Handout 1: Review

• Basic Circuit Variables
  • Voltage:

    Voltage is an ''across variable'' as it measures the difference between two electrical potentials at two points. When a reference point is used (e.g., the ''ground''), the voltage at a certain point is defined as the potential difference between this point and the reference point.

  • Current:

    Current is a ''through variable'' as it measures the amount of electrical charge that flow through a pass per unit time.

• Basic Elements
  • Resistor:

    
    

    $$V=RI,\;\;\;I=\frac{V}{R},\;\;\;R=\frac{V}{I}$$
    
    

  • Capacitor:

    
    

    $$V=\frac{Q}{C}, \;\;\;i(t)=C\frac{d}{dt} v(t),\;\;\; v(t)=\frac{1}{C}\int_{-\infty}^t i(\tau) d\tau$$
    
    

  • Inductor:

    
    

    $$V=\frac{W}{Q};\;\;\;\;\;v(t)=\frac{d}{dt} \psi(t)=L\frac{d}{dt} i(t),\;\;\; i(t)=\frac{1}{L}\int_{-\infty}^t v(\tau) d\tau$$
    
    

• Series and Parallel Connections
  • Series Connections:

    
    

    $$R_{s}=R_1+\cdots+R_n=\sum_{i=1}^n R_i$$
    
    
    
    
    $$L_{s}=L_1+\cdots+L_n=\sum_{i=1}^n L_i$$
    
    
    
    
    $$\frac{1}{C_{s}}=\frac{1}{C_1}+\cdots+\frac{1}{C_n} =\sum_{i=... ..._i},\;\;\;or\;\;\; C_{s}=\frac{1}{\sum_{i=1}^n \frac{1}{C_i}}$$
    
    
    If $n=2$,
    
    $$C_s=\frac{1}{\frac{1}{C_1}+\frac{1}{C_2}}=\frac{C_1C_2}{C_1+C_2}$$
    
    

  • Parallel Connections:

    
    

    $$\frac{1}{R_{p}}=\frac{1}{R_1}+\cdots+\frac{1}{R_n} =\sum_{i=... ..._i},\;\;\;or\;\;\; R_{p}=\frac{1}{\sum_{i=1}^n \frac{1}{R_i}}$$
    
    
    If $n=2$,
    
    $$R_p=\frac{1}{\frac{1}{R_1}+\frac{1}{R_2}}=\frac{R_1R_2}{R_1+R_2}$$
    
    

    
    

    $$\frac{1}{L_{p}}=\frac{1}{L_1}+\cdots+\frac{1}{L_n} =\sum_{i=... ..._i},\;\;\;or\;\;\; L_{p}=\frac{1}{\sum_{i=1}^n \frac{1}{L_i}}$$
    
    
    If $n=2$,
    
    $$L_p=\frac{1}{\frac{1}{L_1}+\frac{1}{L_2}}=\frac{L_1L_2}{L_1+L_2}$$
    
    

    
    

    $$C_{p}=C_1+C_2$$
    
    

  If $n=2$, we have:

  	Resistor	Inductor	Capacitor
  v-i relation	$v=R\;i$	$v=L \;\frac{di}{dt}$	$v=\frac{1}{C}\int i\;dt$
  Series	$R_s=R_1+R_2$	$L_s=L_1+L_2$	$C_s=C_1C_2/(C_1+C_2)$
  Parallel	$R_p=R_1R_2/(R_1+R_2)$	$L_p=L_1L_2/(L_1+L_2)$	$C_p=C_1+C_2$

  Q: Why are $R$ and $L$ similar to each other while $C$ is different?

  A: Observe how $R$, $L$ and $C$ are related to voltage $v$ and current $i$ differently.

• Energy and Power

  
  

  $$V=\frac{W}{Q},\;\;\;W=VQ,\;\;\;P=\frac{dW}{dt}=V \frac{dQ}{dt}=VI=\frac{V^2}{R}=RI^2$$
  
  

• Voltage/Current Dividers

  

  • Voltage Divider
    
    $$I=\frac{V}{R_1+R_2},\;\;\;\;V_1=IR_1=V\;\frac{R_1}{R_1+R_2},\;\;\;\; V_2=IR_2=V\;\frac{R_2}{R_1+R_2}$$
    
    
    Voltage across $R$ is proportional to its own resistance.

    When there are more than two resistors in series, we simply have:
    

    $$V_i=IR_i=V\;\frac{R_i}{\sum_j R_j}$$
    
    

  • Current Divider
    
    $$V=I\frac{R_1R_2}{R_1+R_2},\;\;\;I_1=\frac{V}{R_1}=I\frac{R_2}{R_1+R_2}, \;\;\;\;I_2=\frac{V}{R_2}=I\frac{R_1}{R_1+R_2}$$
    
    
    Current through $R$ is proportional to the other resistance.

    When there are more than twor esistors in parallel, we have:
    

    $$V=I\frac{1}{\sum_j 1/R_j},\;\;\;I_i=\frac{V}{R_i}=I\frac{1/R_i}{\sum_j 1/R_j} =I\frac{G_i}{\sum_j G_j}$$
    
    
    where $G_i=1/R_i$ is the conductance of the ith resistor.

• Kirchhoff's Laws

  

  • KCL: Due to conservation of electric charge, Kirchhoff current Law (KCL) states:

    The algebraic sum of all currents into a node is zero $\sum_k I_k =0$ (Currents leaving the node take negative values.)

  • KVL: Due to conservation of electric energy, Kirchhoff voltage Law (KCL) states:

    The algebraic sum of voltage around a loop is zero $\sum_k V_k =0$ (Voltages with opposite polarity take negative values.)

• Reference Point or Ground

  The voltage between any two nodes in a circuit is the difference between the potentials at the two nodes. However, a particular node in the circuit is usually chosen as the reference point, called the ground. The voltage at any node in the circuit is therefore defined as the voltage between this node and the common ground.