How do you handle dependent sources and controlled elements in nodal and mesh analysis?

1 Dependent sources and controlled elements

A dependent source is a voltage or current source that depends on another voltage or current in the circuit. A controlled element is a resistor, capacitor, or inductor that has a variable value that depends on another voltage or current in the circuit. For example, a voltage-controlled voltage source (VCVS) is a dependent source that produces a voltage proportional to another voltage in the circuit. A current-controlled resistor (CCR) is a controlled element that has a resistance proportional to another current in the circuit.

3 Mesh analysis with dependent sources

To successfully perform mesh analysis with dependent sources, you must first identify all the meshes and assign a clockwise current for each mesh. Next, write the KVL equations for each mesh, using the mesh currents as variables. If the circuit has a CCVS, write an additional equation relating the controlling current and the controlled voltage. If the circuit has a CCCS, include the controlled current as a term in the KVL equations of the meshes connected to the source. Likewise, if the circuit has a VCCS or VCVS, introduce dummy variables for the controlling voltage and/or controlled voltage and write additional equations relating them. Finally, solve the system of equations for the mesh currents.

4 Controlled elements in nodal and mesh analysis

To handle controlled elements in nodal and mesh analysis, you need to modify the equations according to the type of element and its dependency. For example, if the circuit has a CCR, you need to replace its resistance value with a function of the controlling current. Similarly, if the circuit has a voltage-controlled capacitor (VCC) or a current-controlled inductor (CCI), you need to replace their reactance values with functions of the controlling voltage or current. You can use Ohm's law, Kirchhoff's laws, and the definitions of capacitance and inductance to derive these functions.

5 Tips and tricks for solving circuits with dependent sources and controlled elements

Solving circuits with dependent sources and controlled elements can be made easier with some tips and tricks. Superposition can be used to simplify the circuit by turning off all independent sources except one and solving for the dependent sources and controlled elements, then repeating for each source and adding up the results. Source transformations can also be utilized to convert dependent sources and controlled elements into equivalent forms that are more manageable. For instance, a VCVS in series with a resistor can be transformed into a CCVS in parallel with a resistor or a CCR into a VCCS in parallel with a resistor. Thevenin's or Norton's theorem can reduce a complex circuit with dependent sources and controlled elements into an independent source and a resistor. Additionally, loop currents or node voltages can be used as controlling variables to avoid introducing dummy variables.

6 Here’s what else to consider

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