TECHNIQUES OF DC CIRCUIT ANALYSIS: SKEE 1023 Superposition Principle Source Transformation Thevenin’s Theorem Norton’s Theorem Maximum Power Transfer SKEE 1023

What do we mean by a linear relationship? Applies only for LINEAR CIRCUIT Circuit containing only linear circuit elements A LINEAR relationship between voltage and current What do we mean by a linear relationship?

What do we mean by a linear relationship? When the relationship fulfilled 2 properties: Homogeneity (scaling) f(x) = y  f(kx) = ky = kf(x) Additivity f(x) = y  f(x1 + x2) = f(x1) + f(x2) = y1 + y2 What do we mean by a linear relationship?

Superposition Principle: The voltage across an element ( or the current through an element) of a linear circuit containing more than one independent source, is the algebraic sum the voltage across that element (or the current through that element) due to each independent source acting alone. All other independent sources are deactivated voltage sources are shorted current sources are opened Note that dependent sources CANNOT be deactivated !

Superposition Principle: The voltage across an element ( or the current through an element) of a linear circuit containing more than one independent source, is the algebraic sum the voltage across that element (or the current through that element) due to each independent source acting alone.

Superposition Principle: The voltage across an element ( or the current through an element) of a linear circuit containing more than one independent source, is the algebraic sum the voltage across that element (or the current through that element) due to each independent source acting alone. may involve MORE work cannot be applied to power calculation – find i or v first (using superposition) before calculating power ! most suitably used when involved with sources of different properties or types, e.g. different frequencies, mixture of DC and AC, etc.

Source Transformation: A tool used to simplify circuit; a process of replacing a voltage source in series with a resistor by a current source in parallel with a resistor or vice versa vs R a b is R a b voc = isR isc = is voc = vs isc = vs/R If the circuit is equivalent at terminal a-b, their open-circuit and short-circuit characteristics are similar

Source Transformation: A tool used to simplify circuit; a process of replacing a voltage source in series with a resistor by a current source in parallel with a resistor or vice versa vs R a b is R a b voc = isR isc = is voc = vs isc = vs/R

VTh= ? RTh= ? In 1883, M.L. Thevenin proposed a theorem ……. Thevenin’s Theorem: A linear two-terminal circuit can be replaced by an equivalent circuit consisting of a voltage source in series with a resistor Linear two-terminal circuit Load + V  I + V  Load VTh RTh I VTh= ? RTh= ?

Thevenin’s Theorem: A linear two-terminal circuit can be replaced by an equivalent circuit consisting of a voltage source in series with a resistor To determine VTh = VTh RTh Load Linear two-terminal circuit Load

Thevenin’s Theorem: A linear two-terminal circuit can be replaced by an equivalent circuit consisting of a voltage source in series with a resistor To determine VTh RTh Load open circuit voltage = Voc +  VTh = VTh Load Linear two-terminal circuit

Thevenin’s Theorem: A linear two-terminal circuit can be replaced by an equivalent circuit consisting of a voltage source in series with a resistor To determine VTh RTh open circuit voltage = Voc +  VTh = VTh Load Linear two-terminal circuit open circuit voltage = Voc + 

Thevenin’s Theorem: A linear two-terminal circuit can be replaced by an equivalent circuit consisting of a voltage source in series with a resistor To determine VTh Linear two-terminal circuit VTh RTh open circuit voltage = Voc +  = VTh VTh = Voc = Open circuit voltage = VTh (Since the circuit is equivalent)

Thevenin’s Theorem: A linear two-terminal circuit can be replaced by an equivalent circuit consisting of a voltage source in series with a resistor To determine RTh - Method 1 Linear two-terminal circuit VTh RTh isc a b Short circuit current, isc =

Thevenin’s Theorem: A linear two-terminal circuit can be replaced by an equivalent circuit consisting of a voltage source in series with a resistor To determine RTh – Method 2 Pre-requisite: circuit with NO dependent sources Deactivate all the independent sources Linear circuit – independent sources killed Rin = RTh Rin = RTh

Thevenin’s Theorem: A linear two-terminal circuit can be replaced by an equivalent circuit consisting of a voltage source in series with a resistor To determine RTh – Method 3 Deactivate all the independent sources - dependent sources stay as they are Linear Circuit – ONLY dependent sources killed vo io + - RTh is calculated as: Introduce a voltage (or current) source.

IN= ? RN= ? 43 years later, E.L. Norton proposed a similar theorem. …. Norton’s Theorem: A linear two-terminal circuit can be replaced by an equivalent circuit consisting of a current source in parallel with a resistor 43 years later, E.L. Norton proposed a similar theorem. …. Linear two-terminal circuit Load + V  I IN= ? + V  I Load IN RN RN= ?

Norton’s Theorem: A linear two-terminal circuit can be replaced by an equivalent circuit consisting of a current source in parallel with a resistor To determine IN IN RN Linear circuit

Norton’s Theorem: A linear two-terminal circuit can be replaced by an equivalent circuit consisting of a current source in parallel with a resistor To determine IN IN IN= Short circuit current RN Linear circuit Short circuit current = IN

Norton’s Theorem: A linear two-terminal circuit can be replaced by an equivalent circuit consisting of a current source in parallel with a resistor To determine IN IN= Short circuit current IN RN Linear circuit Short circuit current = IN IN = Isc = Short circuit current

SIMILAR METHOD AS HOW TO OBTAIN RTh Norton’s Theorem: A linear two-terminal circuit can be replaced by an equivalent circuit consisting of a current source in parallel with a resistor To determine RN SIMILAR METHOD AS HOW TO OBTAIN RTh RN = RTh

Relationship between Norton’s and Thevenin’s equivalents RN b a Linear two-terminal circuit b a OR VTh RTh b a

Since both circuits are equivalent, voc must be the same Relationship between Norton’s and Thevenin’s equivalents IN RN b a VTh RTh +  Since both circuits are equivalent, voc must be the same + 

Maximum Power Transfer Linear circuit RL What would be the value of RL for power delivered to it become MAXIMUM?

Maximum Power Transfer VTh RTh Linear circuit RL What would be the value of RL for power delivered to it become MAXIMUM?

Maximum Power Transfer RL p Maximum power Rl=linspace(1,60,500); Vth=10; Rth=12; p=((Vth./(Rl+Rth)).^2).*Rl; plot(Rl,p,'r'); grid; RL = 12 

Maximum Power Transfer Mathematically, we evaluate RL when

Using PSpice to verify Norton’s and Thevenin’s Theorems Find Thevenin equivalent at terminals a-b

Using PSpice to verify Norton’s and Thevenin’s Theorems

Using PSpice to verify Norton’s and Thevenin’s Theorems

Using PSpice to verify Norton’s and Thevenin’s Theorems

Using PSpice to verify Norton’s and Thevenin’s Theorems

Using PSpice to verify Norton’s and Thevenin’s Theorems RTh = 6/1 = 6

Using PSpice to verify Norton’s and Thevenin’s Theorems VTh = 20V