thevenin - norton equivalents and maximum power transfer
TRANSCRIPT
Circuit Theorems:Thevenin and Norton Equivalents,
Maximum Power Transfer
Dr. Mustafa Kemal Uyguroğlu
Thevenin’s Theorem
Any circuit with sources (dependent and/or independent) and resistors can be replaced by an equivalent circuit containing a single voltage source and a single resistor.Thevenin’s theorem implies that we can replace arbitrarily complicated networks with simple networks for purposes of analysis.
Independent Sources (Thevenin)
Circuit with independent sources
RTh
VTh
Thevenin equivalent circuit
+–
No Independent Sources
Circuit without independent sources
RTh
Thevenin equivalent circuit
Thevenin Equivalent Circuit
Basic steps to determining Theveninequivalent are– Find vTh RTh
VTh+–
i = 0 i = 0
+
Voc
-
+
Voc
-
Linear
Two-terminal circuit
Voc = VTh
Thevenin Equivalent Circuit
- Compute the Thevenin equivalent resistance, RTh
(a) If there are only independent sources, then short circuit all the voltage sources and open circuit the current sources (just like superposition).
RTh
Linear circuit with all independent sources set equal to zero.
Rin = RTh
RinRin
Thevenin Equivalent Circuit
(b) If there are only dependent sources, then must use a test voltage or current source in order to calculate
RTh = VTest/Itest
RTh
Linear circuit with dependent sources
Itest
+
Vtest
-
Linear circuit with dependent sources
Vtest
+
-
Itest
Thevenin Equivalent Circuit
(c) If there are both independent and dependent sources, then compute(i) RTh = VTest/Itest (all independent sources set equal to zero)(ii) compute RTh from VOC/ISC.
RTh
+–
Linear
Two-terminal circuit
i = isc i = iscVTh
isc = VTh / RTh
Example
Example cont.
Example cont.
Ref.
5 3 5.52 2oc oc
ocV V V V−
+ = ⇒ =KCL at Voc :
Example cont.
Example cont.
Problem: for the following circuit , determine the Thevenin equivalent circuit.
Solution:Step 1: In this circuit, we have a dependent source. Hence, we start by finding the open circuit voltage Voc = Vab. KCL at node C (5 - Voc)/2 + Voc/4 = 0 Voc = 10 V
Step 2: We obtain the short circuit current Isc by shorting nodes a-b and finding the current through it.
5 = 2 Isc + 3 Isc => Isc = 5/5 Isc = 1 A
Step 3: Find the equivalent Thevenin Voltage and Resistance Vth = Voc = Vab = 10V Rth = Voc/Isc => Rth = 10/1 ΩVth = 10VRth = 10 Ω
Norton Equivalent Circuit
Any Thevenin equivalent circuit is in turn equivalent to a current source in parallel with a resistor [source transformation].A current source in parallel with a resistor is called a Norton equivalent circuit.
Norton Equivalent Circuit
Finding a Norton equivalent circuit requires essentially the same process as finding a Thevenin equivalent circuit.
VTh
RTh
IN RN
Th N NV R I= ThN
Th
VIR
=
+–
Th NR R=
Thevenin/Norton Analysis
1. Pick a good breaking point in the circuit (cannot split a dependent source and its control variable).
2. Thevenin: Compute the open circuit voltage, VOC.Norton: Compute the short circuit current, ISC.
If there is not any independent source then both VOC=0 and ISC=0 [so skip step 2]
Thevenin/Norton Analysis
3. Calculate RTh(RN) = Voc / Isc4. Thevenin: Replace circuit with VOC in series with RTh
Norton: Replace circuit with ISC in parallel with RTh
Note: for circuits containing no independent sources the equivalent network is merely RTh , that is, no voltage (or current) source.
Only steps 2 & 4 differ from Thevenin & Norton!
Maximum Power Transfer
I
Maximum Power Transfer
Power delivered to the load as a function of RL.
Maximum Power Transfer
Maximum Power Transfer
Example
Example cont.
Example
Example cont.
17 6 5 172 2 33 2 6 3
6 32 64 12.85 3 5
Th ThTh
Th
V V V
V V
+ ++ = − ⇒ = − − −
− = = − = −
KCL at the supernode:
Example cont.
Example cont.
Example cont.