Input impedance and admittance
We now introduce a generalization of the familiar impedance and admittance of electrical circuit analysis, in which system behavior can be expressed algebraically instead of differentially. We begin with generalized input impedance.
Consider a system with a source, as shown in figure 12.1. The source can be either an across- or a through-variable source. The ideal source specifies either \(\mathcal{V}_\text{in}\) or \(\mathcal{F}_\text{in}\), and the other variable depends on the system.
Let a source variables have Laplace transforms \(\mathcal{V}_\text{in}(s)\) and \(\mathcal{F}_\text{in}(s)\). We define the system’s input impedance \(Z\) and input admittance \(Y\) to be the Laplace-domain ratios \[\begin{aligned} Z(s) = \frac{\mathcal{V}_\text{in}(s)}{\mathcal{F}_\text{in}(s)} \quad \text{and} \quad Y(s) = \frac{\mathcal{F}_\text{in}(s)}{\mathcal{V}_\text{in}(s)}. \end{aligned}\] Clearly, \[\begin{aligned} Y(s) = \frac{1} {Z(s)}.\end{aligned}\] Both \(Z\) and \(Y\) can be considered transfer functions: for a through-variable source \(\mathcal{F}_\text{in}\), the impedance \(Z\) is the transfer function to across-variable \(\mathcal{V}_\text{in}\); for an across-variable source \(\mathcal{V}_\text{in}\), the admittance \(Y\) is the transfer function to through-variable \(\mathcal{F}_\text{in}\). Often, however, we use the more common impedance \(Z\) to characterize systems with either type of source.
Note that \(Z\) and \(Y\) are system properties, not properties of the source. An impedance or admittance can characterize a system of interconnected elements, or a system of a single element, as the next section explores.
Impedance of ideal passive elements
The impedance and admittance of a single, ideal, one-port element is defined from the Laplace transform of its elemental equation.
- Generalized capacitors
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A generalized capacitor has elemental equation \[\begin{aligned} \frac{d \mathcal{V}_C(t)}{d t} = \frac{1} {C} \mathcal{F}_C(t), \end{aligned}\] the Laplace transform of which is \[\begin{aligned} s \mathcal{V}_C(s) = \frac{1} {C} \mathcal{F}_C(s), \end{aligned}\] which can be solved for impedance \(Z_C = \mathcal{V}_C/\mathcal{F}_C\) and admittance \(Y_C = \mathcal{F}_C/\mathcal{V}_C\): \[\begin{aligned} Z_C(s) = \frac{1} {C s} \quad \text{and} \quad Y_C(s) = C s.\end{aligned}\]
- Generalized inductors
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A generalized inductor has elemental equation \[\begin{aligned} \frac{d \mathcal{F}_L(t)}{d t} = \frac{1} {L} \mathcal{F}_L(t), \end{aligned}\] the Laplace transform of which is \[\begin{aligned} s \mathcal{F}_L(s) = \frac{1} {L} \mathcal{V}_L(s), \end{aligned}\] which can be solved for impedance \(Z_L = \mathcal{V}_L/\mathcal{F}_L\) and admittance \(Y_L = \mathcal{F}_L/\mathcal{V}_L\): \[\begin{aligned} Z_L(s) = L s \quad \text{and} \quad Y_C(s) = \frac{1} {L s}.\end{aligned}\]
- Generalized resistors
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A generalized resistor has elemental equation \[\begin{aligned} \mathcal{V}_R(t) = \mathcal{F}_R(t) R, \end{aligned}\] the Laplace transform of which is \[\begin{aligned} \mathcal{V}_R(s) = \mathcal{F}_R(s) R, \end{aligned}\] which can be solved for impedance \(Z_R = \mathcal{V}_R/\mathcal{F}_R\) and admittance \(Y_R = \mathcal{F}_R/\mathcal{V}_R\): \[\begin{aligned} Z_R(s) = R \quad \text{and} \quad Y_R(s) = \frac{1} {R}.\end{aligned}\]
Impedance of interconnected elements
As with electrical circuits, impedances of linear graphs of interconnected elements can be combined in two primary ways: in parallel or in series.
Elements sharing the same through-variable are said to be in series connection. \(N\) elements connected in series have equivalent impedance \(Z\) and admittance \(Y\): \[\begin{aligned} Z(s) = \sum_{i=1}^N Z_i(s) \quad \text{and} \quad Y(s) = \left. 1 \middle/ \sum_{i=1}^N 1/Y_i(s) \right. \end{aligned}\]
Conversely, elements sharing the same across-variable are said to be in parallel connection. \(N\) elements connected in parallel have equivalent impedance \(Z\) and admittance \(Y\): \[\begin{aligned} Z(s) = \left. 1 \middle/ \sum_{i=1}^N 1/Z_i(s) \right. \quad \text{and} \quad Y(s) = \sum_{i=1}^N Y_i(s). \end{aligned}\]
For the circuit shown, find the input impedance.
The input impedance is the equivalent impedance of the combination of parallel and series connections: $$\begin{aligned} Z(s) = Z_{R_1} + Z_C + \frac{ Z_{R_2} Z_L } { Z_{R_2}+Z_L} , \end{aligned}$$ where $$\begin{aligned} Z_{R_1} = R_1 \text{, } Z_{R_2} = R_2 \text{, } Z_C = 1/(C s) \text{, and } Z_L = L s. \end{aligned}$$
Online Resources for Section 12.1
No online resources.