Analyze Circuits with Dependent Sources

You can analyze circuits with dependent sources using node-voltage analysis, source transformation, and the Thévenin technique, among others. For analyzing circuits that have dependent sources, each technique has particular advantages.

Utilize node-voltage analysis to analyze circuits with dependent sources

Using node voltage methods to analyze circuits with dependent sources follows much the same approach as for independent sources. Consider the circuit shown here. What is the relationship between the output voltage v_o and i_s?

The first step is to label the nodes. Here, the bottom node is your reference node, and you have Node A (with voltage v_A) at the upper left and Node B (with voltage v_B) at the upper right. Now you can formulate the node voltage equations.

Using node-voltage analysis involves Kirchhoff’s current law (KCL), which says the sum of the incoming currents is equal to the sum of the outgoing currents. At Node A, use KCL and substitute in the current expressions from Ohm’s law (i = v/R). The voltage of each device is the difference in node voltages, so you get the following:

Rearranging gives you the node voltage equation:

At Node B, again apply KCL and plug in the current expressions from Ohm’s law:

Rearranging the preceding equation gives you the following node voltage equation at Node B:

The two node voltage equations give you a system of linear equations. Put the node voltage equations in matrix form:

You can solve for the unknown node voltages v_A and v_B using matrix software. After you have the node voltages, you can set the output voltage v_o equal to v_B. You can then use the ever-faithful Ohm’s law to find the output current i_o:

Utilize source transformation to analyze circuits with dependent sources

To see the source transformation technique for circuits with dependent circuits, consider Circuit A as shown here.

Suppose you want to find the voltage across resistor R₃. To do so, you can perform a source transformation, changing Circuit A (with an independent voltage source) to Circuit B (with an independent current source). You now have all the devices connected in parallel, including the dependent and independent current sources.

Don’t use source transformation for dependent sources, because you may end up changing or losing the dependency. You need to make sure the dependent source is a function of the independent source.

Here’s the equation for the voltage source and current source transformation:

The independent current source i_s and the dependent current source gv_x point in the same direction, so you can add these two current sources to get the total current i_eq going through the resistor combination R₁ and R₂. The total current i_eq is i_eq = i_s + g_mv_x. Because v_x is the voltage across R₂, v_x is also equal to v_o in Circuit B: v_o = v_x.

Resistors R₁ and R₂ are connected in parallel, giving you an equivalent resistance R_eq:

The output voltage is equal to the voltage across R_eq, using Ohm’s law and i_eq. You see the equivalent circuit with i_eq and R_eq in Circuit C. Because the dependent current source is dependent on v_x, you need to replace the voltage v_x with v_o:

Solving for the output voltage v_o gives you

See how the output voltage is a function of the input source? The final expression of the output should not have a dependent variable.

Utilize the Thévenin technique to analyze circuits with dependent sources

The Thévenin approach reduces a complex circuit to one with a single voltage source and a single resistor. Independent sources must be turned on because the dependent source relies on the excitation due to an independent source.

To find the Thévenin equivalent for a circuit, you need to find the open-circuit voltage and the short-circuit current at the interface. In other words, you need to find the i-v relationship at the interface.

To see how to get the Thévenin equivalent for a circuit having a dependent source, look at this example. It shows how to find the input resistance and the output Thévenin equivalent circuit at interface points A and B.

The input resistance is

Using Ohm’s law, the current i_in through R₁ is

Solving for i_in, you wind up with

Substituting i_in into the input-resistance equation gives you

Here, the dependent source increases the input resistance by approximately multiplying the resistor R₁ by the dependent parameter μ. R₁ is the input resistance without the dependent source. To find the Thévenin voltage v_T and the Thévenin resistance R_T, you have to find the open-circuit voltage v_oc and short-circuit current i_sc. The resistance R_T is given by the following relationship:

Based on the sample circuit, the open-circuit voltage is v_oc = μv_x. You find that the short-circuit current gives you

After finding v_oc and i_sc, you find the Thévenin resistance:

The output resistance R_o and Thévenin resistance R_T are equal. Based on Kirchhoff’s voltage law (KVL), you have the following expression for v_x:

Substituting v_x into the equation for the open-circuit voltage v_oc, you wind up with

The open-circuit voltage, v_oc, equals the Thévenin voltage, v_T. The nitty-gritty analysis leaves you with Thévenin voltage v_T and Thévenin resistance R_T, entailing a dependent voltage gain of μ:

When μ is very large, the Thévenin voltage v_T equals the source voltage v_s.