Network Analysis

Superposition Theorem

Superposition Theorem statement

The theorem states: “In a network with two or more sources, the current or voltage for any component is the algebraic sum of the effects produced by each source acting separately”

• This means that regardless of the source, we have to analyze them one at a time.

• Things to remember:

– Voltage sources will be shorted

– Current sources will be opened

• For these examples, we will stay with voltage sources

First circuit to consider:

V124 V

V212 V

R1

1kΩ

R2

1kΩ

P

Gnd

Short V2, then calculate VP

• VR2 = VP = 𝑅2

𝑅1:𝑅2𝑉1 =

1×103

(1×103):(1×103)(−24) =

1×103

2×103(−24) = -12V

V124 V

R11kΩ

R21kΩ

P

Short V1, then calculate VP

• VR1 = VP = 𝑅1

𝑅1:𝑅2𝑉2 =

1×103

(1×103):(1×103)(12) =

1×103

2×103(12) = 6V

V212 V

R11kΩ

R21kΩ

P

Algebraically sum the results

•VP = VR1 + VR2 = 6V + -12V = 6V – 12V = -6V

Second circuit to consider:

V150 V

V250 V

R1

150Ω

R2

100Ω

P

Gnd

Short V2 and calculate VP

• VR2 = VP = 𝑅2

𝑅1:𝑅2𝑉1 =

100

100:150(50) =

100

250(50)

= 20V

V150 V

R1150Ω

R2100Ω

P

Short V1 and calculate VP

• VR1 = VP = 𝑅1

𝑅1:𝑅2𝑉2 =

150

100:150(−50) =

150

250(−50) = -30V

V250 V

R1150Ω

R2100Ω

P

Algebraically add the values

• VP = VR1 + VR2 = -30V + 20V = -10V

If the resistors were reversed, the overall value of VP would remain the same but the polarities would be reversed. This is due to R2 now being the larger in the voltage divider ratio when calculating the values.

Third circuit to consider:

V19 V

V227 V

R1

220Ω

R2

680Ω

A

B

Short V2, then calculate VA

• VR2 = VA = 𝑅2

𝑅1:𝑅2𝑉1 =

680

220:680(9) =

680

900(9) =

6.8V

V19 V

R1220Ω

R2680Ω

A

Short V1, then recalculate VA

• VR1 = VA = 𝑅1

𝑅1:𝑅2𝑉2 =

220

220:680(27) =

220

900(27)

= 6.6V

V227 V

R1220Ω

R2680Ω

A

Algebraically determine VAB

• VAB = VR1 + VR2 = 6.8V + 6.6V = 13.4V

If we reverse the polarity of V2, this will necessitate a recalculation of the value from A to B (VAB).

Third circuit, V2 reversed:

V19 V

V227 V

R1

220Ω

R2

680Ω

A

B

Short V2 and determine VA

• VR2 = VA = 𝑅2

𝑅1:𝑅2𝑉1 =

680

220:680(9) =

680

900(9) =

6.8V

– Since there was nothing changed from the first condition, this step is only necessary if you want to check your work.

V19 V

R1220Ω

R2680Ω

A

Short V1 and determine VA (again…)

• VR1 = VA = 𝑅1

𝑅1:𝑅2𝑉2 =

220

220:680(−27) =

220

900(−27) = -6.6V

– The only thing that changed here from the original condition is the polarity of VA.

V227 V

R1220Ω

R2680Ω

A

Find the value of VAB

• VAB = VR1 + VR2 = -6.6V +6.8V = 0.2V

As you can see, this causes the value of VAB to decrease significantly (13.4 V as opposed to 0.2V).

With a special thanks to http://www.weirdity.com/internet/eoti.html for the template to make this possible…