3.6 The Norton Equivalent Network

The Norton equivalent circuit for any linear network at a given pair of terminals consists of a current source iN in parallel with a resistor RN. The current iN and resistance RN can be obtained as follows:

1. iN can be found by applying a short at the designated terminal pair on the original network and calculating or measuring the current through the short circuit.

2. RN can be found in the same manner as RTH, that is, by calculating or measuring the resistance of the open-circuit network seen from the designated terminal pair with all independent sources internal to the network set to zero; that is, with voltage sources replaced with short circuits, and current sources replaced with open circuits.

3.5 Thevenin's Theorem

The Thévenin equivalent circuit for any linear network at a given pair of terminals consists of a voltage source vTH in series with a resistor RTH. The voltage vTH and resistance RTH can be obtained as follows:

1. vTH can be found by calculating or measuring the open-circuit voltage at the designated terminal pair on the original network.

2. RTH can be found by calculating or measuring the resistance of the open-circuit network seen from the designated terminal pair with all independent sources internal to the network set to zero. That is, with independent voltage sources replaced with short circuits, and independent current sources replaced with open circuits. (Dependent sources must be left intact, however.)

3.4 Superposition

Superposition principle states that for all linear systems:

The net response at a given place and time caused by two or more stimuli is the sum of the responses which would have been caused by each stimulus individually.

Superposition method:

1. For each independent source, form a subcircuit with all other independent sources set to zero. Setting a voltage source to zero implies replacing the voltage source with a short circuit, and setting a current source to zero implies replacing the current source with an open circuit.

2. From each subcircuit corresponding to a given independent source, find the response to that independent source acting alone. This step results in a set of individual responses.

3. Obtain the total response by summing together each of the individual responses.

3.3 Node Method

Steps of the node method can be written as:

1. Select a reference node, called ground, from which all other voltages will be measured. Define its potential to be 0 V.

2. Label the potentials of the remaining nodes with respect to the ground node. Any node connected to the ground node through either an independent or a dependent voltage source should be labeled with the voltage of that source. The voltages of the remaining nodes are the primary unknowns and should be labeled accordingly. Since there are generally far fewer nodes than branches in a circuit, there will be far fewer primary unknowns to determine in a node analysis.

3. Write KCL for each of the nodes that has an unknown node voltage (in other words, the ground node and nodes with voltage sources connected to ground are excluded), using KVL and element laws to obtain the currents directly in terms of the node voltage differences and element parameters. Thus, one equation is written for each unknown node voltage.

4. Solve the equations resulting from Step 3 for the unknown node voltages. This is the most difficult step in the analysis.

5. Back-solve for the branch voltages and currents. More specifically, use node voltages and KVL to determine branch voltages as desired. Then, use the branch voltages, the element laws, and KCL to determine the branch currents, again as desired.

3.2 The Node Voltage

• A node voltage is the potential difference between the given node and some other node that has been chosen as a reference node. The reference node is called the ground.

3.1 Network Theorems

• The basic network analysis method is fundamental but unfortunately often insufficient. The problem is that frequently we deal with complicated circuits in which we are interested in relating only one output variable to one input variable.
• There exist better approaches to the organization of circuit analysis which greatly simplify circuit analysis and provide substantial insight about how circuits behave, ie Node Method

2.6 Dependent Sources and The Control Concept

• Voltage-controlled current source:

• Linear voltage-controlled current source: iout = g*vin
• g is a constant coefficient called transconductance with unit of conductance
• Current-controlled current source:

• Linear current-controlled current source: (coefficient = current transfer ratio)
• Linear voltage-controlled voltage source: (coefficient = voltage transfer ratio)
• Linear current-controlled voltage source: vout = r*iIN (r = transresistance)
• VCCS, CCCS, VCVS, CCVS:

2.4 Series and Parallel Simplication

• Voltage Divider: v2 = (R2/(R1 + R2))*V
• Resistors in series: Rs = R1 + R2
• Current Divider: i2 = (G2/(G1 + G2))*I
• Resistors in parallel: Rp = R1*R2/(R1+R2)

2.3 Circuit Analysis: Basic Method

Basic method of circuit analysis:

1. Define each branch current and voltage in the circuit (follow associated variables convention)
2. Assemble the element laws for the elements
3. Apply Kirchhoff’s current and voltage laws
4. Jointly solve the equations assembled in Steps 2 and 3 for the branch variables defined in Step 1

Summary:

1. Define voltages and currents for each element
2. Write KVL
3. Write KCL
4. Write constituent relations
5. Solve