Chapter 7 Review

Key Terms

Cramer’s rule
An explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution.

linear circuit
An electronic circuit in which, for a sinusoidal input voltage of frequency f, any steady-state output of the circuit (the current through any component, or the voltage between any two points) is also sinusoidal with frequency f.

load resistor
An electrical component or portion of a circuit that consumes electric power. Normally, the term “load” is used to refer specifically to the active component for which power consumption is mainly intended, as opposed to internal resistance or resistors used in conjunction with capacitors or inductors for timing.

maximum power transfer theorem
Maximum power is transferred to a load when the internal resistance equals the load resistance; for a Thévenin-equivalent circuit, this is true when the load resistance equals the Thévenin resistance.

mesh analysis
A method that is used to solve planar circuits for the currents (and indirectly the voltages) at any place in an electrical circuit.

mesh current
The currents defined in mesh analysis as flowing around each loop of a planar circuit. The actual current through each branch of the circuit is found as a combination of the mesh currents in that branch.

planar circuit (mesh)
An electrical circuit that can be drawn on a plane surface with no wires crossing each other.

superposition theorem
The response (voltage or current) in any branch of a linear circuit having more than one independent source equals the algebraic sum of the responses caused by each independent source acting alone, where all the other independent sources are replaced by short circuits.

Thévenin-equivalent circuit
A simple circuit, equivalent to any more complex linear circuit, involving the Thévenin voltage and Thévenin resistance relative to a particular load. In the Thévenin-equivalent circuit, the Thévenin voltage and resistance are connected in series with the load.

Thévenin resistance
One component of a linear circuit’s Thévenin-equivalent, found by removing the load resistor from the original circuit and calculating the total equivalent resistance between the two load connection points.

Thévenin’s theorem
An electrical circuit theorem through which any complex linear circuit may be replaced by its Thévenin-equivalent with respect to a given load.

Thévenin voltage
One component of a linear circuit’s Thévenin-equivalent, found by removing the load resistor from the original circuit and calculating the potential difference from one load connection point to the other.


Key Equations

Cramer’s rule I_i=\frac{\mathrm{det}(\mathbf{R}_i)}{\mathrm{det}(\mathbf{R})}
Current through a load resistor \frac{V_{\mathrm{th}}}{R_{\mathrm{th}}+R_L}
Voltage across a load resistor \frac{R_LV_{\mathrm{th}}}{R_{\mathrm{th}}+R_L}
Power dissipated in a load resistor P_L=I_L^2R_L=\frac{V_L^2}{R_L}=\frac{R_LV_{\mathrm{th}}^2}{(R_{\mathrm{th}}+R_L)^2}

Summary

7.1 Mesh Analysis

  • Steps in mesh analysis method:
    1. Draw mesh current loops, ensuring:
      1. each loop is unique; and
      2. all circuit elements—voltage sources, resistors, capacitors, inductors, etc. and short circuits—are covered by at least one loop.
    2. Apply loop rule as described in Kirchhoff’s Rules (particularly with reference to Figure 6.3.5) and solve simultaneous equations.
    3. Add or subtract mesh currents in branches that are covered by multiple loops, depending on the direction of each loop and the sign of each current calculated in step 2.
  • When \mathbf{V} and \mathbf{R} are known, the elements of \mathbf{I} are given by Cramer’s rule:

        \[I_i=\frac{\mathrm{det}(\mathbf{R}_i)}{\mathrm{det}(\mathbf{R})},\]

    where I_i is the i-th element of \mathbf{I} and \mathbf{R}_i is the matrix formed by replacing the i-th column of \mathbf{R} with \mathbf{V}.

  • Steps to read off \mathbf{R} and \mathbf{V} directly from a planar circuit:
    1. Draw one mesh current loop inside each loop of the circuit.
    2. Work your way around each loop I_i, reading off terms V_i and R_{ij} as:
      1. V_i is the sum -V_a-V_b-\ldots for each voltage source V_a, V_b,\ldots that I_i passes from negative to positive, and +V_m+V_n+\ldots for each voltage source V_m, V_n,\ldots that I_i passes through from positive to negative,
      2. R_{ii} is the sum -R_a-R_b-\ldots for each resistor R_a, R_b,\ldots that I_i passes, and
      3. R_{ij} is the sum +R_m+R_n+\ldots for each resistor R_m, R_n,\ldots passed by a mesh current I_j adjacent to I_i.

7.2 Superposition Theorem

  • Linear circuits can be analysed by calculating contributions to current for each voltage source independently
  • Steps in superposition method:
    1. Replace all potential sources but one with a short circuit; find the voltage/current through each branch of the network.
    2. Repeat for each potential source.
    3. Add up all the separate voltages/currents in each branch.

7.3 Thévenin’s Theorem

  • Any linear circuit containing several voltage sources and resistors can be simplified to an equivalent circuit with a single voltage source and resistance connected in series with a load. Specifically, the three components connected in series are:
    1. Load resistor, R_L;
    2. Thévenin voltage V_{\mathrm{th}}, found by removing R_L from the original circuit and calculating the potential difference from one load connection point to the other;
    3. Thévenin resistance R_{\mathrm{th}}, found by removing R_L from the original circuit and calculating the total equivalent resistance between the two load connection points.
  • Once the Thévenin voltage and resistance are determined, the Thévenin-equivalent circuit is redrawn with the Thévenin voltage attached to the Thévenin resistance in series, and any load resistance attached between the two connection points
  • Maximum power transfer theorem: maximum power is transferred to a load when the internal resistance equals the load resistance; for a Thévenin-equivalent circuit, this is true when the load resistance equals the Thévenin resistance

Answers to Check Your Understanding

7.1 a.

    \[\mathbf{V}=\left(\begin{array}{c}-V_1\\V_1-V_2\\V_3\end{array}\right),~~~\mathbf{R}=\left(\begin{array}{ccc}-(R_1+R_2+R_3)&R_3&0\\R_3&-(R_3+R_5)&R_5\\0&R_5&-(R_4+R_5)\end{array}\right)\]

b.

    \[\mathbf{V}=\left(\begin{array}{c}V_1-V_2\\V_2\\-V_3\end{array}\right),~~~\mathbf{R}=\left(\begin{array}{ccc}-(R_1+R_4)&0&R_4\\0&-(R_2+R_3)&0\\R_4&0&-(R_4+R_5+R_6)\end{array}\right)\]

c.

    \[\mathbf{V}=\left(\begin{array}{c}-V\\0\\0\end{array}\right),~~~\mathbf{R}=\left(\begin{array}{ccc}-(R_1+R_4)&R_1&R_4\\R_1&-(R_1+R_2+R_3)&R_3\\R_4&R_3&-(R_3+R_4+R_5)\end{array}\right)\]

7.2 Replacing the 6.00~\mathrm{V} source with a short means current from the 9.00~\mathrm{V} source passes through the short. With the 9.00~\mathrm{V} source replaced with a short, the current through the 7.00~\Omega resistor due to the 6.00~\mathrm{V} source is

    \[I_{7\Omega}=\frac{6.00~\mathrm{V}}{7.00~\Omega}=\frac{6.00}{7.00}~\mathrm{A}.\]

Therefore, the current through the 7.00~\Omega resistor is I_{7\Omega}=\frac{6.00}{7.00}~\mathrm{A}, and the power dissipated is P_{7\Omega}=\left(\frac{6.00}{7.00}~\mathrm{A}\right)^2(7.00~\Omega)=5.14~\mathrm{W}. In fact, since the 7~\Omega-resistor is connected directly to the terminals of the 6.00~\mathrm{V} source, the current is I_{7\Omega}=\frac{6.00}{7.00}~\mathrm{A} regardless of the value of the 9.00~\mathrm{V} battery.

7.3. V_{\mathrm{th}}=V_1, R_{\mathrm{th}}=R_3.


Conceptual Questions

7.1 Mesh Analysis

1 Sally and Frank are asked to find the \mathbf{R} and \mathbf{V} matrices for the circuit below. They both follow the procedure for reading off matrix elements described in Mesh Analysis. When comparing results, they find that they have the same elements in the first rows of both \mathbf{R} and \mathbf{V}, but the elements in the second and third rows of their matrices are swapped. How did this happen? Did one of them make an error? If given values for the voltages and resistances, should they expect to find the same currents through each component? Explain.

7.2 Superposition Theorem

2. How much power is used by the 9~\Omega-resistor in the circuit below? How much power is used by the 3~\Omega-resistor? Explain why the answers are obvious, with no need to do any work. (Hint: this problem is best approached using superposition method).

3. (a) Use superposition theorem to determine the potential difference across the resistor in the circuit below. (b) Explain why mesh analysis is a bad approach to use for this problem (e.g. you could demonstrate this by applying the mesh strategy and Cramer’s rule to calculate mesh currents and interpret the result).

7.3 Thévenin’s Theorem

4. Efficiency. The efficiency of a circuit is defined as the ratio between the power used by a load and the total power used. (a) Show that the general expression for efficiency can be written as

    \[\eta=\frac{R_L}{R_{\mathrm{th}}+R_L}.\]

For any given V_{\mathrm{th}} and R_{\mathrm{th}}, show that: (b) the condition for minimum efficiency (R_L\gg R_{\mathrm{th}}) corresponds to maximum current, but the voltage across and power dissipated at the load are both small; (c) the condition for maximum efficiency (R_L\gg R_{\mathrm{th}}) corresponds to maximum voltage across the load, but current through and power dissipated at the load are both small; (d) the condition of maximum power transfer (R_L=R_{\mathrm{th}}), while only moderately efficient, corresponds to both moderate voltage across and moderate current through the load. (Hint: You may find it useful to refer to Equations 7.3.1-7.3.3).


Problems

7.1 Mesh Analysis

5. Apply Cramer’s rule to find expressions for I_1 and I_2 in terms of the given resistances and voltages.

6. Refer to the circuit in Problem 5, with V_1=1~\mathrm{V}, V_2=2~\mathrm{V}, R_1=3~\Omega, R_2=5~\Omega, and R_3=7~\Omega. Apply Cramer’s rule to calculate the values of I_1 and I_2.

7. Apply Cramer’s rule to find expressions for I_1 and I_2 in terms of the given resistances and voltages.

8. Refer to the circuit in Problem 7, with V_1=1~\mathrm{V}, V_2=2~\mathrm{V}, V_3=3~\mathrm{V}, R_1=5~\Omega, R_2=7~\Omega, R_3=11~\Omega, and R_4=13~\Omega. Apply Cramer’s rule to calculate the values of I_1 and I_2.

9. Apply Cramer’s rule to find expressions for I_1, I_2, and I_3 in terms of the given resistances and voltages.

10. Refer to the circuit in Problem 9, with V_1=1~\mathrm{V}, V_2=2~\mathrm{V}, R_1=3~\Omega, R_2=5~\Omega, R_3=7~\Omega, R_4=11~\Omega, R_5=13~\Omega, and R_6=17~\Omega. Apply Cramer’s rule to calculate the values of I_1, I_2, and I_3.

11. Apply Cramer’s rule to find expressions for I_1, I_2, and I_3 in terms of the given resistances and voltages.

12. Refer to the circuit in Problem 11, with V_1=1~\mathrm{V}, V_2=2~\mathrm{V}, R_1=3~\Omega, R_2=5~\Omega, R_3=7~\Omega, and R_4=11~\Omega. Apply Cramer’s rule to calculate the values of I_1, I_2, and I_3.

13. Find expressions for the total power supplied by all voltage sources and the total power dissipated at all resistors, and confirm that these are equal.

14. Refer to the circuit in Problem 13, with V_1=1~\mathrm{V}, V_2=2~\mathrm{V}, R_1=3~\Omega, R_2=5~\Omega, and R_3=7~\Omega. Calculate the total power supplied by all voltage sources and the total power dissipated at all resistors, and confirm that the two values are equal.

15. Find expressions for the total power supplied by all voltage sources and the total power dissipated at all resistors, and confirm that these are equal.

16. Refer to the circuit in Problem 15, with V_1=1~\mathrm{V}, V_2=2~\mathrm{V}, V_3=3~\Omega, R_1=5~\Omega, R_2=7~\Omega, R_3=11~\Omega, and R_4=13~\Omega. Calculate the total power supplied by all voltage sources and the total power dissipated at all resistors, and confirm that the two values are equal.

17. Find expressions for the total power supplied by all voltage sources and the total power dissipated at all resistors, and confirm that these are equal.

18. Refer to the circuit in Problem 17, with V_1=1~\mathrm{V}, V_2=2~\mathrm{V}, R_1=3~\Omega, R_2=5~\Omega, R_3=7~\Omega, R_4=11~\Omega, and R_5=13~\Omega. Calculate the total power supplied by all voltage sources and the total power dissipated at all resistors, and confirm that the two values are equal.

19.  Find expressions for the total power supplied by all voltage sources and the total power dissipated at all resistors, and confirm that these are equal.

20. Refer to the circuit in Problem 19, with V_1=1~\mathrm{V}, V_2=2~\mathrm{V}, R_1=3~\Omega, R_2=5~\Omega, R_3=7~\Omega, R_4=11~\Omega, and R_5=13~\Omega. Calculate the total power supplied by all voltage sources and the total power dissipated at all resistors, and confirm that the two values are equal.

21. Find expressions for the current through and power dissipated at R_5 in terms of the given resistances and voltages.

22. Refer to the circuit in Problem 21. Calculate (a) the mesh current values and (b) the power supplied by each voltage source, when V_1=1~\mathrm{V}, V_2=2~\mathrm{V}, V_3=3~\Omega, R_1=5~\Omega, R_2=7~\Omega, R_3=11~\Omega, R_4=13~\Omega, and R_5=17~\Omega.

23. Find the potential difference V_{ab}=V_a-V_b between points a and b in the following circuit diagram.

24. (a) Use a computer to calculate the mesh currents I_1-I_6 in the following circuit diagram, when V_1=1~\mathrm{V}, V_2=2~\mathrm{V}, V_3=3~\Omega, R_1=5~\Omega, R_2=7~\Omega, R_3=11~\Omega, R_4=13~\OmegaR_5=17~\OmegaR_6=19~\OmegaR_7=23~\OmegaR_8=29~\Omega, and R_9=31~\Omega. (b) Calculate the potential difference V_{ab}=V_a-V_b across R_5 and R_6, confirming that V_{ab}=V_2.

7.2 Superposition Theorem

25. Use superposition method to find an expression for the current through and power dissipated by R_2 in the circuit below.

26. Use superposition method to calculate the current through and power dissipated by R_2 in the circuit from problem 25, when V_1=1~\mathrm{V}, V_2=2~\mathrm{V}, R_1=3~\Omega, and R_2=5~\Omega.

27. Use superposition method to find expressions for (a) the currents through each resistor in the circuit below, and (b) the potential difference V_{ab}=V_a-V_b between points a and b.

28. Refer to the circuit in problem 27, with V_1=1~\mathrm{V}, V_2=2~\mathrm{V}, R_1=3~\Omega, R_2=5~\Omega, and R_3=7~\Omega. Use superposition method to calculate (a) the current through each resistor, and (b) the potential difference V_{ab}=V_a-V_b between points a and b.

29. Use superposition method to find an expression for the current through and power dissipated by R_1 in the circuit below.

30. Use superposition method to determine the current through and power dissipated by R_2 in the circuit from problem 29, when V_1=1~\mathrm{V}, V_2=2~\mathrm{V}, V_3=3~\mathrm{V}, R_1=5~\Omega, R_2=7~\Omega, R_3=11~\Omega, and R_4=13~\Omega.

31. Use superposition method to find expressions for (a) the current through each resistor in the circuit from Problem 29, and (b) the potential difference V_{ab}=V_a-V_b between points a and b.

32. Refer to the circuit in problem 29, withV_1=1~\mathrm{V}, V_2=2~\mathrm{V}, V_3=3~\mathrm{V}, R_1=5~\Omega, R_2=7~\Omega, R_3=11~\Omega, and R_4=13~\Omega. Use superposition method to calculate (a) the current through each resistor, and (b) the potential difference V_{ab}=V_a-V_b between points a and b.

7.3 Thévenin’s Theorem

33. Find expressions for R_{\mathrm{th}} and V_{\mathrm{th}} with respect to R_L in the circuit diagram below.

34. Refer to the circuit diagram in Problem 33, with V_1=1~\mathrm{V}, R_1=2~\Omega, and R_2=3~\Omega. Calculate (a) R_{\mathrm{th}} and V_{\mathrm{th}} with respect to R_L, and (b) the maximum amount of power that can be used by R_L.

35. Find expressions for R_{\mathrm{th}} and V_{\mathrm{th}} with respect to R_L in the circuit diagram below.

36. Refer to the circuit diagram in Problem 35, with V_1=1~\mathrm{V}, R_1=2~\Omega, R_2=3~\Omega, and R_3=5~\Omega. Calculate (a) R_{\mathrm{th}} and V_{\mathrm{th}} with respect to R_L, and (b) the maximum amount of power that can be used by R_L.

37. Find expressions for R_{\mathrm{th}} and V_{\mathrm{th}} with respect to R_L in the circuit diagram below.

38. Refer to the circuit diagram in Problem 37, with V_1=1~\mathrm{V}, V_2=2~\mathrm{V}, R_1=3~\Omega, R_2=5~\Omega, and R_3=7~\Omega. Calculate (a) R_{\mathrm{th}} and V_{\mathrm{th}} with respect to R_L, and (b) the maximum amount of power that can be used by R_L.

39. Find expressions for R_{\mathrm{th}} and V_{\mathrm{th}} with respect to R_L in the circuit diagram below.

40. Refer to the circuit diagram in Problem 39, with V_1=1~\mathrm{V}, V_2=2~\mathrm{V}, R_1=3~\Omega, R_2=5~\Omega, and R_3=7~\Omega. Calculate (a) R_{\mathrm{th}} and V_{\mathrm{th}} with respect to R_L, and (b) the maximum amount of power that can be used by R_L.

41. Find expressions for R_{\mathrm{th}} and V_{\mathrm{th}} with respect to R_L in the circuit diagram below.

42. Refer to the circuit diagram in Problem 41, with V_1=1~\mathrm{V}, R_1=2~\Omega, R_2=3~\Omega, and R_3=5~\Omega, and R_4=7~\Omega. Calculate (a) R_{\mathrm{th}} and V_{\mathrm{th}} with respect to R_L, and (b) the maximum amount of power that can be used by R_L.

Additional Problems

43. Find the current through, and the power dissipated by R_3 in the circuit below using (a) mesh analysis method and (b) superposition method.

44. Find the current through V_1 and V_2 in the circuit below using (a) mesh analysis method and (b) superposition method.

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