Linearity, superposition, Thevenin Equivalent

Linearity theorem

For any circuit containing resistors and independent voltage or current sources,

every node voltage and branch current is a linear funtion of the source values and has the form sigma aiUi where Ui are the source values and ai are constants.

Using Linearity theorem

Use variables instead of fixed values for independent v and I sources:

1) Label all nodes

2) KCL equations

3) Solve for node voltage

effect voltage given by…

(a1 x cause_1) + (a2 x cause_2)…

proportionality

multiplying cause and effect by same factor

superposition

find effects produced by two causes seperately and add them together to find the effect of both

combining linearity and superposition

gives linearity property

zero-value voltage source

• zero volts between terminals for any current

• equivalent to a short circuit

• if u have voltage source with fixed Vs value, regard it as =0 or replace with wire

zero-value current source

no current flowing between terminals

• equivalent to open circuit or broken wire

Superposition

find the effect of each source on its own by setting other sources to zero, then sum.

superposition steps

1) consider source U and zero V and W

2) Zero V means replacing it with a wire, zero W means removing W

3) resistor 1 II 6 = 6/7, Xu = 0.3U

voltage divider circuit

voltage x R interest/R total

Equivalent networks

Relationship between V and I being exactly the same in shaded region

Thévenin theorem

two-terminal network consisting of resistors, fixed voltage sources

Thevenin equivalent circuit

voltage and current in unshaded part of circuit is same in both circuits

• new components are called equivalent resistance, RTh and equivalent voltage VTh

• Useful way to simplify complicated circuit, if we dont wanna know about V and I in shaded part

Thevenin circuit properties

V = RthI + Vth

• open circuit voltage: I = 0, Voc = Vth

• short circuit current: V = 0, Isc = -Vth/Rth

• thevenin resistance: slope of characteristics is dI/dV = 1/Rth

Thevenin: in any two-terminal circuit …

the three quantities will have same values

• If we can work out two of them, we can work out thevenin equivalent

Determining thevenin values

V = Rth*I + Vth

1) open circuit output