# 10.12: Equivalent Circuit Model for Reception, Redux

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Section 10.9 provides an informal derivation of an equivalent circuit model for a receiving antenna. This model is shown in Figure $$\PageIndex{1}$$. Figure $$\PageIndex{1}$$: Thévenin equivalent circuit model for an antenna in the presence of an incident electric field $$\mathbf{E}^{i}$$. (CC BY-SA 4.0; S. Lally)

The derivation of this model was informal and incomplete because the open-circuit potential $${\bf E}^i\cdot{\bf l}_e$$ and source impedance $$Z_A$$ were not rigorously derived in that section. While the open-circuit potential was derived in Section 10.11 (“Potential Induced in a Dipole”), the source impedance has not yet been addressed. In this section, the source impedance is derived, which completes the formal derivation of the model. Before reading this section, a review of Section 10.10 (“Reciprocity”) is recommended.

The starting point for a formal derivation is the two-port model shown in Figure $$\PageIndex{2}$$. Figure $$\PageIndex{2}$$: Two-port system. (public domain (modified); Inductiveload)

If the two-port is passive, linear, and time-invariant, then the potential $$v_2$$ is a linear function of potentials and currents present at ports 1 and 2. Furthermore, $$v_1$$ must be proportional to $$i_1$$, and similarly $$v_2$$ must be proportional to $$i_2$$, so any pair of “inputs” consisting of either potentials or currents completely determines the two remaining potentials or currents. Thus, we may write:

$v_2 = Z_{21} i_1 + Z_{22} i_2 \label{m0216_ev2}$

where $$Z_{11}$$ and $$Z_{12}$$ are, for the moment, simply constants of proportionality. However, note that $$Z_{11}$$ and $$Z_{12}$$ have SI base units of $$\Omega$$, and so we refer to these quantities as impedances. Similarly we may write:

$v_1 = Z_{11} i_1 + Z_{12} i_2 \label{m0216_ev1}$

We can develop expressions for $$Z_{12}$$ and $$Z_{21}$$ as follows. First, note that $$v_{1}=Z_{12}i_2$$ when $$i_1=0$$. Therefore, we may define $$Z_{12}$$ as follows:

$Z_{12} \triangleq \left.\frac{v_1}{i_2}\right|_{i_1=0} \nonumber$

The simplest way to make $$i_1=0$$ (leaving $$i_2$$ as the sole “input”) is to leave port 1 open-circuited. In previous sections, we invoked special notation for these circumstances. In particular, we defined $$\widetilde{I}_2^t$$ as $$i_2$$ in phasor representation, with the superscript “$$t$$” (“transmitter”) indicating that this is the sole “input”; and $$\widetilde{V}_1^r$$ as $$v_1$$ in phasor representation, with the superscript “$$r$$” (“receiver”) signaling that port 1 is both open-circuited and the “output.” Applying this notation, we note:

$Z_{12} = \frac{\widetilde{V}_1^r}{\widetilde{I}_2^t} \label{m0216_eZ12}$

Similarly:

$Z_{21} = \frac{\widetilde{V}_2^r}{\widetilde{I}_1^t} \label{m0216_eZ21}$

In Section 10.10 (“Reciprocity”), we established that a pair of antennas could be represented as a passive linear time-invariant two-port, with $$v_1$$ and $$i_1$$ representing the potential and current at the terminals of one antenna (“antenna 1”), and $$v_2$$ and $$i_2$$ representing the potential and current at the terminals of another antenna (“antenna 2”). Therefore for any pair of antennas, quantities $$Z_{11}$$, $$Z_{12}$$, $$Z_{22}$$, and $$Z_{21}$$ can be identified that completely determine the relationship between the port potentials and currents.

We also established in Section 10.10 that:

$\widetilde{I}_1^t \widetilde{V}_1^r = \widetilde{I}_2^t \widetilde{V}_2^r \label{m0216_eRIV}$

Therefore,

$\frac{\widetilde{V}_1^r}{\widetilde{I}_2^t} = \frac{\widetilde{V}_2^r}{\widetilde{I}_1^t} \label{m0216_eR1}$

Referring to Equations \ref{m0216_eZ12} and \ref{m0216_eZ21}, we see that Equation \ref{m0216_eR1} requires that:

$Z_{12} = Z_{21} \nonumber$

This is a key point. Even though we derived this equality by open-circuiting ports one at a time, the equality must hold generally since Equations \ref{m0216_ev2} and \ref{m0216_ev1} must apply – with the same values of $$Z_{12}$$ and $$Z_{21}$$ – regardless of the particular values of the port potentials and currents.

We are now ready to determine the Thévenin equivalent circuit for a receiving antenna. Let port 1 correspond to the transmitting antenna; that is, $$i_1$$ is $$\widetilde{I}_1^t$$. Let port 2 correspond to an open-circuited receiving antenna; thus, $$i_2=0$$ and $$v_2$$ is $$\widetilde{V}_2^r$$. Now applying Equation \ref{m0216_ev2}:

\begin{aligned} v_2 &= Z_{21} i_1 + Z_{22} i_2 \nonumber \\ &= \left(\widetilde{V}_2^r/\widetilde{I}_1^t\right) \widetilde{I}_1^t + Z_{22} \cdot 0 \nonumber \\ &= \widetilde{V}_2^r\end{aligned} \nonumber

We previously determined $$\widetilde{V}_2^r$$ from electromagnetic considerations to be (Section 10.11):

$\widetilde{V}_2^r = \widetilde{\bf E}^i\cdot{\bf l}_e \nonumber$

where $$\widetilde{\bf E}^i$$ is the field incident on the receiving antenna, and $${\bf l}_e$$ is the vector effective length as defined in Section 10.11. Thus, the voltage source in the Thévenin equivalent circuit for the receive antenna is simply $$\widetilde{\bf E}^i\cdot{\bf l}_e$$, as shown in Figure $$\PageIndex{1}$$.

The other component in the Thévenin equivalent circuit is the series impedance. From basic circuit theory, this impedance is the ratio of $$v_2$$ when port 2 is open-circuited (i.e., $$\widetilde{V}_2^r$$) to $$i_2$$ when port 2 is short-circuited. This value of $$i_2$$ can be obtained using Equation \ref{m0216_ev2} with $$v_2=0$$:

$0 = Z_{21} \widetilde{I}_1^t + Z_{22} i_2 \nonumber$

Therefore:

$i_2 = - \frac{Z_{21}}{Z_{22}} \widetilde{I}_1^t \nonumber$

Now using Equation \ref{m0216_eZ21} to eliminate $$\widetilde{I}_1^t$$, we obtain:

$i_2 = - \frac{\widetilde{V}_2^r}{Z_{22}} \nonumber$

Note that the reference direction for $$i_2$$ as defined in Figure $$\PageIndex{2}$$ is opposite the reference direction for short-circuit current. That is, given the polarity of $$v_2$$ shown in Figure $$\PageIndex{2}$$, the reference direction of current flow through a passive load attached to this port is from “$$+$$” to “$$-$$” through the load. Therefore, the source impedance, calculated as the ratio of the open-circuit potential to the short circuit current, is:

$\frac{\widetilde{V}_2^r}{+\widetilde{V}_2^r/Z_{22}} = Z_{22} \nonumber$

We have found that the series impedance $$Z_A$$ in the Thévenin equivalent circuit is equal to $$Z_{22}$$ in the two-port model.

To determine $$Z_{22}$$, let us apply a current $$i_2=\widetilde{I}_2^t$$ to port 2 (i.e., antenna 2). Equation \ref{m0216_ev2} indicates that we should see:

$v_2 = Z_{21} i_1 + Z_{22} \widetilde{I}_2^t \nonumber$

Solving for $$Z_{22}$$:

$Z_{22} = \frac{v_2}{\widetilde{I}_2^t} - Z_{21} \frac{i_1}{\widetilde{I}_2^t} \label{m0216_eZ22exact}$

Note that the first term on the right is precisely the impedance of antenna 2 in transmission. The second term in Equation \ref{m0216_eZ22exact} describes a contribution to $$Z_{22}$$ from antenna 1. However, our immediate interest is in the equivalent circuit for reception of an electric field $$\widetilde{\bf E}^i$$ in the absence of any other antenna. We can have it both ways by imagining that $$\widetilde{\bf E}^i$$ is generated by antenna 1, but also that antenna 1 is far enough away to make $$Z_{21}$$ – the factor that determines the effect of antenna 1 on antenna 2 – negligible. Then we see from Equation \ref{m0216_eZ22exact} that $$Z_{22}$$ is the impedance of antenna 2 when transmitting.

Summarizing:

The Thévenin equivalent circuit for an antenna in the presence of an incident electric field $$\widetilde{\bf E}^i$$ is shown in Figure $$\PageIndex{1}$$. The series impedance $$Z_A$$ in this model is equal to the impedance of the antenna in transmission.

## Mutual coupling

This concludes the derivation, but raises a follow-up question: What if antenna 2 is present and not sufficiently far away that $$Z_{21}$$ can be assumed to be negligible? In this case, we refer to antenna 1 and antenna 2 as being “coupled,” and refer to the effect of the presence of antenna 1 on antenna 2 as coupling. Often, this issue is referred to as mutual coupling, since the coupling affects both antennas in a reciprocal fashion. It is rare for coupling to be significant between antennas on opposite ends of a radio link. This is apparent from common experience. For example, changes to a receive antenna are not normally seen to affect the electric field incident at other locations. However, coupling becomes important when the antenna system is a dense array; i.e., multiple antennas separated by distances less than a few wavelengths. It is common for coupling among the antennas in a dense array to be significant. Such arrays can be analyzed using a generalized version of the theory presented in this section.

This page titled 10.12: Equivalent Circuit Model for Reception, Redux is shared under a CC BY-SA license and was authored, remixed, and/or curated by Steven W. Ellingson (Virginia Tech Libraries' Open Education Initiative) .