Added content

acdcconv.mdown

@@ -390,8 +390,7 @@ $$ % Rectifier model, by Diego Trapero
  (2,2) to[C](2,0)
  (2,2) to[short](3,2)
  (3,0) to[R](3,2)
- (2,0) to[short](3,0)
- (0,0) to[short](2,0)
+ (0,0) to[short](3,0)
  (0,0) node[ground](){}
  ;
 \end{circuitikz}
@@ -409,8 +408,7 @@ $$ % Rectifier model, by Diego Trapero
  (2,2) to[C](2,0)
  (2,2) to[short](3,2)
  (3,0) to[R](3,2)
- (2,0) to[short](3,0)
- (0,0) to[short](2,0)
+ (0,0) to[short](3,0)
  (0,0) node[ground](){}
  ;
 \end{circuitikz}
@@ -430,8 +428,7 @@ $$ % Rectifier model, by Diego Trapero
  (2,2) to[C](2,0)
  (2,2) to[short](3,2)
  (3,0) to[R](3,2)
- (2,0) to[short](3,0)
- (0,0) to[short](2,0)
+ (0,0) to[short](3,0)
  (0,0) node[ground](){}
  ;
 \end{circuitikz}
@@ -494,7 +491,9 @@ The ripple of the wave using the triangular approximation is
 $$\Delta v_o = \frac{V \Delta T}{C}$$
 $$\Delta v_o = \frac{V_i T }{2RC}$$
 
-**
+**Mean value**
+
+******
 
 **Advantages of a smoothing condenser**
 
@@ -526,99 +525,128 @@ $$ % AC to DC Full Wave Rectifier with RLC load, by Diego Trapero
 \end{circuitikz}
 $$
 
+**$v_R$ mean value**
+
+The mean value of $v_o$ can be calculated by using superposition with the circuit:
+
+$$ % Rectifier model, by Diego Trapero
+\begin{circuitikz}[scale = 1]
+ \draw
+ (0,0) to[voltage source, l=$v_o$](0,2)
+ (0,2) to[diode](2,2)
+ (2,2) to[L](4,2)
+ (4,2) to[C](4,0)
+ (4,2) to[short](5,2)
+ (5,0) to[R](5,2)
+ (0,0) to[short](5,0)
+ (0,0) node[ground](){}
+ ;
+\end{circuitikz}
+$$
+
+For the mean values, the resulting DC circuit is:
+
+$$ % Rectifier model, by Diego Trapero
+\begin{circuitikz}[scale = 1]
+ \draw
+ (0,0) to[voltage source, l=$\bar{v_o}$](0,2)
+ (0,2) to[diode](2,2)
+ (2,2) to[short](4,2)
+ (4,2) to[open](4,0)
+ (4,2) to[short](5,2)
+ (5,0) to[R](5,2)
+ (0,0) to[short](5,0)
+ (0,0) node[ground](){}
+ ;
+\end{circuitikz}
+$$
+
+The resulting $\bar{v_R}$ is
+
+$$\bar{v_R} = \bar{v_o} = \frac{2V_{ip}}{\pi}$$
+
+
 **RLC circuit transfer function**
 
 $$ \mathbf{H} = \frac{\mathbf{V_o}}{\mathbf{V_i}} = \frac{1}{1 + \frac{L}{R}j\omega - LC \omega^2} $$
 
 $$ H = | \frac{\mathbf{V_o}}{\mathbf{V_i}} | = \frac{1}{ \sqrt{(1- LC \omega^2 )^2 + (\frac{L}{R} \omega )^2 } } $$
 
 
-**Advantages of LC filtering**
+**$v_R$ ripple**
 
-* No current peaks in the diodes.
-* Less condenser ripple.
-* Less capacity and current capacitor required.
+Ripple is due to the first harmonic:
 
-**Disadvantages of LC filtering**
+$$\Delta v_R = H_{filter} \cdot |V_{o1}|$$
 
-* Size and weight of the inductor
+$$ H = \frac{1}{ \sqrt{(1- LC \omega^2 )^2 + (\frac{L}{R} \omega )^2 } } $$
 
-### RL load
+$$|V_{o1}| = \frac{4}{\pi} \frac{1}{3} V_{ip}$$
 
-### Loads table
 
-\tabulinesep=1mm
-\begin{longtabu} to\linewidth{|X[1,m,c]|X[2,m,c]|X[2,m,c]|X[2,m,c]|}
+**Inductor current, $i_L$**
 
- \hline
- What & R load & RC load & RLC load \\
- \hline
+Inductor current can be approximated with its mean value and ripple.
 
- Circuit Diagram
- &
- $$ % AC to DC Full Wave Rectifier with R load, by Diego Trapero
- \begin{circuitikz}[scale = 0.6, transform shape]
-  \draw
-  (0,0) node[fullWaveRectifier](FWR){}
-  (-19mm, -5mm) to[sinusoidal voltage source](-19mm, 5mm)
-  (9mm, 5mm) to[short](19mm,5mm)
-  (19mm, 5mm) to[open](19mm,-5mm)
-  (29mm, 5mm) to[R](29mm,-5mm)
-  (-19mm, 5mm) -- (FWR.1)
-  (-19mm, -5mm) -- (FWR.2)
-  (19mm,5mm) -- (29mm, 5mm)
-  (FWR.3) -- (29mm,-5mm)
-  ;
- \end{circuitikz}
- $$ 
- &
- $$ % AC to DC Full Wave Rectifier with RC load, by Diego Trapero
- \begin{circuitikz}[scale = 0.6, transform shape]
-  \draw
-  (0,0) node[fullWaveRectifier](FWR){}
-  (-19mm, -5mm) to[sinusoidal voltage source](-19mm, 5mm)
-  (9mm, 5mm) to[short](19mm,5mm)
-  (19mm, 5mm) to[C](19mm,-5mm)
-  (29mm, 5mm) to[R](29mm,-5mm)
-  (-19mm, 5mm) -- (FWR.1)
-  (-19mm, -5mm) -- (FWR.2)
-  (19mm,5mm) -- (29mm, 5mm)
-  (FWR.3) -- (29mm,-5mm)
-  ;
- \end{circuitikz}
- $$
- &
- $$ % AC to DC Full Wave Rectifier with RLC load, by Diego Trapero
- \begin{circuitikz}[scale = 0.6, transform shape]
-  \draw
-  (0,0) node[fullWaveRectifier](FWR){}
-  (-19mm, -5mm) to[sinusoidal voltage source](-19mm, 5mm)
-  (9mm, 5mm) to[L](19mm,5mm)
-  (19mm, 5mm) to[C](19mm,-5mm)
-  (29mm, 5mm) to[R](29mm,-5mm)
-  (-19mm, 5mm) -- (FWR.1)
-  (-19mm, -5mm) -- (FWR.2)
-  (19mm,5mm) -- (29mm, 5mm)
-  (FWR.3) -- (29mm,-5mm)
-  ;
- \end{circuitikz}
- $$
- \\
- \hline
+* **Mean value**. Using superposition
 
- $v_g$ vs $i_g$
- &
- \vspace{20mm}
- &
- RC
- &
- RLC
- \\
+$$ % Rectifier model, by Diego Trapero
+\begin{circuitikz}[scale = 1]
+ \draw
+ (0,0) to[voltage source, l=$\bar{v_o}$](0,2)
+ (0,2) to[diode](2,2)
+ (2,2) to[short](4,2)
+ (4,2) to[open](4,0)
+ (4,2) to[short](5,2)
+ (5,0) to[R](5,2)
+ (0,0) to[short](5,0)
+ (0,0) node[ground](){}
+ ;
+\end{circuitikz}
+$$
 
-
-\end{longtabu}
+$$i_L = \frac{\bar{v_o}}{R}$$
+
+* **Ripple**. As in the $v_R$ ripple, maximum variation of the current is due to the 1st harmonic of $v_o$. Thus, ripple can be calculated:
+
+$$ % Rectifier model, by Diego Trapero
+\begin{circuitikz}[scale = 1]
+ \draw
+ (0,0) to[voltage source, l=$v_o$](0,2)
+ (0,2) to[diode](2,2)
+ (2,2) to[L](4,2)
+ (4,2) to[C](4,0)
+ (4,2) to[short](5,2)
+ (5,0) to[R](5,2)
+ (0,0) to[short](5,0)
+ (0,0) node[ground](){}
+ ;
+\end{circuitikz}
+$$
+
+$$I_{1p} = \frac{V_{o1}}{Z_1}$$
+
+If $\Delta v_R$ is already calculated, it can be used to compute the ripple:
+
+$$\Delta i_L = \frac{\Delta v_R}{Z_{RC}}$$
+
+
+
+The rectifier is in CCM if the inductor is always conducting some current:
 
-\vspace{80mm}
+$$\frac{1}{2}\Delta i_l < \bar{i_L} $$
+
+**Advantages of LC filtering**
+
+* No current peaks in the diodes.
+* Less condenser ripple.
+* Less capacity and current capacitor required.
+
+**Disadvantages of LC filtering**
+
+* Size and weight of the inductor
+
+### RL load
 
 
 **Flyback Diode**