# Series Circuit and Voltage Division

## Series Circuit and Voltage Division

• Circuit elements are in series if they carry the identical current. The equivalent resistance of resistors connected in series is the sum of the individual resistances.

Example 1: Find the total equivalent resistance in the following circuit

 Solution $R_{T}=R_{1}+R_{2}+R_{3}$ $=20+10+40$ $=70\Omega$

Example 2: For the following circuit:

1. Find the total resistance
2. Find the current i
3. Find the voltage accross the 10Ω resistor
 Solution Total resistance $R_{T}=20+10+40$ $=70\Omega$ The current can be calculated as $i=\frac{v}{R_{T}}=\frac{12}{70}=0.1714A$ The voltage across the 10Ω resistor $v_{10\Omega}=i10=1.714V$

Example 3: For the following circuit:

1. Find the total resister value RT
2. Find the current i
3. Find the voltage across the individual resistors
4. Verify Kirchhoff’s voltage law
 Solution Total resistor value $R_{T}=1+2+4+1=8k\Omega$ The current can be calculated a $i=\frac{v}{R_{T}}=\frac{12}{8k}=1.5mA$ The voltage over the resistors $v_{1}=i1k=1.5(10^{-3})10^{3}=1.5V$, $v_{2}=i2k=1.5(10^{-3})(2)10^{3}=3V$ $v_{3}=i4k=1.5(10^{-3})(4)10^{3}=6V$, $v_{4}=i1k=1.5(10^{-3})10^{3}=1.5V$ To verify the Kirshhoff's Voltage   $12=v_{1}+v_{2}+v_{3}+v_{4}$ $=1.5+3+6+1.5$ $=12V$

Example 4: For the following circuit:

1. Find V1
2. Find V2
3. Verify Kirchhoff Voltage Law around a closed loop
 Solution $R_{T}=R_{1}+R_{2}+R_{3}+R_{4}$ $=2k\Omega+4k\Omega+1k\Omega+3k\Omega=10k\Omega$ $I=\frac{V}{R_{T}}=\frac{40}{10k}=4mA$ $V_{1}=(R_{1}+R_{2})I=(2k\Omega+4k\Omega)*4mA$ $V_{1}=24V$ $V_{2}=(R_{3}+R_{4})I=(1k\Omega+3k\Omega)*4mA$ $V_{2}=16V$ $40v=V_{1}+V_{2}$ $=24v+16v$ $=40V$

#### Voltage Division:

 In the following circuit, the current through all the resistor in series is The equivalent resistor Req is sum of the resistor value. To find the voltage drop vi across the resistor Ri, we use current and resistor value The right side of the equation gives us Voltage division equation.

### Examples:

Example 5: For the following circuit

1. Find the total resister value RT
2. Find the current i
3. Find the voltage over resistors
4. Verify Kirchhoff’s voltage law
 Solution $R_T=10+20+40+20=90\Omega$

Example 6: Find the current for the following circuit

 Solution $R_T=20k$

### Practice Problems:

(Click image to view solution)

Problem 1: Find $V_1,V_2,V_3$ for the following circuit.

#### View Solution

 Solution: Apply Kirchhoff’s Voltage Law $-V_1+12+V_2-4+V_3=0$ $-V_1+V_2+V_3=-8$ $-2(2k)+I(4k)+I(2k)=-8$ $4kI=-8$ $I=-2mA$

Problem 2: Calculate the voltages $V_1,V_2,V_g$ for the following circuit.

#### View Solution

 Solution: the current, $I=5mA$ <$V_1=I_1\bullet&space;R_1&space;=2k\times5mA=10V$ $V_2=I_1\bullet&space;R_2=3k\times5mA=15V$ Applying Kirchhoff’s Law : $\sum_s(V_S)=0$ $V_g-V_1-V_2=0$ $V_g-10-15=0$ $V_g=25$

Problem 3: Find the applied voltage to the following circuit using the information provided

#### View Solution

 Solution: $R_T=220\Omega+330\Omega+880\Omega$

Problem 4: Find the resistor R value using the information provided

#### View Solution

 Solution: $I=\frac{V}{R_T}=\frac{60}{120+R}$

Problem 5: Find Vx

#### View Solution

 Solution: $-10+V_{300}+V_{200\Omega}+5=0$