[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 $$\begin{array}{rl}{\phi }_n& ={\theta }_{Vn}-{\theta }_{In}\\ \bar{P}& =\overline{P_{dc}}+\overline{P_{ac}}\\ \overline{P_{ac}}& =\sum _{n=1}^N{V}_{n,rms}{I}_{n,rms}cos{\phi }_n\\ & =\frac{1}{2}\sum _{n=1}^N{V}_n{I}_{n}cos{\phi }_n\\ & =\frac{1}{2}\sum _{n=1}^N\mathrm{RE}[{\mathbf{V}}_{\mathbf{n}}{\mathbf{I}}_{\mathbf{n}}^{\mathbf{\ast }}]\end{array}$$

$$\begin{array}{rl}\bar{P}& =\frac{1}{T}{\int }_0^{T}VIdt\\ & =\frac{1}{T}{\int }_0^T(V_{dc}+v_{ac})(I_{dc}+i_{ac})dt\\ & =\frac{1}{T}{\int }_0^T(V_{dc}{I}_{dc}+v_{ac}{I}_{dc}+V_{dc}{i}_{ac}+v_{ac}{i}_{ac})dt\\ & =\frac{1}{T}({\int }_0^T{V}_{dc}{I}_{dc}dt+\underset{0}{\underbrace{{\int }_0^T{v}_{ac}{I}_{dc}dt}}+\underset{0}{\underbrace{{\int }_0^T{V}_{dc}{i}_{ac}dt}}+{\int }_0^T{v}_{ac}{i}_{ac}dt)\\ & =\frac{1}{T}({\int }_0^T{V}_{dc}{I}_{dc}dt+{\int }_0^T{v}_{ac}{i}_{ac}dt)\\ & =\frac{1}{T}{\int }_0^T{V}_{dc}{I}_{dc}dt+\frac{1}{T}{\int }_0^T{v}_{ac}{i}_{ac}dt\\ & =V_{dc}{I}_{dc}+\frac{1}{T}{\int }_0^T{v}_{ac}{i}_{ac}dt\\ & =\overline{P_{dc}}+\overline{P_{ac}}\\ & =\overline{P_{dc}}+S\cdot PF\end{array}$$ $$\begin{gathered} THD=\sqrt{\frac{\sum _{n=2}^N{I}_{n,rms}}{I_{1,rms}}} \\ PF=\frac{1}{\sqrt{1+TH{D}^2}}cos\phi =\frac{I_{1,rms}}{I_{rms}}DPF \end{gathered}$$ 利用 Fourier series (傅立葉級數) 展開
$$\begin{array}{rl}\overline{P_{ac}}& =\frac{1}{T}{\int }_0^T{v}_{ac}(t)i_{ac}(t)dt\\ & =\frac{1}{T}{\int }_0^T\sum _{n=1}^N{V}_{n}sin(n{w}_{n}t+{\alpha }_n)\sum _{n=1}^N{I}_{n}sin(n{w}_{n}t+{\beta }_n)dt\\ & =\frac{1}{T}({\int }_0^T\sum _{n=1}^N\sum _{n\ne p}^N{V}_{p}sin(p{w}_{p}t+{\alpha }_p)I_{n}sin(n{w}_{n}t+{\beta }_n)dt+{\int }_0^T\sum _{n=1}^N{V}_{n}sin(n{w}_{n}t+{\alpha }_n)I_{n}sin(n{w}_{n}t+{\beta }_n)dt)\\ & =\frac{1}{T}(\sum _{n=1}^N\sum _{p\ne n}^N\underset{0}{\underbrace{{\int }_0^T{V}_{p}sin(p{w}_{p}t+{\alpha }_p)I_{n}sin(n{w}_{n}t+{\beta }_n)dt}}+\sum _{n=1}^N{\int }_0^T{V}_n{I}_{n}sin(n{w}_{n}t+{\alpha }_n)sin(n{w}_{n}t+{\beta }_n)dt)\\ & =\frac{1}{T}\sum _{n=1}^N{V}_n{I}_n{\int }_0^{T}sin(n{w}_{n}t+{\alpha }_n)sin(n{w}_{n}t+{\beta }_n)dt\\ & =\frac{1}{T}\sum _{n=1}^N{V}_n{I}_n{\int }_0^T\frac{-1}{2}[cos(n{w}_{n}t+{\alpha }_n+n{w}_{n}t+{\beta }_n)-cos(n{w}_{n}t+{\alpha }_n-n{w}_{n}t-{\beta }_n)]dt\\ & =\frac{1}{T}\sum _{n=1}^N{V}_n{I}_n\frac{-1}{2}{\int }_0^{T}cos(2n{w}_{n}t+{\alpha }_n+{\beta }_n)-cos({\alpha }_n-{\beta }_n)dt\\ & =\frac{1}{T}\sum _{n=1}^N{V}_n{I}_n\frac{-1}{2}(\underset{0}{\underbrace{{\int }_0^{T}cos(2n{w}_{n}t+{\alpha }_n+{\beta }_n)}}-\underset{Tcos({\alpha }_n-{\beta }_n)}{\underbrace{{\int }_0^{T}cos({\alpha }_n-{\beta }_n)dt}})\\ & =\frac{1}{2}\sum _{n=1}^N{V}_n{I}_{n}cos({\alpha }_n-{\beta }_n)\\ & =\sum _{n=1}^N{V}_{n,rms}{I}_{n,rms}cos{\phi }_n\\ & =\frac{1}{2}\sum _{n=1}^N{V}_n{I}_{n}cos{\phi }_n\\ & =\frac{1}{2}\sum _{n=1}^N\mathrm{RE}[{\mathbf{V}}_{\mathbf{n}}{\mathbf{I}}_{\mathbf{n}}^{\mathbf{\ast }}]\end{array}$$

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 $v_{a{c}_{r}ms}$ 或 $i_{ac,rms}$ 求解即可

$$\begin{array}{rl}\bar{P}& =\overline{P_{dc}}+\overline{P_{ac}}=\frac{V_{rms}^2}{R}=I_{rms}^{2}R\\ \overline{P_{ac}}& =\frac{v_{ac,rms}^2}{R}=i_{ac,rms}^{2}R\end{array}$$

$$\begin{array}{rl}\bar{P}& =\frac{1}{T}{\int }_0^T\frac{V^2}{R}dt\\ & =\frac{1}{T}{\int }_0^T\frac{(V_{dc}+v_{ac}{)}^2}{R}dt\\ & =\frac{1}{T}{\int }_0^T\frac{(V_{dc}^2+2V_{dc}{v}_{ac}+v_{ac}^2)}{R}dt\\ & =\frac{1}{T}{\int }_0^T\frac{V_{dc}^2}{R}dt+\underset{0}{\underbrace{\frac{1}{T}{\int }_0^T\frac{2V_{dc}{v}_{ac}}{R}dt}}+\frac{1}{T}{\int }_0^T\frac{v_{ac}^2}{R}dt\\ & =\frac{1}{T}{\int }_0^T\frac{V_{dc}^2}{R}dt+\frac{1}{T}{\int }_0^T\frac{v_{ac}^2}{R}dt\\ & =\overline{P_{dc}}+\overline{P_{ac}}\\ & =\overline{P_{dc}}+\frac{v_{ac,rms}^2}{R}\end{array}$$ $$\begin{array}{rl}\bar{P}& =\frac{1}{T}{\int }_0^T{I}^{2}Rdt\\ & =\frac{1}{T}{\int }_0^T(I_{dc}+i_{ac}{)}^{2}Rdt\\ & =\frac{1}{T}{\int }_0^T(I_{dc}^2+2I_{dc}{i}_{ac}+i_{ac}^2)Rdt\\ & =\frac{1}{T}{\int }_0^T{I}_{dc}^{2}Rdt+\underset{0}{\underbrace{\frac{1}{T}{\int }_0^{T}2I_{dc}{i}_{ac}Rdt}}+\frac{1}{T}{\int }_0^T{i}_{ac}^{2}Rdt\\ & =\frac{1}{T}{\int }_0^T{I}_{dc}^{2}Rdt+\frac{1}{T}{\int }_0^T{i}_{ac}^{2}Rdt\\ & =\overline{P_{dc}}+\overline{P_{ac}}\\ & =\overline{P_{dc}}+i_{ac,rms}^{2}R\end{array}$$ 利用 Fourier series (傅立葉級數) 展開
$$\begin{array}{rl}{v}_{ac,rms}^2& =\frac{1}{T}{\int }_0^T{v}_{ac}^2(t)dt\\ & =\frac{1}{T}{\int }_0^T\sum _{n=1}^N{V}_{n}sin(n{w}_{n}t+{\alpha }_n)\sum _{n=1}^N{V}_{n}sin(n{w}_{n}t+{\alpha }_n)dt\\ & =\frac{1}{T}({\int }_0^T\sum _{n=1}^N\sum _{n\ne p}^N{V}_{n}sin(n{w}_{n}t+{\alpha }_n)V_{p}sin(p{w}_{p}t+{\alpha }_p)dt+{\int }_0^T\sum _{n=1}^N{V}_n^{2}si{n}^2(n{w}_{n}t+{\alpha }_n)dt)\\ & =\frac{1}{T}(\sum _{n=1}^N\sum _{p\ne n}^N\underset{0}{\underbrace{{\int }_0^T{V}_{n}sin(n{w}_{n}t+{\alpha }_n)V_{p}sin(p{w}_{p}t+{\alpha }_p)dt}}+\sum _{n=1}^N{\int }_0^T{V}_n^{2}si{n}^2(n{w}_{n}t+{\alpha }_n)dt)\\ & =\frac{1}{T}\sum _{n=1}^N{V}_n^2{\int }_0^{T}si{n}^2(n{w}_{n}t+{\alpha }_n)dt\\ & =\frac{1}{T}\sum _{n=1}^N{V}_n^2{\int }_0^T\frac{1-cos(2n{w}_{n}t+2{\alpha }_n)}{2}dt\\ & =\sum _{n=1}^N\frac{V_n^2}{2}\\ & =\sum _{n=1}^N{V}_{n,rms}^2\end{array}$$ 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")