E72 (P)review
or
Things you should know

E72 takes off where E11 left off.  Because of this, there is certain knowledge  from E11 that will be assumed.  You don't have to be able to easily apply all the various methods of solution that you used in those courses -- the ones that are important will come back to you as you use them.  There are, however, some fundamental things that you should know.  

The rest of this indroduction is devoted to a re-statement and review of the salient material from E11.  Look it over to see if there are any topics with which you feel uncomfortable.  If there are any, you may want to break out your E11 (and/or E12) book and review.  Again - you don't need to be facile with all the fine points, but you should feel at ease with each topic.

The preview  is broken up into four parts:

• Circuit Elements - Basic circuit elements and the equations governing them.
• Circuit Analysis Methods - The methods used to analyze circuit behavior.
• System Behavior - How systems behave, and tools for observing that behavior.
• Sample Problems - If you feel unsure about things, try these problems.

Passive Linear Circuit Elements

	Resistor	Capacitor	Inductor
Circuit symbol
Current-Voltage Characteristic
Parallel Equivalent
Series Equivalent
Complex Impedance (Sinusoidal input)
Other	.	high frequency - acts as short circuit low frequency - open circuit	high frequency - open circuit low frequency - short circuit
Time Constant (with Resistor)	.	t=RC	t=L/R
Resonance

Voltage and Current Sources

	Circuit Symbol
Independent DC voltage source		Output voltage is constant (at any current)
Independent AC voltage source		Output voltage varies sinusoidally.
Current Source		Output current is constant (at any voltage).
Voltage dependent voltage source (Voltage Amplifier)		Output voltage is proportional to a control voltage, Vc (with proportionality constant Av -- Av is unitless)
Current dependent current source (Current Amplifier)		Output current is proportional to a control current, Ic (with proportionality constant Ai -- Ai is unitless)
Current dependent voltage source (Transresistance Amplifier)		Output voltage is proportional to a control current, Ic (with proportionality constant Rm -- Rm has units of ohms)
Voltage dependent current source (Transconductance Amplifier)		Output current is proportional to a control voltage, Vc (with proportionality constant Gm -- Gm has units of Siemens (Conductivity))

Circuit Analysis Methods

• Kirchoff's Voltage Law
• Kirchoff's Current Law
• Voltage Divider
• Current Divider
• The Superposition Theorem
• Thevenin's Theorem
• Norton's Theorem

Kirchoff's Voltage Law

The sum of voltages around a loop is zero.

v1-v2-v3-v4=0, or v1=v2+v3+v4

Kirchoff's Current Law

The sum of currents into a node is zero.

       i1+i2-i3=0, or i1+i2=i3

Voltage Divider

The principal of voltage division is shown below for a circuit with resistors. A more general solution can be obtained by replacing the resistors with complex impedances.  Students often have a hard time remembering if R1 or R2 goes in the numerator -- if you can't remember, just let one of the resistors go to zero (or infinity).  In this case, if R2=0, then v2=0.  Which is obviously the correct answer (if R1 had been in the numerator, the result would be v2=v1; clearly incorrect).

Current Divider

Current division is demonstrated below.  The same trick can be used for remembering whether R2 or R1 belongs in the numerator.  We won't use current divider as often as voltage divider.

Superposition Theorem

The superposition theorem states that for linear circuits, the total effect of several sources acting simultaneously is equal to the sum of the effects of the individual sources acting one at a time.

As an example, consider the circuit below:

We can find the voltage across the 1kW resistor by consisdering the voltage source and the current source independently.

Consider Voltage Source Alone (Current Source set to zero -- an open circuit)	Consider Current Source Alone (Voltage  Source set to zero -- a short circuit)
Simplify Circuit Voltage = 1.333 volts (by voltage divider)	Simplify Circuit (333W=1k in parallel with 500 -- 333=1k||500) Voltage = 0.333 volts (by Ohm's law)
Result Therefore, the total voltage (by superposition)=0.333+1.333=1.666 volts.

 

Thevenin's Theorem

The Thevenin theorem states that any linear network can be represented by a voltage source in series with an impedance.  We will generally use a simpler version: any circuit made up of voltage sources, current sources and resistors can be replaced by a voltage and a resistor.

When applying the Thevenin Theorem there are three cases.  

• Case 1: Only independent sources - the typical case.  In the typical case, there are no dependent sources in the circuit to be Thevenized.  To find the Thevenin equivalent, first find the open circuit voltage, V_oc, this is the Thevenin voltage.  To find the Thevenin resistance, set all sources to zero and find the resistance of the resulting circuit

Consider again the circuit from above,

and try to find the Thevenin circuit at the terminals (i.e., across the 1k resistor).  From the discussion of superposition, we know the open circuit voltage, V_oc, is 1.666 volts.  The Thevenin resistance, R_T, is found by finding the equivalent resistance of the circuit with all source set to zero, as shown below

         

Obviously the Thevenin resistance, R_T,  is 1k||500=333W.  Therefore the resulting circuit is:

• Case 2:  Independent and Dependent Sources.  If the circuit to be Thevenized has both dependent and independent source, the method described above cannot be used to find the Thevenin resistance.  Instead, you must find the short circuit current, I_sc (current through short circuit at terminals).  Then the Thevenin resistance is given by  R_T=V_oc/I_sc.
• Case 3: Only Dependent Sources.  If only dependent sources are present, then the Thevenin voltage is zero, and the Thevenin resistance is determine by applying a test voltage V_test and the terminals and determining the resulting current, I_test. The Thevenin resistance is given by R_T=V_test/I_test.  (Likewise, for this third case, you can apply a test current and measure the resulting voltage).

Norton's Theorem

The Norton theorem states that any linear network can be represented by a current source in parallel with an impedance.  Methodologies are similar to those for Thevenin voltages, but your are typically interested in the short circuit current, rather than the open circuit voltage.  We won't use Norton's Theorem as much as the Thevenin's.

On to System Behavior

On to Problems