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4: Linearity and Superposition • Linearity Theorem • Zero-value sources • Superposition • Superposition Calculation • Superposition and dependent sources • Single Variable Source • Superposition and Power • Proportionality • Summary 4: Linearity and Superposition

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 1 / 10 Linearity Theorem

4: Linearity and Superposition Suppose we use variables instead of fixed values for all of the independent • Linearity Theorem voltage and current sources. We can then use nodal analysis to find all • Zero-value sources • Superposition node voltages in terms of the source values. • Superposition Calculation • Superposition and dependent sources • Single Variable Source • Superposition and Power • Proportionality • Summary

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 2 / 10 Linearity Theorem

4: Linearity and Superposition Suppose we use variables instead of fixed values for all of the independent • Linearity Theorem voltage and current sources. We can then use nodal analysis to find all • Zero-value sources • Superposition node voltages in terms of the source values. • Superposition Calculation • Superposition and dependent sources (1) Label all the nodes • Single Variable Source • Superposition and Power • Proportionality • Summary

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 2 / 10 Linearity Theorem

4: Linearity and Superposition Suppose we use variables instead of fixed values for all of the independent • Linearity Theorem voltage and current sources. We can then use nodal analysis to find all • Zero-value sources • Superposition node voltages in terms of the source values. • Superposition Calculation • Superposition and dependent sources (1) Label all the nodes • Single Variable Source (2) KCL equations • Superposition and Power X−U X X−Y • Proportionality 1 + + = 0 • Summary 2 1 3 Y −X 3 +(−U2) = 0

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 2 / 10 Linearity Theorem

4: Linearity and Superposition Suppose we use variables instead of fixed values for all of the independent • Linearity Theorem voltage and current sources. We can then use nodal analysis to find all • Zero-value sources • Superposition node voltages in terms of the source values. • Superposition Calculation • Superposition and dependent sources (1) Label all the nodes • Single Variable Source (2) KCL equations • Superposition and Power X−U X X−Y • Proportionality 1 + + = 0 • Summary 2 1 3 Y −X 3 +(−U2) = 0 (3) Solve for the node voltages 1 2 1 11 X = 3 U1 + 3 U2, Y = 3 U1 + 3 U2

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 2 / 10 Linearity Theorem

4: Linearity and Superposition Suppose we use variables instead of fixed values for all of the independent • Linearity Theorem voltage and current sources. We can then use nodal analysis to find all • Zero-value sources • Superposition node voltages in terms of the source values. • Superposition Calculation • Superposition and dependent sources (1) Label all the nodes • Single Variable Source (2) KCL equations • Superposition and Power X−U X X−Y • Proportionality 1 + + = 0 • Summary 2 1 3 Y −X 3 +(−U2) = 0 (3) Solve for the node voltages 1 2 1 11 X = 3 U1 + 3 U2, Y = 3 U1 + 3 U2

Steps (2) and (3) never involve multiplying two source values together, so:

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 2 / 10 Linearity Theorem

4: Linearity and Superposition Suppose we use variables instead of fixed values for all of the independent • Linearity Theorem voltage and current sources. We can then use nodal analysis to find all • Zero-value sources • Superposition node voltages in terms of the source values. • Superposition Calculation • Superposition and dependent sources (1) Label all the nodes • Single Variable Source (2) KCL equations • Superposition and Power X−U X X−Y • Proportionality 1 + + = 0 • Summary 2 1 3 Y −X 3 +(−U2) = 0 (3) Solve for the node voltages 1 2 1 11 X = 3 U1 + 3 U2, Y = 3 U1 + 3 U2

Steps (2) and (3) never involve multiplying two source values together, so: Linearity Theorem: For any circuit containing resistors and independent voltage and current sources, every node voltage and branch current is a linear of the source values and has the form P aiUi where the Ui are the source values and the ai are suitably dimensioned constants.

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 2 / 10 Linearity Theorem

4: Linearity and Superposition Suppose we use variables instead of fixed values for all of the independent • Linearity Theorem voltage and current sources. We can then use nodal analysis to find all • Zero-value sources • Superposition node voltages in terms of the source values. • Superposition Calculation • Superposition and dependent sources (1) Label all the nodes • Single Variable Source (2) KCL equations • Superposition and Power X−U X X−Y • Proportionality 1 + + = 0 • Summary 2 1 3 Y −X 3 +(−U2) = 0 (3) Solve for the node voltages 1 2 1 11 X = 3 U1 + 3 U2, Y = 3 U1 + 3 U2

Steps (2) and (3) never involve multiplying two source values together, so: Linearity Theorem: For any circuit containing resistors and independent voltage and current sources, every node voltage and branch current is a of the source values and has the form P aiUi where the Ui are the source values and the ai are suitably dimensioned constants. Also true for a circuit containing dependent sources whose values are proportional to voltages or currents elsewhere in the circuit.

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 2 / 10 Zero-value sources

4: Linearity and Superposition A zero-valued voltage source has zero volts • Linearity Theorem between its terminals for any current. It is • Zero-value sources • Superposition equivalent to a short-circuit or piece of wire • Superposition Calculation • Superposition and or resistor of 0Ω (or ∞ S). dependent sources • Single Variable Source • Superposition and Power • Proportionality • Summary

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 3 / 10 Zero-value sources

4: Linearity and Superposition A zero-valued voltage source has zero volts • Linearity Theorem between its terminals for any current. It is • Zero-value sources • Superposition equivalent to a short-circuit or piece of wire • Superposition Calculation • Superposition and or resistor of 0Ω (or ∞ S). dependent sources • Single Variable Source • Superposition and Power • Proportionality • Summary

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 3 / 10 Zero-value sources

4: Linearity and Superposition A zero-valued voltage source has zero volts • Linearity Theorem between its terminals for any current. It is • Zero-value sources • Superposition equivalent to a short-circuit or piece of wire • Superposition Calculation • Superposition and or resistor of 0Ω (or ∞ S). dependent sources • Single Variable Source • Superposition and Power • Proportionality • Summary A zero-valued current source has no current flowing between its terminals. It is equivalent to an open-circuit or a broken wire or a resistor of ∞ Ω (or 0 S).

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 3 / 10 Zero-value sources

4: Linearity and Superposition A zero-valued voltage source has zero volts • Linearity Theorem between its terminals for any current. It is • Zero-value sources • Superposition equivalent to a short-circuit or piece of wire • Superposition Calculation • Superposition and or resistor of 0Ω (or ∞ S). dependent sources • Single Variable Source • Superposition and Power • Proportionality • Summary A zero-valued current source has no current flowing between its terminals. It is equivalent to an open-circuit or a broken wire or a resistor of ∞ Ω (or 0 S).

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 3 / 10 Superposition

4: Linearity and Superposition We can use nodal analysis to find X in terms of U, V and W . • Linearity Theorem • Zero-value sources • Superposition • Superposition Calculation • Superposition and dependent sources • Single Variable Source • Superposition and Power • Proportionality • Summary

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 4 / 10 Superposition

4: Linearity and Superposition We can use nodal analysis to find X in terms of U, V and W . • Linearity Theorem X−U X−V X • Zero-value sources KCL: + + − W = 0 • Superposition 2 6 1 • Superposition Calculation • Superposition and dependent sources • Single Variable Source • Superposition and Power • Proportionality • Summary

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 4 / 10 Superposition

4: Linearity and Superposition We can use nodal analysis to find X in terms of U, V and W . • Linearity Theorem X−U X−V X • Zero-value sources KCL: + + − W = 0 • Superposition 2 6 1 • Superposition Calculation • Superposition and 10X − 3U − V − 6W = 0 dependent sources • Single Variable Source • Superposition and Power • Proportionality • Summary

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 4 / 10 Superposition

4: Linearity and Superposition We can use nodal analysis to find X in terms of U, V and W . • Linearity Theorem X−U X−V X • Zero-value sources KCL: + + − W = 0 • Superposition 2 6 1 • Superposition Calculation • Superposition and 10X − 3U − V − 6W = 0 dependent sources • Single Variable Source • Superposition and Power X = 0.3U + 0.1V + 0.6W • Proportionality • Summary

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 4 / 10 Superposition

4: Linearity and Superposition We can use nodal analysis to find X in terms of U, V and W . • Linearity Theorem X−U X−V X • Zero-value sources KCL: + + − W = 0 • Superposition 2 6 1 • Superposition Calculation • Superposition and 10X − 3U − V − 6W = 0 dependent sources • Single Variable Source • Superposition and Power X = 0.3U + 0.1V + 0.6W • Proportionality • Summary From the linearity theorem, we know anyway that X = aU + bV + cW so all we need to do is find the values of a, b and c.

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 4 / 10 Superposition

4: Linearity and Superposition We can use nodal analysis to find X in terms of U, V and W . • Linearity Theorem X−U X−V X • Zero-value sources KCL: + + − W = 0 • Superposition 2 6 1 • Superposition Calculation • Superposition and 10X − 3U − V − 6W = 0 dependent sources • Single Variable Source • Superposition and Power X = 0.3U + 0.1V + 0.6W • Proportionality • Summary From the linearity theorem, we know anyway that X = aU + bV + cW so all we need to do is find the values of a, b and c. We find each coefficient in turn by setting all the other sources to zero:

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 4 / 10 Superposition

4: Linearity and Superposition We can use nodal analysis to find X in terms of U, V and W . • Linearity Theorem X−U X−V X • Zero-value sources KCL: + + − W = 0 • Superposition 2 6 1 • Superposition Calculation • Superposition and 10X − 3U − V − 6W = 0 dependent sources • Single Variable Source • Superposition and Power X = 0.3U + 0.1V + 0.6W • Proportionality • Summary From the linearity theorem, we know anyway that X = aU + bV + cW so all we need to do is find the values of a, b and c. We find each coefficient in turn by setting all the other sources to zero:

We have XU = aU + b × 0 + c × 0 = aU.

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 4 / 10 Superposition

4: Linearity and Superposition We can use nodal analysis to find X in terms of U, V and W . • Linearity Theorem X−U X−V X • Zero-value sources KCL: + + − W = 0 • Superposition 2 6 1 • Superposition Calculation • Superposition and 10X − 3U − V − 6W = 0 dependent sources • Single Variable Source • Superposition and Power X = 0.3U + 0.1V + 0.6W • Proportionality • Summary From the linearity theorem, we know anyway that X = aU + bV + cW so all we need to do is find the values of a, b and c. We find each coefficient in turn by setting all the other sources to zero:

We have XU = aU + b × 0 + c × 0 = aU.

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 4 / 10 Superposition

4: Linearity and Superposition We can use nodal analysis to find X in terms of U, V and W . • Linearity Theorem X−U X−V X • Zero-value sources KCL: + + − W = 0 • Superposition 2 6 1 • Superposition Calculation • Superposition and 10X − 3U − V − 6W = 0 dependent sources • Single Variable Source • Superposition and Power X = 0.3U + 0.1V + 0.6W • Proportionality • Summary From the linearity theorem, we know anyway that X = aU + bV + cW so all we need to do is find the values of a, b and c. We find each coefficient in turn by setting all the other sources to zero:

We have XU = aU + b × 0 + c × 0 = aU.

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 4 / 10 Superposition

4: Linearity and Superposition We can use nodal analysis to find X in terms of U, V and W . • Linearity Theorem X−U X−V X • Zero-value sources KCL: + + − W = 0 • Superposition 2 6 1 • Superposition Calculation • Superposition and 10X − 3U − V − 6W = 0 dependent sources • Single Variable Source • Superposition and Power X = 0.3U + 0.1V + 0.6W • Proportionality • Summary From the linearity theorem, we know anyway that X = aU + bV + cW so all we need to do is find the values of a, b and c. We find each coefficient in turn by setting all the other sources to zero:

We have XU = aU + b × 0 + c × 0 = aU. Similarly, XV = bV and XW = cW ⇒ X = XU + XV + XW .

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 4 / 10 Superposition Calculation

4: Linearity and Superposition Superposition: • Linearity Theorem • Zero-value sources Find the effect of each source on its own • Superposition • Superposition Calculation by setting all other sources to zero. Then • Superposition and dependent sources add up the results. • Single Variable Source • Superposition and Power • Proportionality • Summary

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 5 / 10 Superposition Calculation

4: Linearity and Superposition Superposition: • Linearity Theorem • Zero-value sources Find the effect of each source on its own • Superposition • Superposition Calculation by setting all other sources to zero. Then • Superposition and dependent sources add up the results. • Single Variable Source • Superposition and Power • Proportionality • Summary 6 7 6 XU = 6 U = U = 0.3U 2+ 7 20

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 5 / 10 Superposition Calculation

4: Linearity and Superposition Superposition: • Linearity Theorem • Zero-value sources Find the effect of each source on its own • Superposition • Superposition Calculation by setting all other sources to zero. Then • Superposition and dependent sources add up the results. • Single Variable Source • Superposition and Power • Proportionality • Summary 6 7 6 XU = 6 U = U = 0.3U 2+ 7 20

2 3 2 XV = 2 V = V = 0.1V 6+ 3 20

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 5 / 10 Superposition Calculation

4: Linearity and Superposition Superposition: • Linearity Theorem • Zero-value sources Find the effect of each source on its own • Superposition • Superposition Calculation by setting all other sources to zero. Then • Superposition and dependent sources add up the results. • Single Variable Source • Superposition and Power • Proportionality • Summary 6 7 6 XU = 6 U = U = 0.3U 2+ 7 20

2 3 2 XV = 2 V = V = 0.1V 6+ 3 20

6 2 12 XW = 2 W × = W = 0.6W 6+ 3 3 20

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 5 / 10 Superposition Calculation

4: Linearity and Superposition Superposition: • Linearity Theorem • Zero-value sources Find the effect of each source on its own • Superposition • Superposition Calculation by setting all other sources to zero. Then • Superposition and dependent sources add up the results. • Single Variable Source • Superposition and Power • Proportionality • Summary 6 7 6 XU = 6 U = U = 0.3U 2+ 7 20

2 3 2 XV = 2 V = V = 0.1V 6+ 3 20

6 2 12 XW = 2 W × = W = 0.6W 6+ 3 3 20

Adding them up: X = XU + XV + XW

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 5 / 10 Superposition Calculation

4: Linearity and Superposition Superposition: • Linearity Theorem • Zero-value sources Find the effect of each source on its own • Superposition • Superposition Calculation by setting all other sources to zero. Then • Superposition and dependent sources add up the results. • Single Variable Source • Superposition and Power • Proportionality • Summary 6 7 6 XU = 6 U = U = 0.3U 2+ 7 20

2 3 2 XV = 2 V = V = 0.1V 6+ 3 20

6 2 12 XW = 2 W × = W = 0.6W 6+ 3 3 20

Adding them up: X = XU + XV + XW = 0.3U + 0.1V + 0.6W

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 5 / 10 Superposition and dependent sources

4: Linearity and Superposition A dependent source is one that is determined by the voltage and/or current • Linearity Theorem elsewhere in the circuit via a known equation. Here V , Y − X. • Zero-value sources • Superposition • Superposition Calculation • Superposition and dependent sources • Single Variable Source • Superposition and Power • Proportionality • Summary

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 6 / 10 Superposition and dependent sources

4: Linearity and Superposition A dependent source is one that is determined by the voltage and/or current • Linearity Theorem elsewhere in the circuit via a known equation. Here V , Y − X. • Zero-value sources • Superposition • Superposition Calculation Step 1: Pretend all sources are independent • Superposition and dependent sources and use superposition to find expressions for • Single Variable Source the node voltages: • Superposition and Power • Proportionality • Summary

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 6 / 10 Superposition and dependent sources

4: Linearity and Superposition A dependent source is one that is determined by the voltage and/or current • Linearity Theorem elsewhere in the circuit via a known equation. Here V , Y − X. • Zero-value sources • Superposition • Superposition Calculation Step 1: Pretend all sources are independent • Superposition and dependent sources and use superposition to find expressions for • Single Variable Source the node voltages: • Superposition and Power 10 • Proportionality X = 3 U1 • Summary Y = 2U1

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 6 / 10 Superposition and dependent sources

4: Linearity and Superposition A dependent source is one that is determined by the voltage and/or current • Linearity Theorem elsewhere in the circuit via a known equation. Here V , Y − X. • Zero-value sources • Superposition • Superposition Calculation Step 1: Pretend all sources are independent • Superposition and dependent sources and use superposition to find expressions for • Single Variable Source the node voltages: • Superposition and Power 10 • Proportionality X = 3 U1 + 2U2 • Summary Y = 2U1 + 6U2

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 6 / 10 Superposition and dependent sources

4: Linearity and Superposition A dependent source is one that is determined by the voltage and/or current • Linearity Theorem elsewhere in the circuit via a known equation. Here V , Y − X. • Zero-value sources • Superposition • Superposition Calculation Step 1: Pretend all sources are independent • Superposition and dependent sources and use superposition to find expressions for • Single Variable Source the node voltages: • Superposition and Power 10 1 • Proportionality X = 3 U1 + 2U2 + 6 V • Summary 1 Y = 2U1 + 6U2 + 2 V

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 6 / 10 Superposition and dependent sources

4: Linearity and Superposition A dependent source is one that is determined by the voltage and/or current • Linearity Theorem elsewhere in the circuit via a known equation. Here V , Y − X. • Zero-value sources • Superposition • Superposition Calculation Step 1: Pretend all sources are independent • Superposition and dependent sources and use superposition to find expressions for • Single Variable Source the node voltages: • Superposition and Power 10 1 • Proportionality X = 3 U1 + 2U2 + 6 V • Summary 1 Y = 2U1 + 6U2 + 2 V Step 2: Express the dependent source values in terms of node voltages: V = Y − X

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 6 / 10 Superposition and dependent sources

4: Linearity and Superposition A dependent source is one that is determined by the voltage and/or current • Linearity Theorem elsewhere in the circuit via a known equation. Here V , Y − X. • Zero-value sources • Superposition • Superposition Calculation Step 1: Pretend all sources are independent • Superposition and dependent sources and use superposition to find expressions for • Single Variable Source the node voltages: • Superposition and Power 10 1 • Proportionality X = 3 U1 + 2U2 + 6 V • Summary 1 Y = 2U1 + 6U2 + 2 V Step 2: Express the dependent source values in terms of node voltages: V = Y − X

Step 3: Eliminate the dependent source values from the node voltage equations: 10 1 X = 3 U1 + 2U2 + 6 (Y − X) 1 Y = 2U1 + 6U2 + 2 (Y − X))

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 6 / 10 Superposition and dependent sources

4: Linearity and Superposition A dependent source is one that is determined by the voltage and/or current • Linearity Theorem elsewhere in the circuit via a known equation. Here V , Y − X. • Zero-value sources • Superposition • Superposition Calculation Step 1: Pretend all sources are independent • Superposition and dependent sources and use superposition to find expressions for • Single Variable Source the node voltages: • Superposition and Power 10 1 • Proportionality X = 3 U1 + 2U2 + 6 V • Summary 1 Y = 2U1 + 6U2 + 2 V Step 2: Express the dependent source values in terms of node voltages: V = Y − X

Step 3: Eliminate the dependent source values from the node voltage equations: 10 1 7 1 10 X = 3 U1 + 2U2 + 6 (Y − X) ⇒ 6 X − 6 Y = 3 U1 + 2U2 1 Y = 2U1 + 6U2 + 2 (Y − X))

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 6 / 10 Superposition and dependent sources

4: Linearity and Superposition A dependent source is one that is determined by the voltage and/or current • Linearity Theorem elsewhere in the circuit via a known equation. Here V , Y − X. • Zero-value sources • Superposition • Superposition Calculation Step 1: Pretend all sources are independent • Superposition and dependent sources and use superposition to find expressions for • Single Variable Source the node voltages: • Superposition and Power 10 1 • Proportionality X = 3 U1 + 2U2 + 6 V • Summary 1 Y = 2U1 + 6U2 + 2 V Step 2: Express the dependent source values in terms of node voltages: V = Y − X

Step 3: Eliminate the dependent source values from the node voltage equations: 10 1 7 1 10 X = 3 U1 + 2U2 + 6 (Y − X) ⇒ 6 X − 6 Y = 3 U1 + 2U2 1 1 1 Y = 2U1 + 6U2 + 2 (Y − X)) ⇒ 2 X + 2 Y = 2U1 + 6U2

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 6 / 10 Superposition and dependent sources

4: Linearity and Superposition A dependent source is one that is determined by the voltage and/or current • Linearity Theorem elsewhere in the circuit via a known equation. Here V , Y − X. • Zero-value sources • Superposition • Superposition Calculation Step 1: Pretend all sources are independent • Superposition and dependent sources and use superposition to find expressions for • Single Variable Source the node voltages: • Superposition and Power 10 1 • Proportionality X = 3 U1 + 2U2 + 6 V • Summary 1 Y = 2U1 + 6U2 + 2 V Step 2: Express the dependent source values in terms of node voltages: V = Y − X

Step 3: Eliminate the dependent source values from the node voltage equations: 10 1 7 1 10 X = 3 U1 + 2U2 + 6 (Y − X) ⇒ 6 X − 6 Y = 3 U1 + 2U2 1 1 1 Y = 2U1 + 6U2 + 2 (Y − X)) ⇒ 2 X + 2 Y = 2U1 + 6U2 X = 3U1 + 3U2

Y = U1 + 9U2

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 6 / 10 Superposition and dependent sources

4: Linearity and Superposition A dependent source is one that is determined by the voltage and/or current • Linearity Theorem elsewhere in the circuit via a known equation. Here V , Y − X. • Zero-value sources • Superposition • Superposition Calculation Step 1: Pretend all sources are independent • Superposition and dependent sources and use superposition to find expressions for • Single Variable Source the node voltages: • Superposition and Power 10 1 • Proportionality X = 3 U1 + 2U2 + 6 V • Summary 1 Y = 2U1 + 6U2 + 2 V Step 2: Express the dependent source values in terms of node voltages: V = Y − X

Step 3: Eliminate the dependent source values from the node voltage equations: 10 1 7 1 10 X = 3 U1 + 2U2 + 6 (Y − X) ⇒ 6 X − 6 Y = 3 U1 + 2U2 1 1 1 Y = 2U1 + 6U2 + 2 (Y − X)) ⇒ 2 X + 2 Y = 2U1 + 6U2 X = 3U1 + 3U2

Y = U1 + 9U2 Note: This is an alternative to nodal anlysis: you get the same answer.

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 6 / 10 Single Variable Source

4: Linearity and Superposition Any current or voltage can be written X = a1U1 + a2U2 + a3U3 + .... • Linearity Theorem • Zero-value sources • Superposition • Superposition Calculation • Superposition and dependent sources • Single Variable Source • Superposition and Power • Proportionality • Summary

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 7 / 10 Single Variable Source

4: Linearity and Superposition Any current or voltage can be written X = a1U1 + a2U2 + a3U3 + .... • Linearity Theorem • Zero-value sources Using nodal analysis (slide 4-2) or else • Superposition • Superposition Calculation superposition: • Superposition and 1 2 dependent sources X = 3 U1 + 3 U2. • Single Variable Source • Superposition and Power • Proportionality • Summary

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 7 / 10 Single Variable Source

4: Linearity and Superposition Any current or voltage can be written X = a1U1 + a2U2 + a3U3 + .... • Linearity Theorem • Zero-value sources Using nodal analysis (slide 4-2) or else • Superposition • Superposition Calculation superposition: • Superposition and 1 2 dependent sources X = 3 U1 + 3 U2. • Single Variable Source • Superposition and Power Suppose we know U • Proportionality 2 = 6mA • Summary

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 7 / 10 Single Variable Source

4: Linearity and Superposition Any current or voltage can be written X = a1U1 + a2U2 + a3U3 + .... • Linearity Theorem • Zero-value sources Using nodal analysis (slide 4-2) or else • Superposition • Superposition Calculation superposition: • Superposition and 1 2 dependent sources X = 3 U1 + 3 U2. • Single Variable Source • Superposition and Power Suppose we know U , then • Proportionality 2 = 6mA • Summary 1 2 1 X = 3 U1 + 3 U2 = 3 U1 + 4.

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 7 / 10 Single Variable Source

4: Linearity and Superposition Any current or voltage can be written X = a1U1 + a2U2 + a3U3 + .... • Linearity Theorem • Zero-value sources Using nodal analysis (slide 4-2) or else • Superposition • Superposition Calculation superposition: • Superposition and 1 2 dependent sources X = 3 U1 + 3 U2. • Single Variable Source • Superposition and Power Suppose we know U , then • Proportionality 2 = 6mA • Summary 1 2 1 X = 3 U1 + 3 U2 = 3 U1 + 4.

If all the independent sources except for U1 have known fixed values, then

X = a1U1 + b

where b = a2U2 + a3U3 + ... .

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 7 / 10 Single Variable Source

4: Linearity and Superposition Any current or voltage can be written X = a1U1 + a2U2 + a3U3 + .... • Linearity Theorem • Zero-value sources Using nodal analysis (slide 4-2) or else • Superposition • Superposition Calculation superposition: • Superposition and 1 2 dependent sources X = 3 U1 + 3 U2. • Single Variable Source • Superposition and Power Suppose we know U , then • Proportionality 2 = 6mA • Summary 1 2 1 X = 3 U1 + 3 U2 = 3 U1 + 4.

If all the independent sources except for U1 have known fixed values, then

X = a1U1 + b

where b = a2U2 + a3U3 + ... . This has a straight graph.

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 7 / 10 Superposition and Power

4: Linearity and Superposition The power absorbed (or dissipated) by a component always equals VI • Linearity Theorem where the measurement directions of V and I follow the passive sign • Zero-value sources • Superposition convention. • Superposition Calculation 2 • Superposition and V 2 dependent sources For a resistor VI = R = I R. • Single Variable Source • Superposition and Power • Proportionality • Summary

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 8 / 10 Superposition and Power

4: Linearity and Superposition The power absorbed (or dissipated) by a component always equals VI • Linearity Theorem where the measurement directions of V and I follow the passive sign • Zero-value sources • Superposition convention. • Superposition Calculation 2 • Superposition and V 2 dependent sources For a resistor VI = R = I R. • Single Variable Source • 2 Superposition and Power (U1+U2) • Proportionality Power in resistor is P = 10 = 6.4W • Summary

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 8 / 10 Superposition and Power

4: Linearity and Superposition The power absorbed (or dissipated) by a component always equals VI • Linearity Theorem where the measurement directions of V and I follow the passive sign • Zero-value sources • Superposition convention. • Superposition Calculation 2 • Superposition and V 2 dependent sources For a resistor VI = R = I R. • Single Variable Source • 2 Superposition and Power (U1+U2) • Proportionality Power in resistor is P = 10 = 6.4W • Summary 2 U1 Power due to U1 alone is P1 = 10 = 0.9W

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 8 / 10 Superposition and Power

4: Linearity and Superposition The power absorbed (or dissipated) by a component always equals VI • Linearity Theorem where the measurement directions of V and I follow the passive sign • Zero-value sources • Superposition convention. • Superposition Calculation 2 • Superposition and V 2 dependent sources For a resistor VI = R = I R. • Single Variable Source • 2 Superposition and Power (U1+U2) • Proportionality Power in resistor is P = 10 = 6.4W • Summary 2 U1 Power due to U1 alone is P1 = 10 = 0.9W 2 U2 Power due to U2 alone is P2 = 10 = 2.5W

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 8 / 10 Superposition and Power

4: Linearity and Superposition The power absorbed (or dissipated) by a component always equals VI • Linearity Theorem where the measurement directions of V and I follow the passive sign • Zero-value sources • Superposition convention. • Superposition Calculation 2 • Superposition and V 2 dependent sources For a resistor VI = R = I R. • Single Variable Source • 2 Superposition and Power (U1+U2) • Proportionality Power in resistor is P = 10 = 6.4W • Summary 2 U1 Power due to U1 alone is P1 = 10 = 0.9W 2 U2 Power due to U2 alone is P2 = 10 = 2.5W

P =6 P1 + P2 ⇒ Power does not obey superposition.

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 8 / 10 Superposition and Power

4: Linearity and Superposition The power absorbed (or dissipated) by a component always equals VI • Linearity Theorem where the measurement directions of V and I follow the passive sign • Zero-value sources • Superposition convention. • Superposition Calculation 2 • Superposition and V 2 dependent sources For a resistor VI = R = I R. • Single Variable Source • 2 Superposition and Power (U1+U2) • Proportionality Power in resistor is P = 10 = 6.4W • Summary 2 U1 Power due to U1 alone is P1 = 10 = 0.9W 2 U2 Power due to U2 alone is P2 = 10 = 2.5W

P =6 P1 + P2 ⇒ Power does not obey superposition. You must use superposition to calculate the total V and/or the total I and then calculate the power.

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 8 / 10 Proportionality

4: Linearity and Superposition From the linearity theorem, all voltages and currents have the form P aiUi • Linearity Theorem where the Ui are the values of the independent sources. • Zero-value sources • Superposition • Superposition Calculation If you multiply all the independent sources by the same factor, k, then all • Superposition and dependent sources voltages and currents in the circuit will be multiplied by k. • Single Variable Source • 2 Superposition and Power The power dissipated in any component will be multiplied by k . • Proportionality • Summary

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 9 / 10 Proportionality

4: Linearity and Superposition From the linearity theorem, all voltages and currents have the form P aiUi • Linearity Theorem where the Ui are the values of the independent sources. • Zero-value sources • Superposition • Superposition Calculation If you multiply all the independent sources by the same factor, k, then all • Superposition and dependent sources voltages and currents in the circuit will be multiplied by k. • Single Variable Source • 2 Superposition and Power The power dissipated in any component will be multiplied by k . • Proportionality • Summary Special Case: If there is only one independent source, U, then all voltages and currents are proportional to U and all power dissipations are proportional to U 2.

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 9 / 10 Summary

4: Linearity and Superposition • Linearity Theorem: X = Pi aiUi over all independent sources Ui • Linearity Theorem • Zero-value sources • Superposition • Superposition Calculation • Superposition and dependent sources • Single Variable Source • Superposition and Power • Proportionality • Summary

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 10 / 10 Summary

4: Linearity and Superposition • Linearity Theorem: X = Pi aiUi over all independent sources Ui • Linearity Theorem • Zero-value sources • Superposition: sometimes simpler than nodal analysis, often more • Superposition insight. • Superposition Calculation • Superposition and ◦ Zero-value voltage and current sources dependent sources • Single Variable Source ◦ Dependent sources - treat as independent and add dependency • Superposition and Power • Proportionality as an extra equation • Summary

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 10 / 10 Summary

4: Linearity and Superposition • Linearity Theorem: X = Pi aiUi over all independent sources Ui • Linearity Theorem • Zero-value sources • Superposition: sometimes simpler than nodal analysis, often more • Superposition insight. • Superposition Calculation • Superposition and ◦ Zero-value voltage and current sources dependent sources • Single Variable Source ◦ Dependent sources - treat as independent and add dependency • Superposition and Power • Proportionality as an extra equation • Summary • If all sources are fixed except for U1 then all voltages and currents in the circuit have the form aU1 + b.

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 10 / 10 Summary

4: Linearity and Superposition • Linearity Theorem: X = Pi aiUi over all independent sources Ui • Linearity Theorem • Zero-value sources • Superposition: sometimes simpler than nodal analysis, often more • Superposition insight. • Superposition Calculation • Superposition and ◦ Zero-value voltage and current sources dependent sources • Single Variable Source ◦ Dependent sources - treat as independent and add dependency • Superposition and Power • Proportionality as an extra equation • Summary • If all sources are fixed except for U1 then all voltages and currents in the circuit have the form aU1 + b. • Power does not obey superposition.

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 10 / 10 Summary

4: Linearity and Superposition • Linearity Theorem: X = Pi aiUi over all independent sources Ui • Linearity Theorem • Zero-value sources • Superposition: sometimes simpler than nodal analysis, often more • Superposition insight. • Superposition Calculation • Superposition and ◦ Zero-value voltage and current sources dependent sources • Single Variable Source ◦ Dependent sources - treat as independent and add dependency • Superposition and Power • Proportionality as an extra equation • Summary • If all sources are fixed except for U1 then all voltages and currents in the circuit have the form aU1 + b. • Power does not obey superposition. • Proportionality: multiplying all sources by k multiplies all voltages and currents by k and all powers by k2.

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 10 / 10 Summary

4: Linearity and Superposition • Linearity Theorem: X = Pi aiUi over all independent sources Ui • Linearity Theorem • Zero-value sources • Superposition: sometimes simpler than nodal analysis, often more • Superposition insight. • Superposition Calculation • Superposition and ◦ Zero-value voltage and current sources dependent sources • Single Variable Source ◦ Dependent sources - treat as independent and add dependency • Superposition and Power • Proportionality as an extra equation • Summary • If all sources are fixed except for U1 then all voltages and currents in the circuit have the form aU1 + b. • Power does not obey superposition. • Proportionality: multiplying all sources by k multiplies all voltages and currents by k and all powers by k2.

For further details see Hayt Ch 5 or Irwin Ch 5.

E1.1 Analysis of Circuits (2018-10340) Linearity and Superposition: 4 – 10 / 10