# I. Impedance of an R-L circuit.

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1 I. Impedance of an R-L circuit. [For inductor in an AC Circuit, see Chapter 31, pg. 1024] Consider the R-L circuit shown in Figure: 1. A current i(t) = I cos(ωt) is driven across the circuit using an AC voltage source. Note that the current across the circuit is changing with time and has an angular frequency ω. 1. Write down the instantaneous voltage drop across the resistor v R (t) in terms of R, ω, t and I. [Recall Ohm s law.] Figure: 1 2. Write down the amplitude of the voltage drop across the resistor (V R ) in terms of R and I. 3. Plot i(t) and v R vs. time in separate graphs. Clearly show the values of v R (t) and i(t) at time-intervals of!!!. 4. Write down the instantaneous voltage drop across the inductor v L (t) in terms of L, ω, t and I. [Hint: Recall self-induced EMF in the inductor is L!". The drop across the inductor v!"! = L!". The extra negative sign in v L!" denotes that the inductor opposes the increase in current.] 5. Write down the amplitude of the voltage drop across the inductor (V L ) in terms of L, ω and I.

2 6. Plot i(t) and v L (t) vs. time in separate graphs. Clearly show the values of v L (t) and i(t) at time-intervals of!!!. Label the first maximum values in terms of I and V L, respectively. 2 We now want to compare the phases of the currents in the circuit elements and the corresponding voltage drops. To do so we need to write all the functions as either sine or cosine. In this example it is best (that is, we make an educated choice) that we write everything as cosine, since current in the circuit is written as i(t) = I cos(ωt). Recall that since this is a series circuit, the current across the resistor i R (t) and the current across the inductor i L (t) are the same and equal to i(t). 7. What is the phase of i(t)? 8. Write v R (t) in the form v! (t) = V! cos ωt + φ!. What is the phase of v R? 9. Write v L (t) in the form v! (t) = V! cos ωt + φ!. What is the phase of v L? 10. Does the phase for v R (t) change as a function of time? 11. Does the phase for v L (t) change as a function of time? 12. What is the phase difference between v R (t) and v L (t)?

3 From questions 7-9 we see that: 3 i(t) = I cos(ωt) v! (t) = V! cos(ωt) v! (t) = V! cos ωt +!! Therefore the instantaneous net voltage drop in the circuit (due to Kirchoff s voltage law) is: v(t) = v! (t) + v! (t). Consider the following pictorial way of depicting the above equations (see Figure: 2). We imagine a vector V R of magnitude V! making an angle ωt with the X-axis (technically this is called a phasor). We also imagine another vector V L with magnitude V! making an angle ωt +! with the X-axis. Then the X-component of the resultant! vector (after we have vectorially added the two phasors) is the instantaneous voltage drop across the circuit. 13. What is the magnitude of the resultant V = V R + V L? Write your answer in terms of L, R, ω and I. 14. One can write the amplitude of the resultant as V = IZ, where Z is the magnitude of the impedance of the circuit. Using your result from question 13, what is the magnitude of the impedance of the series R-L circuit? Write your answer in terms of L, R and ω. The relative angle (φ) between V and I remain constant at all times and determines how much power is dissipated in the circuit (This angle in the diagram is the actual physical phase difference between the net voltage drop and the net current in the circuit). In fact average power dissipated in an AC circuit over one cycle is P!" =! VI cos φ! Figure: For the R-L circuit in Figure 1 calculate tan φ. Write your answer in terms of L, R and ω. [Hint: Look at Figure: 2. Use the definition of tan φ in the right-angled triangle involving angle φ.] 16. How much power is dissipated in an R-L circuit with zero resistance? [Hint: What is φ in this circuit?]

4 II. Impedance of an R-C circuit. [For capacitor in an AC Circuit, see Chapter 31, pg. 1026] Consider the R-C circuit shown in Figure: 3. Similar to the R-L circuit, a current i(t) = I cos(ωt) is driven across the circuit using an AC voltage source Write down the instantaneous voltage drop across the capacitor v C (t) in terms of C, ω, t and I. [Hint: Recall i(t) =!"!" and q = Cv!] Figure: 3 2. Write down the amplitude of the voltage drop across the capacitor (V C ) in terms of C, ω and I. 3. Write v C (t) in the form v! (t) = V! cos ωt + φ!. What is the phase of v C? What is the phase difference of v C (t) and i(t)? 4. Construct a diagram similar to Figure 2 for V! and V!. Label all of the vectors, the resultant and the angles ωt and φ. Assume 0 < ωt < π/2. 5. One can write the amplitude of the resultant as V = IZ, where Z is the magnitude of the impedance of the circuit. This is similar to what we have already seen in the series R-L circuit. Using your result from question 11, what is the magnitude of the impedance of the series R-C circuit? Write your answer in terms of C, R and ω.

5 III. Impedance of an R-L-C circuit. 5 Consider the series R-L-C circuit as shown in Figure: 4. You know all the phasors one needs to consider (See question 4 of part II and Figure: 2). 1. Draw a phasor diagram. Calculate the amplitude of the net voltage drop phasor in terms of R, L, C, ω and I. [Assume V L >V C and 0 < ωt < π/2.] Figure: 4 2. Writing V = IZ, read off the magnitude of the impedance Z of the circuit. Write your answer in terms of R, L, C and ω. Is the magnitude of the impedance smallest or largest when ω =!!"? [Recall the frequency ω =!!" is called the resonance frequency.] 3. Calculate tan φ, where φ is the phase difference between the net voltage drop and the net current in the circuit. [Hint: Revisit answer to question 1.]

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