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Circuit Analysis Techniques
Thévenin & Norton Equivalent Circuits
• Circuit simplification techniques that focus on terminal behaviour
Used to represent circuit made of linear elements
Replace with VTH and RTH equivalent to original circuit
VTH is the open circuit voltage at the terminal
RTH is the ratio of the open circuit voltage to the short circuit current
•𝑅𝑇𝐻 =
Eg. Determine Thévenin Equivalent
• Find equivalent Thévenin circuit
• Find the Thévenin equivalent circuit
Note: short circuit bypasses the resistors. Current source continues to
supply constant current!
Norton Equivalent Circuit
• Derive from Thévenin by source transformation
Special considerations
• When there are dependent sources, pay attention to the value of the
controlling current/voltage. (it is not constant!)
I depends on v and vice versa.
For short circuit, note that isc is
-20i while 𝑣 = 𝑣𝑎𝑏 becomes zero,
so the dependent voltage source
in first loop goes to zero as well!
Note: v= -20i*25 and -5+2000i+3v=0
At short cct: v=0, i= 5/2k, Isc= -20i= -20*5/2k
Techniques for quickly determining Rth
• If circuit contains only independent sources, ‘zero’ out all sources
• Short voltage sources, open current sources. Compute the R looking into the
• If circuit has dependent sources
• Deactivate the independent sources
• Apply a test voltage (or test current source).
• Rth is the ratio of test voltage to the current delivered by the test source
• Rth= Vt/It
Just to determine RTH
Determine VTH as usual – open circuit voltage
Determine Thévenin and Norton’s Equivalent
Try Assessment Problems 4.17, 4.21
Vth= 8V, RTh = 1ohm, IN=8A
Maximum Power Transfer
• Comms & instrumentation systems: transfer as much of the signal to
the load as possible
• Showing that for max power transfer RTH= RL:
• Expressing power in RL as fxn of variables:
• Find value of RL that maximizes p:
Max power transfer….II
• Set to zero
• Simplifies to RTh= RL
• Find value of R for max power transfer
Hint: find thevenin equivalent first
Also look at full Example 4.12 in NR
• When a linear system is excited by more than one source, the net
response is the sum of the independent responses.
• Useful in complex circuits. Can simplify no. of equations to solve
• Examples to date have been simple.
• Following is for illustrative purposes
• Find the branch currents in this circuit
Find the branch currents
• Open I, short V
I1= I1’ + I1’’