text:
South African National Lottery
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South African National Lottery
Logo used 2015 - present
Region
South Africa
First draw
11 March 2000 (2000-03-11)
Operator
Ithuba
Regulated By
National Lotteries Commission
Highest Jackpot
R102,016,595
Odds of winning jackpot
20,358,520 to 1 (Lotto)
20,358,520 to 1 (Lotto Plus)
24,435,180 to 1 (PowerBall)
24,435,180 to 1 (PowerBall Plus)
531,441 to 1 (SportStake)
4 to 1 (EAZiWIN)
Number of Games
6
Shown on
E.tv, SABC 2
Website
www.nationallottery.co.za
The National Lottery is operated by Camelot Group, to whom the licence was granted in 2015. The lottery is regulated by the National Lottery Commission, and was established in 2000.
Lottery tickets may be bought only by people of at least 18 years of age.
In the 2007 fiscal year transaction values totalled R3.972 billion, with an average of five million transactions per week. In the 2012 National lottery generated R4.7 billion in sales of Lotto and Powerball tickets.[1]
Lotto is the most popular type of gambling in South Africa[2] but Powerball has been the faster-growing for last years due to its high payouts.[1]
Contents
[hide]
1 History
2 Eligibility
3 Games
3.1 Current games
3.2 LOTTO
3.2.1 LOTTO PLUS 1
3.2.2 LOTTO PLUS 2
3.2.3 PowerBall
3.2.4 PowerBall PLUS
3.2.5 SPORTSTAKE 13
3.2.6 EAZiWIN
3.3 PICK 3
3.4 49s
3.5 Discontinued games
3.5.1 Wina Manje
3.5.2 Raffle
4 Other ways to play
4.1 Online
5 Record Jackpots
6 Operators
7 Socio-economic impact
8 Revenue distribution
9 Previous lotteries in South Africa
10 See also
11 External links
12 References
History[edit]
The National lottery was introduced to South Africa on 11 March 2000. At the time it was run by Uthingo.
After a marketing effort that aimed to reach 80 percent of South African homes directly[3] more than 800,000 tickets were sold in the first day of availability[4] Nearly R70 million worth of tickets were sold in the first three weeks of operation.[5]
In October 2002 operator Uthingo suggested a daily lottery to supplement the weekly draw. The concept, called Keno, was rejected by the trade and industry ministry in March 2003.[6] In November 2003 the Lotto Plus game was launched, acting as a supplementary weekly lottery available on the purchase of a primary lottery ticket, with an entry fee of R1.[7]
In July 2006 the Gidani consortium, featuring Greek company Intralot as a technical partner, was judged the preferred bidder to operate the lottery for seven years starting April 2007. The operating licence was awarded in October 2006.[8] In March 2007 the Pretoria High Court set aside that award on application by incumbent Uthingo, finding that the failure to adequately investigate the shareholders in some bidding consortia left room for conflicts of interest.[9]
Following the final draw by incumbent Uthingo, the lottery was indefinitely suspended in April 2007.[10]
In September the operating licence was awarded to Gidani again.[11] When ticket sales re-opened in October more than 200,000 tickets were sold within the first three hours.[12] Gidani introduced scratch cards, but they were discontinued for several months when they lost their licence to Ithuba.
In 2015, Lotto licences were awarded to Ithuba. In 2015, 2 new games were introduced by Ithuba: EAZiWIN, an instant win game consisting of four types of indigenous inspired games; Morabaraba, Fafi Fortune, 4 Siya Wina and Popa Feela and PowerBall Plus.
Eligibility[edit]
Players must be 18 years or older
Tickets may be bought in person at approved retailers & banks in South Africa.
Online purchase of tickets are only available to people who have an FNB or Nedbank bank account have registered on the South African National Lottery website and are residents of South Africa with a South African ID.
Games[edit]
LOTTO and LOTTO PLUS odds
Match
Odds of winning
% of total winning pool
2 and bonus 1 : 96 R20.00 FIXED
3 1 : 72 R50.00 FIXED
3 and bonus 1 : 1028 7.3%
4 1 : 1,371 8.4%
4 and bonus 1 : 30,161 5.0%
5 1 : 75,402 4.0%
5 and bonus 1 : 3,393,087 2.3%
All 6 1 : 20,358,520 73.0%
9 games operate under the South African National Lottery brand:
Current games[edit]
LOTTO[edit]
LOTTO and LOTTO PLUS odds
Match
Odds of winning
% of total winning pool
2 and bonus 1 : 96 R20.00 FIXED
3 1 : 72 R50.00 FIXED
3 and bonus 1 : 1028 7.3%
4 1 : 1,371 8.4%
4 and bonus 1 : 30,161 5.0%
5 1 : 75,402 4.0%
5 and bonus 1 : 3,393,087 2.3%
All 6 1 : 20,358,520 73.0%
Lotto and Lotto Plus coupon 2015 - present
Players buy tickets with their choice of six different numbers between 1 and 52; there is provision for random numbers to be generated automatically for those who do not wish to choose, known as 'Quick Pick'.
In the draw, six numbered balls are drawn without replacement from a set of 52 balls numbered from 1 to 52. A further Bonus Ball is also drawn, which affects only players who match five numbers.
Prizes are awarded to players who match at least three of the six drawn numbers, with prizes increasing for matching more of the drawn numbers. All players who match all six drawn numbers win equal shares of the jackpot; the chance of doing so is 1 in 20,358,520. If four, five, or six balls are matched, the relevant prize is divided equally between all who match that many balls. If no player matches all six numbers, the jackpot is added to that of the next Lotto draw—a Rollover.
The entry fee to the LOTTO draw is set at R5.00 per board.
The draw is conducted on Wednesdays and Saturdays on SABC 2 at 20:56.
PowerBall and PowerBall PLUS odds
Match
Odds of winning
% of total winning pool
PowerBall 1 : 37 Fixed (R10 for PowerBall/ R5
for Power Ball Plus)
1 ball and PowerBall 1 : 53 Fixed (R15 for PowerBall/ R7.50
for Power Ball Plus)
2 balls and PowerBall 1 : 247 5.1%
3 balls 1 : 165 7.6%
3 balls and PowerBall 1 : 3,133 8.9%
4 balls 1 : 6,430 9.3%
4 balls and PowerBall 1 : 122,176 4.9%
5 balls 1 : 1,286,062 9.7%
All balls including PowerBall 1 : 24,435,180 54.4%
LOTTO PLUS 1[edit]
LOTTO PLUS 1 is exactly the same as LOTTO, but gives the player a second chance to win. When buying a LOTTO ticket, the player must pay an extra R2.50 per board to enter the LOTTO PLUS 1 draw. Odds are the same, while prizes are usually slightly lower.
The draw is conducted on Wednesdays and Saturdays on SABC 2 at 20:56.
LOTTO PLUS 2[edit]
LOTTO PLUS 2 is exactly the same as LOTTO, but gives the player a third chance to win. When buying a LOTTO ticket, the player must pay an extra R2.50 per board to enter the LOTTO PLUS 2 draw. Odds are the same, while prizes are usually slightly lower.
The draw is conducted on Wednesdays and Saturdays on SABC 2 at 20:56.
PowerBall[edit]
Powerball and Powerball Plus coupon 2015 - present
The PowerBall jackpot draw requires players to pick five main numbers from 1 to 45 and one 'PowerBall' number from 1 to 20 for an entry fee of R5 per board. Prizes may be won by matching the main numbers, with matches of the PowerBall number winning higher prizes. The top prize of the game is won by matching all five main numbers as well as the PowerBall. Draws take place on Tuesdays and Fridays on E.tv at 20:56.
After 28 November 2015, odds were as more possibilities to win were introduced. An example of the recent additions were, if the player matched only the Powerball, he or she would win money, while before, that ticket would not win anything.
The record prize for any Lottery game in South Africa was in PowerBall at R102,016,595. This prize was won in the Free State but never collected. The highest claimed prize was also from PowerBall at R91,068,427
PowerBall PLUS[edit]
PowerBall PLUS is exactly the same as PowerBall, but gives the player a second chance to win. When buying a PowerBall ticket, the player must pay an extra R2.50 per board to enter the PowerBall PLUS draw. Odds are the same, while prizes are usually slightly lower.
Powerball PLUS was a more recent game added by the National Lottery of South Africa and is currently the newest game. It was introduced on 28 November 2015.
Draws take place on Tuesdays and Fridays on E.tv at 20:56.
SPORTSTAKE 13[edit]
Sportstake coupon 2015 - present
A player can play SPORTSTAKE 13 by predicting outcomes of 13 predetermined match fixtures drawn from English and other identified professional soccer fixtures.
For each fixture the player chooses their prediction by marking on the bet slip;
[1] – for a home win
[x] – for a draw
[2] – for a home loss (Away win)
You can select a single or multiple outcomes result for each fixture.
A valid wager must consist of at least one selection per fixture.
A minimum price per wager per board is R2.00 vat incl, and a maximum wager per Betslip is R2000.00
Each wager will cost you R2.00 vat incl.
You can play PROPICK™ as a quick play option
There is no Multi Draw option.
SPORTSTAKE 13 fixture results are considered as the final score after 90 minutes of play or after extra time where applicable, however excludes penalty shoot-out.
EAZiWIN odds
Name of Game
Game based on
Price of Card
Highest Prize
Odds of winning
Popa Feela Unknown R3 R30,000 1 : 4
4 Siya Wina Unknown R5 R100,000
Fafi Fortune Fafi
Morabaraba Morabaraba
EAZiWIN[edit]
All 4 EAZiWIN play cards
As well as tickets for the Draw Games, the National Lottery also sells instant win digital scratch cards.
They are digital scratch cards, where a card is compared against a slip and not scratched like traditional scratchcards. These are called 'EAZiWIN and are based on traditional African games.
Cards range from R3 to R5.
PICK 3[edit]
Lotto's newest game was announced to launch on 3 December 2016. It will be a daily game with a top prize of R10,000.
49s[edit]
Lottery betting game operated by Coral & Ladbrokes, very popular game in south africa, 49s takes place twice a day at 12:49 and 17:49.
Discontinued games[edit]
Wina Manje[edit]
Wina Manje was a scratchcard game that consisted of many types of scratch cards. This was replaced by EAZiWIN in 2015.
Raffle[edit]
South African National Lottery announced that there would be a once-off raffle for Christmas 2016. The draw took place on 30 December 2016. It was officially named "Raffle". Prizes included R1,000, R10,000, R100,000, with the top prize being a Mercedes Benz C200 Cabriolet
Other ways to play[edit]
As well as by purchasing a ticket at a shop, tickets can be purchased many other ways.
Online[edit]
All National Lottery games can be played online after registering. There are two ways of playing the lotto online.
FNB & Nedbank: LOTTO, LOTTO PLUS 1, LOTTO PLUS 2, PowerBall and PowerBall PLUS are available for play through the bank's apps.
National Lottery website: LOTTO, LOTTO PLUS 1, LOTTO PLUS 2, PowerBall, PowerBall PLUS and SPORTSTAKE 13 are available for play through the South African National lottery website.
Record Jackpots[edit]
Below lists the highest 11 jackpots from the National Lottery of South Africa.
Rank
Jackpot
Game
Draw number
Date
Notes
1 R110,000,000.00[13] Lotto 1783 2018-01-27 4 Winners won R27.5 million each
2 R102,016,595.00 PowerBall 166 2011-06-03 Not collected
3 R91,068,427.00 PowerBall 32 2010-02-12 Highest collected jackpot
4 R70,031,306.00 PowerBall N/A 2018-01-15
5 R69,542,295.00 PowerBall 74 2010-07-13
6 R64,547,102.00 PowerBall 100 2010-10-12
7 R62,996,195 PowerBall N/A 2016-09-18
8 R61,261,924.29 PowerBall 632 2015-12-11
9 R60,432,313.00 PowerBall 429 2013-12-20
10 R51,844,047.00 PowerBall 564 2015-04-14
11 R44,941,661.00 PowerBall 483 2014-07-01
Operators[edit]
Operator
Year begun
Year stopped
Ithuba 2015 Present
Gidani 2007 2015
Utingo 2000 2007
Socio-economic impact[edit]
In June 2003 it was reported that 27 percent of lottery players were unemployed[14] and that 43 percent of players earned less than R2,000 a month.[15] It was also reported that legalised gambling had created 50 673 jobs in 2000, although it may have redirected spendings from other industries.[16]
2006 research found that 82 percent of South Africans played the lottery once a week and that 53 percent of the population did not engage in any other form of gambling. The average player spent R81 per month on the lottery.[17]
Revenue distribution[edit]
Under the current operator, Ithuba, 34 percent of revenue is paid to a central charitable distribution fund, up from an initial 28 percent.[18] Six percent of revenue is paid as retail commission, ten percent is retained as operational costs and 50 percent is paid in prizes.[19]
Previous lotteries in South Africa[edit]
The now defunct homeland of Ciskei established a lottery in 1984 and operated by Score-A-Lot. In 1991 Score-A-Lot was the first Lottery in Africa to operate Video Lottery Terminals (VLT) in Africa and the first totally cashless operation using smart card technology. After lengthy negotiations with South Africans "Department of Trade and Industry" (DTI) Score-A-Lot closed in Dec 2001[20] [21]
A lottery was established by decree in the former homeland of Transkei in 1989 and operated by Score-A-Lot. In 1991 Score-A-Lot was the first Lottery in Africa to operate Video Lottery Terminals (VLT) in Africa and the first totally cashless operation using smart card technology. After lengthy negotiations with South Africans "Department of Trade and Industry" (DTI) Score-A-Lot closed in Dec 2001[22]
The Natal Lotto (also referred to as the KZN Lotto) was launched in the KwaZulu-Natal province in 1992.[23] During eight years of operation it raised R869 million and paid R345 million to charities and R448 million in prize money.[24]
See also[edit]
Uthingo
External links[edit]
National Lotteries Board
National Lottery website
UK49
References[edit]
^ Jump up to: a b "Betting on Africa. 2nd annual edition, November 2013. Gambling Outlook: 2013-2017 (South Africa - Nigeria - Kenya)" (PDF). PwC South Africa. 2013. Retrieved 5 July 2016.
Jump up ^ "National Lotteries Board Annual Report, 2007" (PDF). National Lotteries Board. Retrieved 9 September 2008.
Jump up ^ "National Lottery geared up to start on Thursday". The Daily Dispatch. Retrieved 9 September 2008.
Jump up ^ "R2m lottery tickets sold". The Daily Dispatch. Retrieved 9 September 2008.
Jump up ^ "R32m won in lottery – so far". Independent Online. Retrieved 9 September 2008.
Jump up ^ "No tata ma chance everyday, rules Erwin". Independent Online. Retrieved 9 September 2008.
Jump up ^ "Big plus for Lotto punters – at just R1 a pop". Independent Online. Retrieved 9 September 2008.
Jump up ^ "Operator for national lottery awarded". Independent Online. Retrieved 9 September 2008.
Jump up ^ "Lottery shareholders under scrutiny". Independent Online. Retrieved 9 September 2008.
Jump up ^ "Lotto debacle leaves public high and dry". Independent Online. Retrieved 9 September 2008.
Jump up ^ "Gidani gets green light for lottery". Mail&Guardian. Retrieved 9 September 2008.
Jump up ^ "Lotta mania hits SA". Independent Online. Retrieved 9 September 2008.
Jump up ^ https://www.nationallottery.co.za/results/lotto/?game_number=1783-lotto
Jump up ^ "'The Gambling Board must come to its senses'". Independent Online. Retrieved 9 September 2008.
Jump up ^ "Lotto ticket sales coin R80m each week – research". Cape Times. Retrieved 9 September 2008.
Jump up ^ "Parliament worries about gambling's growth". Business Report. Retrieved 9 September 2008.
Jump up ^ "SA starting to gamble sensibly, says report". Independent Online. Retrieved 9 September 2008.
Jump up ^ "Gidani set for lotto high stakes". Independent Online. Retrieved 15 September 2008.
Jump up ^ "Gidani boss strikes family off Lotto roll". Independent Online. Retrieved 15 September 2008.
Jump up ^ "Annual Report of the National Lotteries Board, 2003" (PDF). National Lotteries Board. Retrieved 9 September 2008.
Jump up ^ "Annual Report of the National Lotteries Board, 2004" (PDF). National Lotteries Board. Retrieved 9 September 2008.
Jump up ^ "Judge calls for action on illegal gambling". The Daily Dispatch. Retrieved 9 September 2008.
Jump up ^ "KZN wishes it took a chance with old Lotto". Independent Online. Retrieved 9 September 2008.
Jump up ^ "Lotto KZN lives on in its charitable work". Independent Online. Retrieved 9 September 2008.
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Root mean square
From Wikipedia, the free encyclopedia
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In statistics and its applications, the root mean square (abbreviated RMS or rms) is defined as the square root of mean square (the arithmetic mean of the squares of a set of numbers).[1] The RMS is also known as the quadratic mean and is a particular case of the generalized mean with exponent 2. RMS can also be defined for a continuously varying function in terms of an integral of the squares of the instantaneous values during a cycle.
For a cyclically alternating electric current, RMS is equal to the value of the direct current that would produce the same average power dissipation in a resistive load.[1] In Estimation theory the root mean square error of an estimator is a measure of the imperfection of the fit of the estimator to the data.
Contents
[hide]
1 Definition
2 In common waveforms
2.1 In waveform combinations
3 Uses
3.1 In electrical engineering
3.1.1 Voltage
3.1.2 Average electrical power
3.2 Speed
3.3 Error
4 In frequency domain
5 Relationship to other statistics
6 See also
7 References
8 External links
Definition[edit]
The RMS value of a set of values (or a continuous-time waveform) is the square root of the arithmetic mean of the squares of the values, or the square of the function that defines the continuous waveform. In Physics, the RMS current is the "value of the direct current that dissipates power in a resistor."
In the case of a set of n values { x 1 , x 2 , … , x n } {\displaystyle \{x_{1},x_{2},\dots ,x_{n}\}} , the RMS
x r m s = 1 n ( x 1 2 + x 2 2 + ⋯ + x n 2 ) . {\displaystyle x_{\mathrm {rms} }={\sqrt {{\frac {1}{n}}\left(x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}\right)}}.}
The corresponding formula for a continuous function (or waveform) f(t) defined over the interval T 1 ≤ t ≤ T 2 {\displaystyle T_{1}\leq t\leq T_{2}} is
f r m s = 1 T 2 − T 1 ∫ T 1 T 2 [ f ( t ) ] 2 d t , {\displaystyle f_{\mathrm {rms} }={\sqrt {{1 \over {T_{2}-T_{1}}}{\int _{T_{1}}^{T_{2}}{[f(t)]}^{2}\,dt}}},}
and the RMS for a function over all time is
f r m s = lim T → ∞ 1 T ∫ 0 T [ f ( t ) ] 2 d t . {\displaystyle f_{\mathrm {rms} }=\lim _{T\rightarrow \infty }{\sqrt {{1 \over {T}}{\int _{0}^{T}{[f(t)]}^{2}\,dt}}}.}
The RMS over all time of a periodic function is equal to the RMS of one period of the function. The RMS value of a continuous function or signal can be approximated by taking the RMS of a sequence of equally spaced samples. Additionally, the RMS value of various waveforms can also be determined without calculus, as shown by Cartwright.[2]
In the case of the RMS statistic of a random process, the expected value is used instead of the mean.
In common waveforms[edit]
Further information: RMS amplitude
Sine, square, triangle, and sawtooth waveforms.
A rectangular pulse wave of duty cycle D, the ratio between the pulse duration ( τ {\displaystyle \tau } ) and the period (T); illustrated here with a = 1.
Graph of a sine wave's voltage vs. time (in degrees), showing RMS, peak (PK), and peak-to-peak (PP) voltages.
If the waveform is a pure sine wave, the relationships between amplitudes (peak-to-peak, peak) and RMS are fixed and known, as they are for any continuous periodic wave. However, this is not true for an arbitrary waveform which may or may not be periodic or continuous. For a zero-mean sine wave, the relationship between RMS and peak-to-peak amplitude is:
Peak-to-peak = 2 2 × R M S ≈ 2.8 × R M S . {\displaystyle =2{\sqrt {2}}\times {RMS}\approx 2.8\times {RMS}.}
For other waveforms the relationships are not the same as they are for sine waves.
Waveform
Equation
RMS
DC, constant y = A 0 {\displaystyle y=A_{0}\,} A 0 {\displaystyle A_{0}\,}
Sine wave y = A 1 sin  ( 2 π f t ) {\displaystyle y=A_{1}\sin(2\pi ft)\,} A 1 2 {\displaystyle {\frac {A_{1}}{\sqrt {2}}}}
Square wave y = { A 1 frac  ( f t ) < 0.5 − A 1 frac  ( f t ) > 0.5 {\displaystyle y={\begin{cases}A_{1}&\operatorname {frac} (ft)<0.5\\-A_{1}&\operatorname {frac} (ft)>0.5\end{cases}}} A 1 {\displaystyle A_{1}\,}
DC-shifted square wave y = A 0 + { A 1 frac  ( f t ) < 0.5 − A 1 frac  ( f t ) > 0.5 {\displaystyle y=A_{0}+{\begin{cases}A_{1}&\operatorname {frac} (ft)<0.5\\-A_{1}&\operatorname {frac} (ft)>0.5\end{cases}}} A 0 2 + A 1 2 {\displaystyle {\sqrt {A_{0}^{2}+A_{1}^{2}}}\,}
Modified sine wave y = { 0 frac  ( f t ) < 0.25 A 1 0.25 < frac  ( f t ) < 0.5 0 0.5 < frac  ( f t ) < 0.75 − A 1 frac  ( f t ) > 0.75 {\displaystyle y={\begin{cases}0&\operatorname {frac} (ft)<0.25\\A_{1}&0.25<\operatorname {frac} (ft)<0.5\\0&0.5<\operatorname {frac} (ft)<0.75\\-A_{1}&\operatorname {frac} (ft)>0.75\end{cases}}} A 1 2 {\displaystyle {\frac {A_{1}}{\sqrt {2}}}}
Triangle wave y = | 2 A 1 frac  ( f t ) − A 1 | {\displaystyle y=\left|2A_{1}\operatorname {frac} (ft)-A_{1}\right|} A 1 3 {\displaystyle A_{1} \over {\sqrt {3}}}
Sawtooth wave y = 2 A 1 frac  ( f t ) − A 1 {\displaystyle y=2A_{1}\operatorname {frac} (ft)-A_{1}\,} A 1 3 {\displaystyle A_{1} \over {\sqrt {3}}}
Pulse train y = { A 1 frac  ( f t ) < D 0 frac  ( f t ) > D {\displaystyle y={\begin{cases}A_{1}&\operatorname {frac} (ft)<D\\0&\operatorname {frac} (ft)>D\end{cases}}} A 1 D {\displaystyle A_{1}{\sqrt {D}}}
Phase-to-phase voltage y = A 1 sin  ( t ) − A 1 sin  ( t − 2 π 3 ) {\displaystyle y=A_{1}\sin(t)-A_{1}\sin \left(t-{\frac {2\pi }{3}}\right)\,} A 1 3 2 {\displaystyle A_{1}{\sqrt {\frac {3}{2}}}}
Notes:
t is time
f is frequency
Ai is amplitude (peak value)
D is the duty cycle or the proportion of the time period (1/f) spent high.
frac(r) is the fractional part of r
In waveform combinations[edit]
Waveforms made by summing known simple waveforms have an RMS that is the root of the sum of squares of the component RMS values, if the component waveforms are orthogonal (that is, if the average of the product of one simple waveform with another is zero for all pairs other than a waveform times itself).[3]
R M S Total = R M S 1 2 + R M S 2 2 + ⋯ + R M S n 2 {\displaystyle RMS_{\text{Total}}={\sqrt {{RMS_{1}}^{2}+{RMS_{2}}^{2}+\cdots +{RMS_{n}}^{2}}}}
A special case of this, particularly helpful in electrical engineering, is given #In electrical engineering below. Another special case, useful in statistics, is given in #Relationship to other statistics.
(If the waveforms are in phase, then their RMS amplitudes sum directly.)
Uses[edit]
In electrical engineering[edit]
Voltage[edit]
Further information: Root mean square AC voltage
In electrical engineering, a special case of #RMS of waveform combinations (see also #Relationship to other statistics) is:
R M S Total = R M S DC 2 + R M S AC 2 {\displaystyle RMS_{\text{Total}}={\sqrt {{RMS_{\text{DC}}}^{2}+{RMS_{\text{AC}}}^{2}}}}
where R M S DC {\displaystyle RMS_{\text{DC}}} refers to the direct current component of the signal and R M S AC {\displaystyle RMS_{\text{AC}}} is the alternating current component of the signal.
Average electrical power[edit]
Further information: AC power
Electrical engineers often need to know the power, P, dissipated by an electrical resistance, R. It is easy to do the calculation when there is a constant current, I, through the resistance. For a load of R ohms, power is defined simply as:
P = I 2 R . {\displaystyle P=I^{2}R.}
However, if the current is a time-varying function, I(t), this formula must be extended to reflect the fact that the current (and thus the instantaneous power) is varying over time. If the function is periodic (such as household AC power), it is still meaningful to discuss the average power dissipated over time, which is calculated by taking the average power dissipation:
P avg {\displaystyle P_{\text{avg}}\,\!} = ⟨ I ( t ) 2 R ⟩ {\displaystyle {}=\left\langle I(t)^{2}R\right\rangle \,\!} (where ⟨ … ⟩ {\displaystyle \langle \ldots \rangle } denotes the mean of a function)
= R ⟨ I ( t ) 2 ⟩ {\displaystyle {}=R\left\langle I(t)^{2}\right\rangle \,\!} (as R does not vary over time, it can be factored out)
= I RMS 2 R {\displaystyle {}=I_{\text{RMS}}^{2}R\,\!} (by definition of RMS)
So, the RMS value, IRMS, of the function I(t) is the constant current that yields the same power dissipation as the time-averaged power dissipation of the current I(t).
Average power can also be found using the same method that in the case of a time-varying voltage, V(t), with RMS value VRMS,
P Avg = V RMS 2 R . {\displaystyle P_{\text{Avg}}={V_{\text{RMS}}^{2} \over R}.}
This equation can be used for any periodic waveform, such as a sinusoidal or sawtooth waveform, allowing us to calculate the mean power delivered into a specified load.
By taking the square root of both these equations and multiplying them together, the power is found to be:
P Avg = V RMS I RMS . {\displaystyle P_{\text{Avg}}=V_{\text{RMS}}I_{\text{RMS}}.}
Both derivations depend on voltage and current being proportional (i.e., the load, R, is purely resistive). Reactive loads (i.e., loads capable of not just dissipating energy but also storing it) are discussed under the topic of AC power.
In the common case of alternating current when I(t) is a sinusoidal current, as is approximately true for mains power, the RMS value is easy to calculate from the continuous case equation above. If Ip is defined to be the peak current, then:
I RMS = 1 T 2 − T 1 ∫ T 1 T 2 [ I p sin  ( ω t ) ] 2 d t . {\displaystyle I_{\text{RMS}}={\sqrt {{1 \over {T_{2}-T_{1}}}\int _{T_{1}}^{T_{2}}\left[I_{\text{p}}\sin(\omega t)\right]^{2}dt}}.\,\!}
where t is time and ω is the angular frequency (ω = 2π/T, where T is the period of the wave).
Since Ip is a positive constant:
I RMS = I p 1 T 2 − T 1 ∫ T 1 T 2 sin 2  ( ω t ) d t . {\displaystyle I_{\text{RMS}}=I_{\text{p}}{\sqrt {{1 \over {T_{2}-T_{1}}}{\int _{T_{1}}^{T_{2}}{\sin ^{2}(\omega t)}\,dt}}}.}
Using a trigonometric identity to eliminate squaring of trig function:
I RMS = I p 1 T 2 − T 1 ∫ T 1 T 2 1 − cos  ( 2 ω t ) 2 d t I RMS = I p 1 T 2 − T 1 [ t 2 − sin  ( 2 ω t ) 4 ω ] T 1 T 2 {\displaystyle {\begin{aligned}I_{\text{RMS}}&=I_{\text{p}}{\sqrt {{1 \over {T_{2}-T_{1}}}{\int _{T_{1}}^{T_{2}}{1-\cos(2\omega t) \over 2}\,dt}}}\\I_{\text{RMS}}&=I_{\text{p}}{\sqrt {{1 \over {T_{2}-T_{1}}}\left[{t \over 2}-{\sin(2\omega t) \over 4\omega }\right]_{T_{1}}^{T_{2}}}}\end{aligned}}}
but since the interval is a whole number of complete cycles (per definition of RMS), the sin terms will cancel out, leaving:
I RMS = I p 1 T 2 − T 1 [ t 2 ] T 1 T 2 = I p 1 T 2 − T 1 T 2 − T 1 2 = I p 2 . {\displaystyle I_{\text{RMS}}=I_{\text{p}}{\sqrt {{1 \over {T_{2}-T_{1}}}\left[{t \over 2}\right]_{T_{1}}^{T_{2}}}}=I_{\text{p}}{\sqrt {{1 \over {T_{2}-T_{1}}}{{T_{2}-T_{1}} \over 2}}}={I_{\text{p}} \over {\sqrt {2}}}.}
A similar analysis leads to the analogous equation for sinusoidal voltage:
V RMS = V p 2 . {\displaystyle V_{\text{RMS}}={V_{\text{p}} \over {\sqrt {2}}}.}
Where IP represents the peak current and VP represents the peak voltage.
Because of their usefulness in carrying out power calculations, listed voltages for power outlets (e.g., 120 V in the USA, or 230 V in Europe) are almost always quoted in RMS values, and not peak values. Peak values can be calculated from RMS values from the above formula, which implies VP = VRMS × √2, assuming the source is a pure sine wave. Thus the peak value of the mains voltage in the USA is about 120 × √2, or about 170 volts. The peak-to-peak voltage, being double this, is about 340 volts. A similar calculation indicates that the peak mains voltage in Europe is about 325 volts, and the peak-to-peak mains voltage, about 650 volts.
RMS quantities such as electric current are usually calculated over one cycle. However, for some purposes the RMS current over a longer period is required when calculating transmission power losses. The same principle applies, and (for example) a current of 10 amps used for 12 hours each day represents an RMS current of 5 amps in the long term.
The term "RMS power" is sometimes erroneously used in the audio industry as a synonym for "mean power" or "average power" (it is proportional to the square of the RMS voltage or RMS current in a resistive load). For a discussion of audio power measurements and their shortcomings, see Audio power.
Speed[edit]
Main article: Root-mean-square speed
In the physics of gas molecules, the root-mean-square speed is defined as the square root of the average squared-speed. The RMS speed of an ideal gas is calculated using the following equation:
v RMS = 3 R T M {\displaystyle {v_{\text{RMS}}}={\sqrt {3RT \over {M}}}}
where R represents the ideal gas constant, 8.314 J/(mol·K), T is the temperature of the gas in kelvins, and M is the molar mass of the gas in kilograms. The generally accepted terminology for speed as compared to velocity is that the former is the scalar magnitude of the latter. Therefore, although the average speed is between zero and the RMS speed, the average velocity for a stationary gas is zero.
Error[edit]
Main article: Root-mean-square error
When two data sets—one set from theoretical prediction and the other from actual measurement of some physical variable, for instance—are compared, the RMS of the pairwise differences of the two data sets can serve as a measure how far on average the error is from 0. The mean of the pairwise differences does not measure the variability of the difference, and the variability as indicated by the standard deviation is around the mean instead of 0. Therefore, the RMS of the differences is a meaningful measure of the error.
In frequency domain[edit]
The RMS can be computed in the frequency domain, using Parseval's theorem. For a sampled signal x [ n ] = x ( t = n T ) {\displaystyle x[n]=x(t=nT)} , where T {\displaystyle T} is the sampling period,
∑ n = 1 N x 2 [ n ] = 1 N ∑ m = 1 N | X [ m ] | 2 {\displaystyle \sum _{n=1}^{N}{x^{2}[n]}={\frac {1}{N}}\sum _{m=1}^{N}{\bigl |}X[m]{\bigr |}^{2}} ,
where X [ m ] = FFT  { x [ n ] } {\displaystyle X[m]=\operatorname {FFT} \{x[n]\}} and N is number of samples and FFT coefficients.
In this case, the RMS computed in the time domain is the same as in the frequency domain:
R M S { x [ n ] } = 1 N ∑ n x 2 [ n ] = 1 N 2 ∑ m | X [ m ] | 2 = ∑ m | X [ m ] N | 2 . {\displaystyle \mathrm {RMS} \{x[n]\}={\sqrt {{\frac {1}{N}}\sum _{n}{x^{2}[n]}}}={\sqrt {{\frac {1}{N^{2}}}\sum _{m}{{\bigl |}X[m]{\bigr |}}^{2}}}={\sqrt {\sum _{m}{\left|{\frac {X[m]}{N}}\right|^{2}}}}.}
Relationship to other statistics[edit]
If x ¯ {\displaystyle {\bar {x}}} is the arithmetic mean and σ x {\displaystyle \sigma _{x}} is the standard deviation of a population or a waveform then:[4]
x r m s 2 = x ¯ 2 + σ x 2 = x 2 ¯ . {\displaystyle x_{\mathrm {rms} }^{2}={\overline {x}}^{2}+\sigma _{x}^{2}={\overline {x^{2}}}.}
From this it is clear that the RMS value is always greater than or equal to the average, in that the RMS includes the "error" / square deviation as well.
Physical scientists often use the term "root mean square" as a synonym for standard deviation when it can be assumed the input signal has zero mean, i.e., referring to the square root of the mean squared deviation of a signal from a given baseline or fit.[5][6] This is useful for electrical engineers in calculating the "AC only" RMS of a signal. Standard deviation being the root mean square of a signal's variation about the mean, rather than about 0, the DC component is removed (i.e. RMS(signal) = Stdev(signal) if the mean signal is 0). A special case of this, particularly helpful in electrical engineering, is given above.
See also[edit]
Central moment
Geometric mean
L2 norm
Least squares
Mean squared displacement
Table of mathematical symbols
True RMS converter
Average rectified value (ARV)
References[edit]
^ Jump up to: a b A Dictionary of Physics (6 ed.). Oxford University Press. 2009. ISBN 9780199233991.
Jump up ^ Cartwright, Kenneth V (Fall 2007). "Determining the Effective or RMS Voltage of Various Waveforms without Calculus" (PDF). Technology Interface. 8 (1): 20 pages.
Jump up ^ Nastase, Adrian S. "How to Derive the RMS Value of Pulse and Square Waveforms". MasteringElectronicsDesign.com. Retrieved 21 January 2015.
Jump up ^ Chris C. Bissell and David A. Chapman (1992). Digital signal transmission (2nd ed.). Cambridge University Press. p. 64. ISBN 978-0-521-42557-5.
Jump up ^ "Root-Mean-Square".
Jump up ^ "ROOT, TH1:GetRMS".
External links[edit]
A case for why RMS is a misnomer when applied to audio power
A Java applet on learning RMS
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List of House of Cards episodes
From Wikipedia, the free encyclopedia
Jump to: navigation, search
House of Cards is an American political drama web television series created by Beau Willimon for Netflix. The first season, based on the BBC miniseries of the same name and the series of books by Michael Dobbs,[1] premiered exclusively via Netflix's web streaming service on February 1, 2013.[2]
Kevin Spacey stars as Representative Francis Underwood, the Majority Whip, who, after being passed up for the position of Secretary of State, initiates an elaborate plan to get himself into a position of greater power, aided by his wife, Claire Underwood (Robin Wright). In January 2016, Netflix renewed the series for a fifth season,[3] which premiered on May 30, 2017.[4] As of May 30, 2017,[update] 65 episodes of House of Cards have been released, concluding the fifth season.
Contents
[hide]
1 Series overview
2 Episodes
2.1 Season 1 (2013)
2.2 Season 2 (2014)
2.3 Season 3 (2015)
2.4 Season 4 (2016)
2.5 Season 5 (2017)
3 References
4 External links
Series overview[edit]
Season
Episodes
Originally released
1 13 February 1, 2013 (2013-02-01)
2 13 February 14, 2014 (2014-02-14)
3 13 February 27, 2015 (2015-02-27)
4 13 March 4, 2016 (2016-03-04)
5 13 May 30, 2017 (2017-05-30)
Episodes[edit]
Season 1 (2013)[edit]
Main article: House of Cards (season 1)
No.
overall
No. in
season
Title
Directed by
Written by
Original release date
Prod.
code
1
1 "Chapter 1" David Fincher Beau Willimon February 1, 2013 (2013-02-01) HOC-101
2
2 "Chapter 2" David Fincher Beau Willimon February 1, 2013 (2013-02-01) HOC-102
3
3 "Chapter 3" James Foley Keith Huff and Beau Willimon February 1, 2013 (2013-02-01) HOC-103
4
4 "Chapter 4" James Foley Rick Cleveland and Beau Willimon February 1, 2013 (2013-02-01) HOC-104
5
5 "Chapter 5" Joel Schumacher Sarah Treem February 1, 2013 (2013-02-01) HOC-105
6
6 "Chapter 6" Joel Schumacher Sam Forman February 1, 2013 (2013-02-01) HOC-106
7
7 "Chapter 7" Charles McDougall Kate Barnow and Beau Willimon February 1, 2013 (2013-02-01) HOC-107
8
8 "Chapter 8" Charles McDougall Beau Willimon February 1, 2013 (2013-02-01) HOC-108
9
9 "Chapter 9" James Foley Beau Willimon and Rick Cleveland February 1, 2013 (2013-02-01) HOC-109
10
10 "Chapter 10" Carl Franklin Sarah Treem February 1, 2013 (2013-02-01) HOC-110
11
11 "Chapter 11" Carl Franklin Keith Huff and Kate Barnow and Beau Willimon February 1, 2013 (2013-02-01) HOC-111
12
12 "Chapter 12" Allen Coulter Gina Gionfriddo and Beau Willimon February 1, 2013 (2013-02-01) HOC-112
13
13 "Chapter 13" Allen Coulter Beau Willimon February 1, 2013 (2013-02-01) HOC-113
Season 2 (2014)[edit]
Main article: House of Cards (season 2)
No.
overall
No. in
season
Title
Directed by
Written by
Original release date
Prod.
code
14
1 "Chapter 14" Carl Franklin Beau Willimon February 14, 2014 (2014-02-14) HOC-201
15
2 "Chapter 15" Carl Franklin Beau Willimon February 14, 2014 (2014-02-14) HOC-202
16
3 "Chapter 16" James Foley Bill Cain February 14, 2014 (2014-02-14) HOC-203
17
4 "Chapter 17" James Foley Laura Eason February 14, 2014 (2014-02-14) HOC-204
18
5 "Chapter 18" John Coles Kenneth Lin February 14, 2014 (2014-02-14) HOC-205
19
6 "Chapter 19" John Coles John Mankiewicz February 14, 2014 (2014-02-14) HOC-206
20
7 "Chapter 20" James Foley Bill Kennedy February 14, 2014 (2014-02-14) HOC-207
21
8 "Chapter 21" James Foley David Manson February 14, 2014 (2014-02-14) HOC-208
22
9 "Chapter 22" Jodie Foster Beau Willimon February 14, 2014 (2014-02-14) HOC-209
23
10 "Chapter 23" Robin Wright Laura Eason and Beau Willimon February 14, 2014 (2014-02-14) HOC-210
24
11 "Chapter 24" John Coles John Mankiewicz and Beau Willimon February 14, 2014 (2014-02-14) HOC-211
25
12 "Chapter 25" James Foley Beau Willimon February 14, 2014 (2014-02-14) HOC-212
26
13 "Chapter 26" James Foley Beau Willimon February 14, 2014 (2014-02-14) HOC-213
Season 3 (2015)[edit]
Main article: House of Cards (season 3)
No.
overall
No. in
season
Title
Directed by
Written by
Original release date
Prod.
code
27
1 "Chapter 27" John David Coles Beau Willimon February 27, 2015 (2015-02-27) HOC-301
28
2 "Chapter 28" John David Coles John Mankiewicz February 27, 2015 (2015-02-27) HOC-302
29
3 "Chapter 29" Tucker Gates Frank Pugliese February 27, 2015 (2015-02-27) HOC-303
30
4 "Chapter 30" Tucker Gates Laura Eason February 27, 2015 (2015-02-27) HOC-304
31
5 "Chapter 31" James Foley Kenneth Lin February 27, 2015 (2015-02-27) HOC-305
32
6 "Chapter 32" James Foley Melissa James Gibson February 27, 2015 (2015-02-27) HOC-306
33
7 "Chapter 33" John Dahl Beau Willimon February 27, 2015 (2015-02-27) HOC-307
34
8 "Chapter 34" John Dahl Bill Kennedy February 27, 2015 (2015-02-27) HOC-308
35
9 "Chapter 35" Robin Wright John Mankiewicz February 27, 2015 (2015-02-27) HOC-309
36
10 "Chapter 36" Agnieszka Holland Frank Pugliese February 27, 2015 (2015-02-27) HOC-310
37
11 "Chapter 37" Agnieszka Holland Melissa James Gibson February 27, 2015 (2015-02-27) HOC-311
38
12 "Chapter 38" Robin Wright Beau Willimon February 27, 2015 (2015-02-27) HOC-312
39
13 "Chapter 39" James Foley Beau Willimon February 27, 2015 (2015-02-27) HOC-313
Season 4 (2016)[edit]
Main article: House of Cards (season 4)
No.
overall
No. in
season
Title
Directed by
Written by
Original release date
Prod.
code
40
1 "Chapter 40" Tucker Gates Beau Willimon March 4, 2016 (2016-03-04) HOC-401
41
2 "Chapter 41" Tucker Gates Melissa James Gibson March 4, 2016 (2016-03-04) HOC-402
42
3 "Chapter 42" Robin Wright Frank Pugliese March 4, 2016 (2016-03-04) HOC-403
43
4 "Chapter 43" Robin Wright John Mankiewicz March 4, 2016 (2016-03-04) HOC-404
44
5 "Chapter 44" Tom Shankland Melissa James Gibson March 4, 2016 (2016-03-04) HOC-405
45
6 "Chapter 45" Tom Shankland Laura Eason March 4, 2016 (2016-03-04) HOC-406
46
7 "Chapter 46" Tom Shankland Bill Kennedy March 4, 2016 (2016-03-04) HOC-407
47
8 "Chapter 47" Alex Graves John Mankiewicz March 4, 2016 (2016-03-04) HOC-408
48
9 "Chapter 48" Robin Wright Frank Pugliese March 4, 2016 (2016-03-04) HOC-409
49
10 "Chapter 49" Robin Wright Melissa James Gibson and Kenneth Lin March 4, 2016 (2016-03-04) HOC-410
50
11 "Chapter 50" Kari Skogland Tian Jun Gu March 4, 2016 (2016-03-04) HOC-411
51
12 "Chapter 51" Jakob Verbruggen Laura Eason and Bill Kennedy March 4, 2016 (2016-03-04) HOC-412
52
13 "Chapter 52" Jakob Verbruggen Beau Willimon March 4, 2016 (2016-03-04) HOC-413
Season 5 (2017)[edit]
Main article: House of Cards (season 5)
No.
overall
No. in
season
Title
Directed by
Written by
Original release date
Prod.
code
53
1 "Chapter 53" Daniel Minahan Frank Pugliese May 30, 2017 (2017-05-30) HOC-501
54
2 "Chapter 54" Daniel Minahan Melissa James Gibson May 30, 2017 (2017-05-30) HOC-502
55
3 "Chapter 55" Alik Sakharov John Mankiewicz May 30, 2017 (2017-05-30) HOC-503
56
4 "Chapter 56" Alik Sakharov Kenneth Lin May 30, 2017 (2017-05-30) HOC-504
57
5 "Chapter 57" Michael Morris Laura Eason May 30, 2017 (2017-05-30) HOC-505
58
6 "Chapter 58" Michael Morris Bill Kennedy May 30, 2017 (2017-05-30) HOC-506
59
7 "Chapter 59" Alik Sakharov Tian Jun Gu May 30, 2017 (2017-05-30) HOC-507
60
8 "Chapter 60" Roxann Dawson John Mankiewicz May 30, 2017 (2017-05-30) HOC-508
61
9 "Chapter 61" Roxann Dawson Bill Kennedy May 30, 2017 (2017-05-30) HOC-509
62
10 "Chapter 62" Agnieszka Holland Kenneth Lin May 30, 2017 (2017-05-30) HOC-510
63
11 "Chapter 63" Agnieszka Holland Laura Eason May 30, 2017 (2017-05-30) HOC-511
64
12 "Chapter 64" Robin Wright Frank Pugliese & Melissa James Gibson May 30, 2017 (2017-05-30) HOC-512
65
13 "Chapter 65" Robin Wright Melissa James Gibson & Frank Pugliese May 30, 2017 (2017-05-30) HOC-513
References[edit]
Jump up ^ Itzkoff, Dave (October 4, 2012). "Netflix Sets February Premiere for ‘House of Cards’". The New York Times. Retrieved February 1, 2013.
Jump up ^ Ryan, Maureen (January 22, 2013). ""House of Cards" On Netflix: Inside Intel on Kevin Spacey's Dark Drama". The Huffington Post. Retrieved January 25, 2013.
Jump up ^ Steiner, Amanda Michelle (January 28, 2016). "House of Cards renewed for season 5, showrunner exits". Entertainment Weekly. Retrieved January 28, 2016.
Jump up ^ Birnbaum, Debra (January 20, 2017). "‘House of Cards’ Season 5 Gets Premiere Date, Teaser". Variety. Retrieved January 20, 2017.
External links[edit]
Official website
List of House of Cards episodes on IMDb
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House of Cards (U.S. TV series)
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List of politicians
See also
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House of Cards (UK TV series)
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