Tuesday, February 17, 2015

Thevenin's Theorem

      Thevenin's Theorem states that it is possible to simplify any linear circuit, no matter how complex, to an equivalent circuit with just a single voltage source and series resistance connected to a load. The qualification of “linear” is identical to that found in the Superposition Theorem, where all the underlying equations must be linear (no exponents or roots). If we're dealing with passive components (such as resistors, and later, inductors and capacitors), this is true. However, there are some components (especially certain gas-discharge and semiconductor components) which are nonlinear: that is, their opposition to current changes with voltage and/or current. As such, we would call circuits containing these types of components, nonlinear circuits.
     
     Thevenin's Theorem is especially useful in analyzing power systems and other circuits where one particular resistor in the circuit (called the “load” resistor) is subject to change, and re-calculation of the circuit is necessary with each trial value of load resistance, to determine voltage across it and current through it.


We can calculate the Thévenin equivalent circuit in two steps:
1. Calculate ZTh. Set all sources to zero (replace voltage sources by short circuits and current sources by open circuits) and then find the total impedance between the two terminals.
2. Calculate VTh. Find the open circuit voltage between the terminals.  


Example
Find the Thévenin equivalent of the network for the points A and B at a frequency: f = 1 kHz, vS(t) = 10 cost V. 
 
The first step is to find the open circuit voltage between points A and B:
The open circuit voltage using voltage division:
= -0.065 - j2.462 = 2.463 e-j91.5º
The second step is to replace the voltage source by a short circuit and to find the impedance between points A and B:
  
Here is the Thévenin equivalent circuit, valid only at a frequency of 1kHz. We must first, however, solve for CT's capacitance. Using the relationship 1/wCT = 304 ohm, we find CT = 0.524 uF
Now we have the solution: RT = 301 ohm and CT = 0.524 m F:

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