[Circuit] 平均功率

電路知識：平均功率 (實功率)
簡介：推導平均功率
  • 實功功率（real power，也稱為有功功率，active power）
    • 以 P 來表示，其單位是瓦特（W）。
  • 視在功率（apparent power）
    • 以 S 來表示，其單位是伏安（VA），是電壓和電流有效值（rms） 的乘積。
    • 有效值（rms） 定義
      • 當一個交流電和另一個直流電分別在相同的條件下作等量的電功
        這個直流電的數值就稱為該交流電的有效值。
        像是通過同樣的電阻，所產生的能量
  • 虛功功率（reactive power）
    • 以 Q 來表示，其單位是無功伏安/瓦爾/乏（var）。
    • 另外三個不正確的寫法也被廣泛使用 VAr, VAR, Var

AC 需轉換為 Fourier Series $$ \begin{align*} \varphi _n &= \theta_{Vn}-\theta_{In}\\ \overline{P} &= \overline{P_{dc}}+\overline{P_{ac}}\\ \overline{P_{ac}} &= \sum_{n=1}^{N}V_{n,rms}I_{n,rms}cos\varphi _n \\ &=\frac{1}{2}\sum_{n=1}^{N}V_{n}I_{n}cos\varphi _n\\ &=\frac{1}{2}\sum_{n=1}^{N}\mathrm{RE}[\mathbf{V_{n}I_{n}^*}]\\ \end{align*} $$
$$ \begin{align*} \overline{P} &= \frac{1}{T}\int_{0}^{T}VIdt \\ &= \frac{1}{T}\int_{0}^{T}(V_{dc}+v_{ac})(I_{dc}+i_{ac})dt \\ &= \frac{1}{T}\int_{0}^{T}(V_{dc}I_{dc}+v_{ac}I_{dc}+V_{dc}i_{ac}+v_{ac}i_{ac})dt \\ &= \frac{1}{T}\left (\int_{0}^{T}V_{dc}I_{dc}dt+\underset{0}{\underbrace{\int_{0}^{T}v_{ac}I_{dc}dt}}+\underset{0}{\underbrace{\int_{0}^{T}V_{dc}i_{ac}dt}}+\int_{0}^{T}v_{ac}i_{ac}dt \right ) \\ &= \frac{1}{T}\left (\int_{0}^{T}V_{dc}I_{dc}dt+\int_{0}^{T}v_{ac}i_{ac}dt \right ) \\ &= \frac{1}{T}\int_{0}^{T}V_{dc}I_{dc}dt+\frac{1}{T}\int_{0}^{T}v_{ac}i_{ac}dt \\ &= V_{dc}I_{dc}+\frac{1}{T}\int_{0}^{T}v_{ac}i_{ac}dt\\ &=\overline{P_{dc}}+\overline{P_{ac}}\\ &=\overline{P_{dc}}+S\cdot PF\\ \end{align*} $$ $$ THD = \sqrt{\frac{\sum_{n=2}^{N}I_{n,rms}}{I_{1,rms}}} \\ PF = \frac{1}{\sqrt{1+THD^2}}cos\varphi =\frac{I_{1,rms}}{I_{rms}}DPF $$ 利用 Fourier series (傅立葉級數) 展開
$$ \begin{align*} \overline{P_{ac}} &= \frac{1}{T}\int_{0}^{T}v_{ac}(t)i_{ac}(t)dt\\ &= \frac{1}{T}\int_{0}^{T}\sum_{n=1}^{N}V_nsin(nw_nt+\alpha _n)\sum_{n=1}^{N}I_nsin(nw_nt+\beta _n)dt\\ &= \frac{1}{T}\left (\int_{0}^{T}\sum_{n=1}^{N}\sum_{n\neq p}^{N}V_psin(pw_pt+\alpha _p)I_nsin(nw_nt+\beta_n)dt +\int_{0}^{T}\sum_{n=1}^{N}V_nsin(nw_nt+\alpha _n)I_nsin(nw_nt+\beta _n)dt \right )\\ &= \frac{1}{T}\left (\sum_{n=1}^{N}\sum_{p\neq n}^{N}\underset{0}{\underbrace{\int_{0}^{T}V_psin(pw_pt+\alpha _p)I_nsin(nw_nt+\beta _n)dt}} +\sum_{n=1}^{N}\int_{0}^{T}V_nI_nsin(nw_nt+\alpha _n)sin(nw_nt+\beta _n)dt \right )\\ &= \frac{1}{T}\sum_{n=1}^{N}V_nI_n\int_{0}^{T}sin(nw_nt+\alpha _n)sin(nw_nt+\beta _n)dt \\ &= \frac{1}{T}\sum_{n=1}^{N}V_nI_n\int_{0}^{T}\frac{-1}{2}[cos(nw_nt+\alpha _n+nw_nt+\beta _n)-cos(nw_nt+\alpha _n-nw_nt-\beta _n)]dt \\ &= \frac{1}{T}\sum_{n=1}^{N}V_nI_n\frac{-1}{2}\int_{0}^{T}cos(2nw_nt+\alpha _n+\beta _n)-cos(\alpha _n-\beta _n)dt \\ &= \frac{1}{T}\sum_{n=1}^{N}V_nI_n\frac{-1}{2}\left (\underset{0}{\underbrace{\int_{0}^{T}cos(2nw_nt+\alpha _n+\beta _n)}}-\underset{Tcos(\alpha _n-\beta _n)}{\underbrace{\int_{0}^{T}cos(\alpha _n-\beta _n)dt}} \right ) \\ &= \frac{1}{2}\sum_{n=1}^{N}V_nI_ncos(\alpha _n-\beta _n) \\ &= \sum_{n=1}^{N}V_{n,rms}I_{n,rms}cos\varphi _n \\ &=\frac{1}{2}\sum_{n=1}^{N}V_{n}I_{n}cos\varphi _n\\ &=\frac{1}{2}\sum_{n=1}^{N}\mathrm{RE}[\mathbf{V_{n}I_{n}^*}]\\ \end{align*} $$

若負載為 R ，那麼可直接求出 \(V_{rms}\) 或 \(I_{rms}\) 再代入功率公式，或是 \(v_{ac_rms}\) 或 \(i_{ac,rms}\) 求解即可
$$ \begin{align*} \overline{P} &= \overline{P_{dc}}+\overline{P_{ac}}= \frac{V_{rms}^2}{R} = I_{rms}^2R \\ \overline{P_{ac}} &= \frac{v_{ac,rms}^2}{R} = i_{ac,rms}^2R \\ \end{align*} $$
$$ \begin{align*} \overline{P} &= \frac{1}{T}\int_{0}^{T}\frac{V^2}{R}dt\\ &= \frac{1}{T}\int_{0}^{T}\frac{(V_{dc}+v_{ac})^2}{R}dt \\ &= \frac{1}{T}\int_{0}^{T}\frac{(V_{dc}^2+2V_{dc}v_{ac}+v_{ac}^2)}{R}dt \\ &= \frac{1}{T}\int_{0}^{T}\frac{V_{dc}^2}{R}dt+\underset{0}{\underbrace{\frac{1}{T}\int_{0}^{T}\frac{2V_{dc}v_{ac}}{R}dt}}+\frac{1}{T}\int_{0}^{T}\frac{v_{ac}^2}{R}dt \\ &= \frac{1}{T}\int_{0}^{T}\frac{V_{dc}^2}{R}dt+\frac{1}{T}\int_{0}^{T}\frac{v_{ac}^2}{R}dt \\ &=\overline{P_{dc}}+\overline{P_{ac}}\\ &=\overline{P_{dc}}+\frac{v_{ac,rms}^2}{R}\\ \end{align*} $$ $$ \begin{align*} \overline{P} &= \frac{1}{T}\int_{0}^{T}I^2Rdt\\ &= \frac{1}{T}\int_{0}^{T}(I_{dc}+i_{ac})^2Rdt \\ &= \frac{1}{T}\int_{0}^{T}(I_{dc}^2+2I_{dc}i_{ac}+i_{ac}^2)Rdt \\ &= \frac{1}{T}\int_{0}^{T}I_{dc}^2Rdt+\underset{0}{\underbrace{\frac{1}{T}\int_{0}^{T}2I_{dc}i_{ac}Rdt}}+\frac{1}{T}\int_{0}^{T}i_{ac}^2Rdt \\ &= \frac{1}{T}\int_{0}^{T}I_{dc}^2Rdt+\frac{1}{T}\int_{0}^{T}i_{ac}^2Rdt \\ &=\overline{P_{dc}}+\overline{P_{ac}}\\ &=\overline{P_{dc}}+i_{ac,rms}^2R\\ \end{align*} $$ 利用 Fourier series (傅立葉級數) 展開
$$ \begin{align*} v_{ac,rms}^2 &= \frac{1}{T}\int_{0}^{T}v_{ac}^2(t)dt\\ &= \frac{1}{T}\int_{0}^{T}\sum_{n=1}^{N}V_nsin(nw_nt+\alpha _n)\sum_{n=1}^{N}V_nsin(nw_nt+\alpha _n)dt\\ &= \frac{1}{T}\left (\int_{0}^{T}\sum_{n=1}^{N}\sum_{n\neq p}^{N}V_nsin(nw_nt+\alpha _n)V_psin(pw_pt+\alpha _p)dt +\int_{0}^{T}\sum_{n=1}^{N}V_n^2sin^2(nw_nt+\alpha _n)dt \right )\\ &= \frac{1}{T}\left (\sum_{n=1}^{N}\sum_{p\neq n}^{N}\underset{0}{\underbrace{\int_{0}^{T}V_nsin(nw_nt+\alpha _n)V_psin(pw_pt+\alpha _p)dt}} +\sum_{n=1}^{N}\int_{0}^{T}V_n^2sin^2(nw_nt+\alpha _n)dt \right )\\ &= \frac{1}{T}\sum_{n=1}^{N}V_n^2\int_{0}^{T}sin^2(nw_nt+\alpha _n)dt \\ &= \frac{1}{T}\sum_{n=1}^{N}V_n^2\int_{0}^{T}\frac{1-cos(2nw_nt+2\alpha _n)}{2}dt \\ &= \sum_{n=1}^{N}\frac{V_n^2 }{2}\\ &= \sum_{n=1}^{N}V_{n,rms}^2 \end{align*} $$ 電流也可以依此推導之

R 程式碼模擬

rms <- function(data) {
  square = data*data
  mean = sum(square)/length(data)
  return (mean^(1/2))
}

n = 1000
theta = seq(0,2*pi,length=n)

#可依需求更改 I V R 的關係
t = 1:n
#v = array(10,n)
v = 7*sin(10*theta+pi/3)+30
i_1 = 10*sin(10*theta)
i_2 = 10*sin(5*theta)
i_3 = 10*sin(15*theta)
i = i_1 + i_2 + i_3 +15
r = v/i

par(mfrow=c(3,1))
plot(t, v, 'l')
plot(t, i, 'l')
plot(t, r, 'l')

v_rms = rms(v)
i_rms = rms(i)
r_rms = rms(r)

v_m = mean(v)
i_m = mean(i)
r_m = mean(r)

p_dc = i_m*v_m
cat(paste("p_dc = i_dc * v_dc = ", p_dc), "\n")

p_ac_avg = mean((i-i_m) * (v-v_m))
cat(paste("p_ac_avg = i_ac * v_ac = ", p_ac_avg), "\n")

p_ac_rms = rms(i-i_m) * rms(v-v_m)
cat(paste("p_ac_rms = i_ac_rms * v_ac_rms = ", p_ac_rms), "\n")

cat('\n')

cat(paste("p_ac_avg+p_dc = ", p_ac_avg+p_dc), "\n")
cat(paste("p_ac_rms+p_dc = ", p_ac_rms+p_dc), "\n")

p_rms = i_rms*v_rms
cat(paste("p_rms = i_rms * v_rms = ", p_rms), "\n")

cat('\n')

p_a = mean(i*v)
cat(paste("p_avg = ", p_a), "\n")

cat('\n')

phase = acos(p_ac_avg/p_ac_rms)*180/pi
cat(paste("phase = ", phase), "\n")

p_ac_r = p_ac_rms * cos(phase/180*pi)
cat(paste("p_ac Real = ", p_ac_r), "\n")

p_ac_q = p_ac_rms * sin(phase/180*pi)
cat(paste("p_ac Reactive = ", p_ac_q), "\n")

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