described by the mystics of all ages and countries.” His final utopian novel, Island (1962), is, in fact, Huxley’s final word on the subject of how to construct the good society. He died in Hollywood on November 22, 1963, the day President John F. Kennedy was assassinated.


9.10: Relationships and Family Life in Middle Adulthood
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Learning Outcomes
- Describe the link between intimacy and subjective well-being
- Discuss issues related to family life in middle adulthood
Types of Relationships
Intimate Relationships
It makes sense to consider the various types of relationships in our lives when trying to determine just how relationships impact our well-being. For example, would you expect a person to derive the same happiness from an ex-spouse as from a child or coworker? Among the most important relationships for most people is their long-time romantic partner. Most researchers begin their investigation of this topic by focusing on intimate relationships because they are the closest form of social bond. Intimacy is more than just physical in nature; it also entails psychological closeness. Research findings suggest that having a single confidante—a person with whom you can be authentic and trust not to exploit your secrets and vulnerabilities—is more important to happiness than having a large social network (Taylor, 2010).
Another important aspect of relationships is the distinction between formal and informal. Formal relationships are those that are bound by the rules of politeness. In most cultures, for instance, young people treat older people with formal respect, avoiding profanity and slang when interacting with them. Similarly, workplace relationships tend to be more formal, as do relationships with new acquaintances. Formal connections are generally less relaxed because they require a bit more work, demanding that we exert more self-control. Contrast these connections with informal relationships—friends, lovers, siblings, or others with whom you can relax. We can express our true feelings and opinions in these informal relationships, using the language that comes most naturally to us, and generally be more authentic. Because of this, it makes sense that more intimate relationships—those that are more comfortable and in which you can be more vulnerable—might be the most likely to translate to happiness.
Marriage and Happiness
One of the most common ways that researchers often begin to investigate intimacy is by looking at marital status. The well-being of married people is compared to that of people who are single or have never been married. In other research, married people are compared to people who are divorced or widowed (Lucas & Dyrenforth, 2005). Researchers have found that the transition from singlehood to marriage brings about an increase in subjective well-being (Haring-Hidore, Stock, Okun, & Witter, 1985; Lucas, 2005; Williams, 2003). In fact, this finding is one of the strongest in social science research on personal relationships over the past quarter of a century.
As is usually the case, the situation is more complex than might initially appear. As a marriage progresses, there is some evidence for a regression to a hedonic set-point— that is, most individuals have a set happiness point or level, and that both good and bad life events – marriage, bereavement, unemployment, births and so on – have some effect for a period of time, but over many months, they will return to that set-point. One of the best studies in this area is that of Luhmann et al (2012), who report a gradual decline in subjective well-being after a few years, especially in the component of affective well-being. A dverse events obviously have an effect on subjective well-being and happiness, and these effects can be stronger than the positive effects of being married in some cases (Lucas, 2005).
Although research frequently points to marriage being associated with higher rates of happiness, this does not guarantee that getting married will make you happy! The quality of one’s marriage matters greatly. When a person remains in a problematic marriage, it takes an emotional toll. Indeed, a large body of research shows that people’s overall life satisfaction is affected by their satisfaction with their marriage (Carr, Freedman, Cornman, Schwarz, 2014; Dush, Taylor, & Kroeger, 2008; Karney, 2001; Luhmann, Hofmann, Eid, & Lucas, 2012; Proulx, Helms, & Buehler, 2007). The lower a person’s self-reported level of marital quality, the more likely he or she is to report depression (Bookwala, 2012). In fact, longitudinal studies—those that follow the same people over a period of time—show that as marital quality declines, depressive symptoms increase (Fincham, Beach, Harold, & Osborne, 1997; Karney, 2001). Proulx and colleagues (2007) arrived at this same conclusion after a systematic review of 66 cross-sectional and 27 longitudinal studies.
Marital satisfaction has peaks and valleys during the course of the life cycle. Rates of happiness are highest in the years prior to the birth of the first child. It hits a low point with the coming of children. Relationships typically become more traditional and there are more financial hardships and stress in living. Children bring new expectations to the marital relationship. Two people who are comfortable with their roles as partners may find the added parental duties and expectations more challenging to meet. Some couples elect not to have children in order to have more time and resources for the marriage. These child-free couples are happy keeping their time and attention on their partners, careers, and interests.
What is it about bad marriages, or bad relationships in general, that takes such a toll on well-being? Research has pointed to conflict between partners as a major factor leading to lower subjective well-being (Gere & Schimmack, 2011). This makes sense. Negative relationships are linked to ineffective social support (Reblin, Uchino, & Smith, 2010) and are a source of stress (Holt-Lunstad, Uchino, Smith, & Hicks, 2007). In more extreme cases, physical and psychological abuse can be detrimental to well-being (Follingstad, Rutledge, Berg, Hause, & Polek, 1990). Victims of abuse sometimes feel shame, lose their sense of self, and become less happy and prone to depression and anxiety (Arias & Pape, 1999). However, the unhappiness and dissatisfaction that occur in abusive relationships tend to dissipate once the relationships end. (Arriaga, Capezza, Goodfriend, Rayl & Sands, 2013).
Typology of Marriage
One way marriages vary is with regard to the reason the partners are married. Some marriages have intrinsic value: the partners are together because they enjoy, love and value one another. Marriage is not thought of as a means to another end, instead it is regarded as an end in itself. These partners look for someone they are drawn to, and with whom they feel a close and intense relationship. Other marriages called utilitarian marriages are unions entered into primarily for practical reasons. For example, the marriage brings financial security, children, social approval, housekeeping, political favor, a good car, a great house, and so on.
There have been a few attempts to establish a typological framework for marriages. The best-known is that of Olson (1993), who referred to five typical kinds of marriage. Using a sample of 6,267 couples, Olson & Fowers (1993) identified eleven relationship domains which covered both areas related to relationship satisfaction, and the more functional areas related to marriage. So, five of the eleven included areas such as marital satisfaction, communication, and, things like financial management, parenting and egalitarian roles. Using these eleven areas they came up with five kinds of marriage. One aspect of this early study is the link between marital satisfaction and income/college education . The link between these factors is now commonplace in the literature. Olson & Fowers (1993) were one of the first studies to point to this link. The less well off are more prone to divorce, as are those with less college-level education. Income and college education are of course linked, and there is now increasing concern that marital dissolution and broader patterns of social inequality are now inextricably linked. [1]
- vitalized: Very high relationship quality. Tend to belong in a higher income bracket. Happy with their spouse across all areas: personality, communication, roles and expectations.
- harmonious relationships: These marriages have some areas of tension and disagreement but there is still broad agreement on major issues. Lack of agreement on parenting was the primary feature of this group, although the couples still scored highly on relationship quality.
- traditional marriages: Much less emphasis on emotional closeness, but still slightly above average. High levels of compatibility in relation to parenting.
- conflicted: These marriages accomplish functional goals such as parenting but are marked by a great deal of interpersonal disagreement. Communication and conflict resolution scores are extremely low.
- devitalized: low scores across all eleven areas – Little interpersonal closeness and little agreement on family roles.
The Family Life Cycle
To better understand patterns of family life and the changes in roles and expectations as a family ages, researchers have theorized about typical stages of family life. Read more about the family life cycle in the following interactive activity.
Marital Communication
Advice on how to improve one’s marriage is centuries old. One of today’s experts on marital communication is John Gottman. Gottman differs from many marriage counselors in his belief that having a good marriage does not depend on compatibility, rather, the way that partners communicate with one another is crucial. At the University of Washington in Seattle, Gottman has measured the physiological responses of thousands of couples as they discuss issues which have led to disagreements. Fidgeting in one’s chair, leaning closer to or further away from the partner while speaking, and increases in respiration and heart rate are all recorded and analyzed, along with videotaped recordings of the partners’ exchanges.
Gottman believes he can accurately predict whether or not a couple will stay together by analyzing their communication. In marriages destined to fail, partners engage in the “marriage killers” such as contempt, criticism, defensiveness, and stonewalling. Each of these undermines the politeness and respect that healthy marriages require. According to Gottman, stonewalling, or shutting someone out, is the strongest sign that a relationship is destined to fail. Perhaps the most interesting aspect of Gottman’s work is the emphasis on the fact that marriage is about constant negotiation rather than conflict resolution.
What Gottman terms perpetual problems, are responsible for 69% of conflicts within marriage. For example, if someone in a couple has said, “I am so sick of arguing over this,” then that may be a sign of perpetual problem. While this may seem problematic, Gottman argues that couples can still be connected despite these perpetual problems if they can laugh about it, treat it as a “third thing” (not reducible to the perspective of either party), and recognize that these are part of relationships that need to be aired and dealt with as best you can. It is somewhat refreshing to hear that differences lie at the heart of marriage, rather than a rationale for its dissolution!
Link to Learning
Listen to NPR’s Act One: What Really Happens in Marriage to hear John Gottman talk about his work.
Parenting in Later Life
Just because children grow up does not mean their family stops being a family, rather the specific roles and expectations of its members change over time. One major change comes when a child reaches adulthood and moves away. When exactly children leave home varies greatly depending on societal norms and expectations, as well as on economic conditions such as employment opportunities and affordable housing options. Some parents may experience sadness when their adult children leave the home—a situation called an empty nest .
Many parents are also finding that their grown children are struggling to launch into independence. It’s an increasingly common story: a child goes off to college and, upon graduation, is unable to find steady employment. In such instances, a frequent outcome is for the child to return home, becoming a “boomerang kid.” The boomerang generation, as the phenomenon has come to be known, refers to young adults, mostly between the ages of 25 and 34, who return home to live with their parents while they strive for stability in their lives—often in terms of finances, living arrangements, and sometimes romantic relationships. These boomerang kids can be both good and bad for families. Within American families, 48% of boomerang kids report having paid rent to their parents, and 89% say they help out with household expenses—a win for everyone (Parker, 2012). On the other hand, 24% of boomerang kids report that returning home hurts their relationship with their parents (Parker, 2012). For better or for worse, the number of children returning home has been increasing around the world. The Pew Research Center (2016) reported that the most common living arrangement for people aged 18-34 was living with their parents (32.1%). [2]
Adult children typically maintain frequent contact with their parents, if for no other reason, money and advice. Attitudes toward one’s parents may become more accepting and forgiving, as parents are seen in a more objective way, as people with good points and bad. As adults children can continue to be subjected to criticism, ridicule, and abuse at the hand of parents. How long are we “adult children”? For as long as our parents are living, we continue in the role of son or daughter. (I had a neighbor in her nineties who would tell me her “boys” were coming to see her this weekend. Her boys were in their 70s-but they were still her boys!) But after one’s parents are gone, the adult is no longer a child; as one 40 year old man explained after the death of his father, “I’ll never be a kid again.”
Family Issues and Considerations
In addition to middle-aged parents spending more time, money, and energy taking care of their adult children, they are also increasingly taking care of their own aging and ailing parents. Middle-aged people in this set of circumstances are commonly referred to as the sandwich generation (Dukhovnov & Zagheni, 2015). Of course, cultural norms and practices again come into play. In some Asian and Hispanic cultures, the expectation is that adult children are supposed to take care of aging parents and parents-in-law. In other Western cultures—cultures that emphasize individuality and self-sustainability—the expectation has historically been that elders either age in place, modifying their home and receiving services to allow them to continue to live independently, or enter long-term care facilities. However, given financial constraints, many families find themselves taking in and caring for their aging parents, increasing the number of multigenerational homes around the world.
Being a midlife child often involves kinkeeping ; organizing events and communication in order to maintain family ties. This role was first defined by Carolyn Rosenthal (1985). Kinkeepers are often midlife daughters (they are the person who tells you what food to bring to a gathering, or makes arrangement for a family reunion). They can often function as “managers” who maintain family ties and lines of communication. This is true for both large nuclear families, reconstituted, and multi-generational families. Rosenthal found that over half of the families she sampled were capable of identifying the individual who performed this role. Often adults at this stage of their lives are pressed into caregiving roles. Often referred to as the “sandwich generation”, they are still looking out for their own children while simultaneously caring for elderly parents. Given shifts in longevity and increasing costs for professional care of the elderly, this role will likely expand, placing ever greater pressure on careers.
Abuse in Family Life
Abuse can occur in multiple forms and across all family relationships. Breiding, Basile, Smith, Black, & Mahendra (2015) define the forms of abuse as:
- Physical abuse: the use of intentional physical force to cause harm. Scratching, pushing, shoving, throwing, grabbing, biting, choking, shaking, slapping, punching, and hitting are common forms of physical abuse
- Sexual abuse: the act of forcing someone to participate in a sex act against his or her will. Such abuse is often referred to as sexual assault or rape. A marital relationship does not grant anyone the right to demand sex or sexual activity from anyone, even a spouse
- Psychological abuse: aggressive behavior that is intended to control someone else. Such abuse can include threats of physical or sexual abuse, manipulation, bullying, and stalking.
Abuse between partners is referred to as intimate partner violence; however, such abuse can also occur between a parent and child (child abuse), adult children and their aging parents (elder abuse), and even between siblings.
The most common form of abuse between parents and children is that of neglect. Neglect refers to a family’s failure to provide for a child’s basic physical, emotional, medical, or educational needs (DePanfilis, 2006). Harry Potter’s aunt and uncle, as well as Cinderella’s stepmother, could all be prosecuted for neglect in the real world.
Abuse is a complex issue, especially within families. There are many reasons people become abusers: poverty, stress, and substance abuse are common characteristics shared by abusers, although abuse can happen in any family. There are also many reasons adults stay in abusive relationships: (a) learned helplessness (the abused person believing he or she has no control over the situation); (b) the belief that the abuser can/will change; (c) shame, guilt, self-blame, and/or fear; and (d) economic dependence. All of these factors can play a role.
Children who experience abuse may “act out” or otherwise respond in a variety of unhealthy ways. These include acts of self-destruction, withdrawal, and aggression, as well as struggles with depression, anxiety, and academic performance. Researchers have found that abused children’s brains may produce higher levels of stress hormones. These hormones can lead to decreased brain development, lower stress thresholds, suppressed immune responses, and lifelong difficulties with learning and memory (Middlebrooks & Audage, 2008).
Happy Healthy Families
Our families play a crucial role in our overall development and happiness. They can support and validate us, but they can also criticize and burden us. For better or worse, we all have a family. In closing, here are strategies you can use to increase the happiness of your family:
- Teach morality—fostering a sense of moral development in children can promote well-being (Damon, 2004).
- Savor the good—celebrate each other’s successes (Gable, Gonzaga & Strachman, 2006).
- Use the extended family network—family members of all ages, including older siblings and grandparents, who can act as caregivers can promote family well-being (Armstrong, Birnie-Lefcovitch & Ungar, 2005).
- Create family identity—share inside jokes, fond memories, and frame the story of the family (McAdams, 1993).
- Forgive—Don’t hold grudges against one another (McCullough, Worthington & Rachal, 1997).
Glossary
- kinkeeping
- ”emotion work”, often undertaken by women, to foster and maintain family relationships
- physical abuse
- the use of intentional physical force to cause harm
- psychological abuse
- aggressive behavior that is intended to control someone else
- sandwich generation
- a cohort of people charged with the dual responsibility of looking after elderly parents while raising their own children
- sexual abuse
- the act of forcing someone to participate in a sex act against his or her will
- D.Olson & B.Fowers (1993) Five Types of Marriage The Family Journal, 1993 Vol. 1, No. 3, 196-207. Retrieved from https://www.prepare-enrich.com/pe/pdf/research/study6.pdf<EMAIL_ADDRESS>- Fry, Richard. For First Time in Modern Era (2016). Living With Parents Edges Out Other Living Arrangements for 18- to 34-Year-Olds. Pew Research Center. Retrieved from https://www.pewsocialtrends.org/2016/05/24/for-first-time-in-modern-era-living-with-parents-edges-out-other-living-arrangements-for-18-to-34-year-olds/<EMAIL_ADDRESS>Contributors and Attributions
- Modification, adaptation, and original content. Authored by : Ronnie Mather and Lumen Learning. Provided by : Lumen Learning. License : CC BY: Attribution
- Family Life Cycle interactive. Authored by : Stephanie Loalada for Lumen Learning. Provided by : Lumen Learning. License : CC BY: Attribution
- Psyc 200 Lifespan Psychology. Authored by : Laura Overstreet. Located at : http://opencourselibrary.org/econ-201/ . License : CC BY: Attribution
- Types of Relationships: Intimate Relationships and Marriage and Happiness. Authored by : Kenneth Tan and Louis Tay . Provided by : Purdue University. Located at : https://nobaproject.com/modules/relationships-and-well-being . Project : The Noba Project. License : CC BY-NC-SA: Attribution-NonCommercial-ShareAlike
- The Family: Parenting in Later Life, Abuse, Happy, Healthy Families. Authored by : Joel A. Muraco . Provided by : University of Wisconsin, Green Bay. Located at : https://nobaproject.com/modules/the-family . Project : The Noba Project. License : CC BY-NC-SA: Attribution-NonCommercial-ShareAlike
- smiling couple. Authored by : Bill Branson. Located at : http://www.freestockphotos.biz/stockphoto/17066 . License : Public Domain: No Known Copyright


12.3: Transmission Lines and Characteristic Impedance
The Parallel Wires of Infinite Length
Suppose, though, that we had a set of parallel wires of infinite length, with no lamp at the end. What would happen when we close the switch? Being that there is no longer a load at the end of the wires, this circuit is open. Would there be no current at all? (Figure below)
Driving an infinite transmission line.
Despite being able to avoid wire resistance through the use of superconductors in this “thought experiment,” we cannot eliminate capacitance along the wires’ lengths. Any pair of conductors separated by an insulating medium creates capacitance between those conductors: (Figure below)
Equivalent circuit showing stray capacitance between conductors.
Voltage applied between two conductors creates an electric field between those conductors. Energy is stored in this electric field, and this storage of energy results in an opposition to change in voltage. The reaction of a capacitance against changes in voltage is described by the equation i = C(de/dt), which tells us that current will be drawn proportional to the voltage’s rate of change over time. Thus, when the switch is closed, the capacitance between conductors will react against the sudden voltage increase by charging up and drawing current from the source. According to the equation, an instant rise in applied voltage (as produced by perfect switch closure) gives rise to an infinite charging current.
Capacitance and Inductance
However, the current drawn by a pair of parallel wires will not be infinite, because there exists series impedance along the wires due to inductance. (Figure below) Remember that current through any conductor develops a magnetic field of proportional magnitude. Energy is stored in this magnetic field, (Figure below) and this storage of energy results in an opposition to change in current. Each wire develops a magnetic field as it carries charging current for the capacitance between the wires, and in so doing drops voltage according to the inductance equation e = L(di/dt). This voltage drop limits the voltage rate-of-change across the distributed capacitance, preventing the current from ever reaching an infinite magnitude:
Equivalent circuit showing stray capacitance and inductance.
Voltage charges capacitance, current charges inductance.
Because the electrons in the two wires transfer motion to and from each other at nearly the speed of light, the “wave front” of voltage and current change will propagate down the length of the wires at that same velocity, resulting in the distributed capacitance and inductance progressively charging to full voltage and current, respectively, like this: (Figures below, below, below, below)
Uncharged transmission line.
Begin wave propagation.
Continue wave propagation.
Propagate at speed of light.
The Transmission Line
The end result of these interactions is a constant current of limited magnitude through the battery source. Since the wires are infinitely long, their distributed capacitance will never fully charge to the source voltage, and their distributed inductance will never allow unlimited charging current. In other words, this pair of wires will draw current from the source so long as the switch is closed, behaving as a constant load. No longer are the wires merely conductors of electrical current and carriers of voltage, but now constitute a circuit component in themselves, with unique characteristics. No longer are the two wires merely a pair of conductors , but rather a transmission line .
As a constant load, the transmission line’s response to applied voltage is resistive rather than reactive, despite being comprised purely of inductance and capacitance (assuming superconducting wires with zero resistance). We can say this because there is no difference from the battery’s perspective between a resistor eternally dissipating energy and an infinite transmission line eternally absorbing energy. The impedance (resistance) of this line in ohms is called the characteristic impedance , and it is fixed by the geometry of the two conductors. For a parallel-wire line with air insulation, the characteristic impedance may be calculated as such:
If the transmission line is coaxial in construction, the characteristic impedance follows a different equation:
In both equations, identical units of measurement must be used in both terms of the fraction. If the insulating material is other than air (or a vacuum), both the characteristic impedance and the propagation velocity will be affected. The ratio of a transmission line’s true propagation velocity and the speed of light in a vacuum is called the velocity factor of that line.
Velocity factor is purely a factor of the insulating material’s relative permittivity (otherwise known as its dielectric constant ), defined as the ratio of a material’s electric field permittivity to that of a pure vacuum. The velocity factor of any cable type—coaxial or otherwise—may be calculated quite simply by the following formula:
The Natural Impedance
Characteristic impedance is also known as natural impedance , and it refers to the equivalent resistance of a transmission line if it were infinitely long, owing to distributed capacitance and inductance as the voltage and current “waves” propagate along its length at a propagation velocity equal to some large fraction of light speed.
It can be seen in either of the first two equations that a transmission line’s characteristic impedance (Z 0 ) increases as the conductor spacing increases. If the conductors are moved away from each other, the distributed capacitance will decrease (greater spacing between capacitor “plates”), and the distributed inductance will increase (less cancellation of the two opposing magnetic fields). Less parallel capacitance and more series inductance results in a smaller current drawn by the line for any given amount of applied voltage, which by definition is a greater impedance. Conversely, bringing the two conductors closer together increases the parallel capacitance and decreases the series inductance. Both changes result in a larger current drawn for a given applied voltage, equating to a lesser impedance.
Barring any dissipative effects such as dielectric “leakage” and conductor resistance, the characteristic impedance of a transmission line is equal to the square root of the ratio of the line’s inductance per unit length divided by the line’s capacitance per unit length:
Review
- A transmission line is a pair of parallel conductors exhibiting certain characteristics due to distributed capacitance and inductance along its length.
- When a voltage is suddenly applied to one end of a transmission line, both a voltage “wave” and a current “wave” propagate along the line at nearly light speed.
- If a DC voltage is applied to one end of an infinitely long transmission line, the line will draw current from the DC source as though it were a constant resistance.
- The characteristic impedance (Z 0 ) of a transmission line is the resistance it would exhibit if it were infinite in length. This is entirely different from leakage resistance of the dielectric separating the two conductors, and the metallic resistance of the wires themselves. Characteristic impedance is purely a function of the capacitance and inductance distributed along the line’s length, and would exist even if the dielectric were perfect (infinite parallel resistance) and the wires superconducting (zero series resistance).
- Velocity factor is a fractional value relating a transmission line’s propagation speed to the speed of light in a vacuum. Values range between 0.66 and 0.80 for typical two-wire lines and coaxial cables. For any cable type, it is equal to the reciprocal (1/x) of the square root of the relative permittivity of the cable’s insulation.