Welcome to our exploration of circuits with dependent voltage sources. A dependent voltage source is a special circuit element whose voltage value depends on another voltage or current somewhere else in the circuit. Unlike independent sources that have fixed values, dependent sources are controlled by other circuit variables. There are two main types: voltage-controlled voltage sources, abbreviated as VCVS, and current-controlled voltage sources, abbreviated as CCVS.
Now let's examine a specific example circuit containing a voltage-controlled voltage source. In this circuit, we have a twelve volt independent source connected to a four ohm resistor and a six ohm resistor. The dependent voltage source has a value of two times Vx, where Vx is the voltage across the four ohm resistor. To analyze this circuit, we first identify the controlling voltage Vx, then apply nodal analysis at node A, express the dependent source in terms of Vx, and finally solve for the node voltage and currents.
Now let's solve this circuit using nodal analysis. We apply Kirchhoff's current law at node A, stating that the sum of currents entering the node equals zero. The current I1 through the four ohm resistor is twelve minus VA divided by four. The current I2 through the six ohm resistor is two Vx minus VA divided by six. Since Vx equals VA in this circuit, we can substitute VA for Vx. This gives us the equation: twelve minus VA over four plus VA over six equals zero.
Now let's solve the equation step by step. Starting with twelve minus VA over four plus VA over six equals zero, we multiply through by twelve to get three times twelve minus VA plus two VA equals zero. Expanding gives us thirty-six minus three VA plus two VA equals zero, which simplifies to thirty-six minus VA equals zero. Therefore, VA equals thirty-six volts. With this result, we can calculate the circuit currents. I1 equals twelve minus thirty-six over four, which is negative six amperes, indicating current flows in the opposite direction. I2 equals thirty-six over six, which is six amperes. The dependent voltage source value is two times thirty-six, which equals seventy-two volts.
To summarize what we've learned about circuits with dependent voltage sources: Dependent sources have values that are controlled by other circuit variables, unlike independent sources with fixed values. VCVS voltage depends on another voltage in the circuit, while CCVS depends on a current. We apply standard analysis methods like nodal or mesh analysis, express the dependent source value in terms of its controlling variable, and solve the resulting system of equations to find all circuit parameters.