SOURCE TRANSFORMATION

Source Transformation An equivalent circuit is one whose v-i characteristics are identical with the original circuit. It is the process of replacing a voltage source vS in series with a resistor R by a current source iS in parallel with a resistor R, or vice versa.

Source Transformation (a) Independent source transform (b) Dependent source transform The arrow of the current source is directed toward the positive terminal of the voltage source. The source transformation is not possible when R = 0 for voltage source and R = ∞ for current source.

Source Transformation Example 4.6 Use source transformation to find v0

Source Transformation

Source Transformation Example 4.7 Find vx using Source Transformation

Source Transformation

Source Transformation Practice Problem 4.6 Find i0 in the circuit below using source transformation

Source Transformation Practice Problem 4.7 Find ix in the circuit below using source transformation

THEVENIN’S THEOREM

Thevenin’s Theorem It states that a linear two-terminal circuit can be replaced by an equivalent circuit consisting of a voltage source VTH in series with a resistor RTH,

Thevenin’s Theorem VTH is the open-circuit voltage at the terminals. RTH is the input or equivalent resistance at the terminals when the independent sources are turned off.

Finding Thevenin resistance, RTH: Thevenin’s Theorem Finding Thevenin resistance, RTH: CASE 1 (If network has NO dependent sources) Turn off all independent sources. RTH is the input resistance of the network looking between its two terminals

Finding Thevenin resistance, RTH: Thevenin’s Theorem Finding Thevenin resistance, RTH: CASE 2 (If network has dependent sources) Turn off all independent sources BUT leave dependent sources intact (similar to superposition theorem) Two approaches: Approach 1 – Apply a voltage source v0 at the terminals and determine the resulting current i0. Then RTH = v0/i0 (figure a) Approach 2 – Insert a current source i0 at the network terminal and find the terminal voltage v0 (figure b)

Thevenin’s Theorem A Thevenin-equivalent circuit is a simple voltage divider, VL by mere inspection

Thevenin’s Theorem Example 4.8 Find the Thevenin equivalent circuit of the circuit below to the left of the terminals a-b. Also find the current when RL is 6, 16 and 36 ohms.

Thevenin’s Theorem Example 4.8

Thevenin’s Theorem Example 4.9 Find the Thevenin equivalent circuit of the circuit below at terminals a-b.

Thevenin’s Theorem Example 4.9

Thevenin’s Theorem Practice Problem 4.8 Using Thevenin’s Theorem, find the equivalent circuit to the left of the terminals in the circuit below. Then find I

Thevenin’s Theorem Practice Problem 4.9 Find the Thevenin equivalent circuit of the circuit below

Thevenin’s Theorem Practice Problem 4.10 Using Thevenin’s Theorem, find the equivalent circuit to the left of the terminals in the circuit below

Tutorial (Source Transformation) Problem 4.27 Apply source transformation to find vx in the circuit below

Tutorial (Source Transformation) Problem 4.32 Use source transformation to find ix in the circuit below

Tutorial (Thevenin’s Theorem) Problem 4.33 Determine RTH and VTH at terminals 1-2 of each of the circuit below:

Tutorial (Thevenin’s Theorem) Problem 4.38 Apply thevenin’s theorem to find Vo in the circuit below:

Tutorial (Thevenin’s Theorem) Problem 4.39 Obtain the thevenin equivalent at terminals a-b of the circuit below:

Tutorial (Thevenin’s Theorem) Problem 4.44 For the circuit below, obtain the thevenin’s equivalent as seen from terminals (a) a-b (b) b-c

Tutorial (Thevenin’s Theorem) Problem 4.40 Find the thevenin equivalent at terminals a-b of the circuit below:

Homework (Thevenin’s Theorem) Problem 4.42 Find the thevenin equivalent between terminals a-b of the circuit below:

Homework (Thevenin’s Theorem) Problem 4.43 Find the thevenin equivalent looking into terminals a-b of the circuit below:

Homework (Thevenin’s Theorem) Problem 4.36 Solve for the current i in the circuit below (Hint: find thevenin equivalent seen by the 12 ohm resistor)

NORTON’S THEOREM

Norton’s Theorem It states that a linear two-terminal circuit can be replaced by an equivalent circuit of a current source IN in parallel with a resistor RN, The Thevenin’s and Norton equivalent circuits are related by a source transformation.

Norton’s Theorem Where IN is the short circuit current through the terminals. RN is the input or equivalent resistance at the terminals when the independent sources are turned off.

Relationship between Thevenin’s Theorem and Norton’s Theorem 𝐼 𝑁 = 𝑉 𝑇𝐻 𝑅 𝑇𝐻 𝑉 𝑇𝐻 = 𝑣 𝑜𝑐 𝐼 𝑁 = 𝑖 𝑠𝑐 𝑅 𝑇𝐻 = 𝑣 𝑜𝑐 𝑖 𝑠𝑐 = 𝑅 𝑁

Norton’s Theorem Example 4.11 Find the Norton equivalent of the circuit in the figure below

Norton’s Theorem 1st approach

Norton’s Theorem 2nd approach

Norton’s Theorem Example 4.12 Find the Norton equivalent of the circuit at terminals a-b in the figure below

Norton’s Theorem

Norton’s Theorem Practice Problem 4.11 Find the Norton equivalent of the circuit at terminals a-b in the figure below

Norton’s Theorem Practice Problem 4.12 Find the Norton equivalent of the circuit at terminals a-b in the figure below

Tutorial (Norton’s Theorem) Problem 4.50 Obtain the Norton equivalent of the circuit below to the left of the terminals a-b. Use the result to find current i

Tutorial (Norton’s Theorem) Problem 4.51 Obtain the norton equivalent as viewed from terminals: a) a-b b) c-d

Tutorial (Norton’s Theorem) Problem 4.53 Find the Norton equivalent at terminals a-b of the circuit below:

Tutorial (Norton’s Theorem) Try these problems: 1. Problem 4.57, ans: RTH = 10 ohm, VTH = 166.67 V, IN = 16.667 A 2. Problem 4.56, ans: V0 = -2.857 V